1 contents · tion consistent with a pharmacophoric or catalytic site model (“template...

Forcefield-Based Simulations/September 1998 i

1 Contents

1. Contents i

1. Introduction 1

Who should use this documentation . . . . . . . . . . . . . . . . . . 2What can simulation engines do? . . . . . . . . . . . . . . . . . . . . 2

Energy minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Other forcefield-based calculations . . . . . . . . . . . . . . . . 3

What are forcefields and simulation engines? . . . . . . . . . . . 4Using this guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Typographical conventions . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Forcefields 9

The potential energy surface . . . . . . . . . . . . . . . . . . . . . . . 10Empirical fit to the potential energy surface . . . . . . . . . . . 11

The forcefield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The energy expression . . . . . . . . . . . . . . . . . . . . . . . . . 16

Forcefields supported by MSI forcefield engines . . . . . . . . 19Main types of forcefields . . . . . . . . . . . . . . . . . . . . . . . 20Advantages of having several forcefields . . . . . . . . . . 23Primary uses of each MSI forcefield . . . . . . . . . . . . . . 24

Second-generation forcefields accurate for many properties27CFF91, PCFF, CFF —consistent forcefields . . . . . . . . . 29MMFF93, the Merck molecular forcefield . . . . . . . . . . 33

Rule-based forcefields broadly applicable to the periodic table35ESFF, extensible systematic forcefield . . . . . . . . . . . . . 36UFF, universal forcefield . . . . . . . . . . . . . . . . . . . . . . . 41VALBOND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Dreiding forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Classical forcefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52AMBER forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53CHARMm forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . 55

ii Forcefield-Based Simulations/September 1998

1. Contents

CVFF, consistent valence forcefield . . . . . . . . . . . . . . . 57Special-purpose forcefields . . . . . . . . . . . . . . . . . . . . . . . . . 60

Glass forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61MSXX forcefield for polyvinylidene fluoride . . . . . . . . 63Zeolite forcefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Forcefields for sorption on zeolites . . . . . . . . . . . . . . . . 66Forcefields for Cerius2•Morphology module. . . . . . . . 66

Archived and untested forcefields . . . . . . . . . . . . . . . . . . . 67

3. Preparing the Energy Expression and the Model 73

Using forcefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Selecting forcefields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Assigning forcefield atom types and charges . . . . . . . . . . . 78

What are atom types in forcefields? . . . . . . . . . . . . . . . 79Assigning atom types to a model . . . . . . . . . . . . . . . . . 79Assigning charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Parameter assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Determination of which parameters are used with which

atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Automatic assignment of values for missing parameters86Manual parameter assignment . . . . . . . . . . . . . . . . . . . 89

Using alternative forms of energy terms . . . . . . . . . . . . . . . 92Removing terms from the energy expression . . . . . . . . 93Scaling or editing any selected type of term . . . . . . . . . 94Alternative bond terms. . . . . . . . . . . . . . . . . . . . . . . . . 94Scaled torsion terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Inversion terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Nonbond functional form . . . . . . . . . . . . . . . . . . . . . . . 96Hydrogen bonds and hydrogen-bond terms . . . . . . . . 96Bond–angle cross terms vs. Urey–Bradley terms . . . . . 98

Applying constraints and restraints . . . . . . . . . . . . . . . . . . 98When to use constraints/restraints. . . . . . . . . . . . . . . 100Fixed atom constraints . . . . . . . . . . . . . . . . . . . . . . . . 102Template forcing, tethering, quartic droplet restraints, and

consensus conformations . . . . . . . . . . . . . . . . . . . 103General internal-coordinate restraints . . . . . . . . . . . . 106Distance and NOE restraints . . . . . . . . . . . . . . . . . . . 106Distance and angle constraints in dynamics simulations .

109Angle restraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Torsion restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Inversion, out-of-plane, and chiral restraints . . . . . . . 112Plane and other geometrical constraints and restraints112

Modeling periodic systems . . . . . . . . . . . . . . . . . . . . . . . . 113Minimum-image model . . . . . . . . . . . . . . . . . . . . . . . 115Explicit-image model . . . . . . . . . . . . . . . . . . . . . . . . . 117

Forcefield-Based Simulations/September 1998 iii

Crystal simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bonds across boundaries . . . . . . . . . . . . . . . . . . . . . . 120

Handling nonbond interactions . . . . . . . . . . . . . . . . . . . . 120Combination rules for van der Waals terms . . . . . . . 124The dielectric constant and the Coulombic term . . . . 125Nonbond cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Cell multipole method. . . . . . . . . . . . . . . . . . . . . . . . 138Ewald sums for periodic systems . . . . . . . . . . . . . . . 143

4. Minimization 153

General minimization process . . . . . . . . . . . . . . . . . . . . . 155Specific minimization example . . . . . . . . . . . . . . . . . 155Line search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Minimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 160Steepest descents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Conjugate gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 164Newton–Raphson methods . . . . . . . . . . . . . . . . . . . . 166

General methodology for minimization . . . . . . . . . . . . . 173Minimizations with MSI simulation engines. . . . . . . 174When to use different algorithms . . . . . . . . . . . . . . . 176Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . 178Significance of minimum-energy structure . . . . . . . . 180

Energy and gradient calculation . . . . . . . . . . . . . . . . . . . 181Vibrational calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Application of minimization to vibrational theory . . 183Vibrational frequencies . . . . . . . . . . . . . . . . . . . . . . . 185General methodology for vibrational calculations . . 187

5. Molecular Dynamics 189

Integration algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Criteria of good integrators in molecular dynamics . 193Integrators in MSI simulation engines. . . . . . . . . . . . 194

The choice of timestep . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Integration errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Example 1—Two colliding hydrogen atoms . . . . . . . 199Example 2—Energy conservation of a harmonic oscillator

204Statistical ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

NVE ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207NVT ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208NPT and NST ensembles . . . . . . . . . . . . . . . . . . . . . . 208NPH and NSH ensembles . . . . . . . . . . . . . . . . . . . . . 209Equilibrium thermodynamic properties . . . . . . . . . . 210

iv Forcefield-Based Simulations/September 1998

1. Contents

Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211How temperature is calculated . . . . . . . . . . . . . . . . . . 212How temperature is controlled . . . . . . . . . . . . . . . . . . 214

Pressure and stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Units and sign conventions for pressure and stress . . 221How pressure and stress are calculated . . . . . . . . . . . 222How pressure and stress are controlled . . . . . . . . . . . 225

Types of dynamics simulations . . . . . . . . . . . . . . . . . . . . . 229Quenched dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 232Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . 232Consensus dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 233Impulse dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Langevin dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 234Stochastic boundary dynamics . . . . . . . . . . . . . . . . . . 234Multibody order-N dynamics . . . . . . . . . . . . . . . . . . . 234

Constraints during dynamics simulations . . . . . . . . . . . . 235The SHAKE algorithm . . . . . . . . . . . . . . . . . . . . . . . . 236The RATTLE algorithm. . . . . . . . . . . . . . . . . . . . . . . . 237

Dynamics trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238General methodology for dynamics calculations . . . . . . . 238

Stages and duration of dynamics simulations . . . . . . 239Dynamics with MSI simulation engines . . . . . . . . . . . 241Restarting a dynamics simulation . . . . . . . . . . . . . . . 246

6. Free Energy 251

Relative free energy—theory and implementation. . . . . . 251Finite difference thermodynamic integration (FDTI) . 251Relative free energy—methodology . . . . . . . . . . . . . . 255

Absolute free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Theory and implementation . . . . . . . . . . . . . . . . . . . . 258Example: Fentanyl . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Analysis of results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

A. References 273

B. Forcefield Terms and Atom Types 283

Forcefield term definitions . . . . . . . . . . . . . . . . . . . . . . . . 284AMBER atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Standard AMBER forcefield . . . . . . . . . . . . . . . . . . . . 287Homan’s carbohydrate forcefield . . . . . . . . . . . . . . . . 290

CFF91 atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291CHARMm atom types. . . . . . . . . . . . . . . . . . . . . . . . . . . . 294CVFF atom types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Forcefield-Based Simulations/September 1998 v

CVFF_aug atom types . . . . . . . . . . . . . . . . . . . . . . . . . . . 301ESFF atom types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303PCFF—additional atom types . . . . . . . . . . . . . . . . . . . . . 308Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

vi Forcefield-Based Simulations/September 1998

1. Contents

Forcefield-Based Simulations/October 1997 1

1 Introduction

This Forcefield-Based Simulations documentation is a general guide to all MSI’s simulation engines, that is, software products whose computational work is based on a forcefield. These include CHARMm®, Discover®, and the Open Force Field™ (OFF) mod-ules, which are run through the molecular modeling programs (i.e., graphical interfaces) shown in Table 1.

aSee definitions under What are forcefields and simulation engines?.bDiscover and OFF each offer a choice of several forcefields (see Table 3); the CHARMm program gives access only to the CHARMm forcefield in QUANTA and standalone, only to MMFF in Cerius2, and is used only in spe-cialized modules in Insight II.cDiscover exists in two versions: one written in FORTRAN (series 2.8.x, 2.9.x and earlier ; referred to as FDiscover) and the other in C (series 3.0, 3.1, 3.0.0, 4.0.0, 95.0, 96.0, 97.0; referred to as CDiscover). CDiscover and FDiscover are specified in this documentation only where the FORTRAN and C Discover programs have different capabilities. FDiscover and CDiscover (in Insight II) are accessed through the Discover and Discover_3 modules, respectively.dCDiscover only.eCHARMm and Discover can also be run without the assistance of a graph-ical molecular modeling program.

Table 1. Simulation enginesa within MSI’s molecular modeling programsb

Molecular modeling program and release

number

Simulation engine

CHARMm Discoverc OFF

Cerius2™ 4.0 √ √d √Insight® II 4.0.0 √Insight® II 97.0 √ √QUANTA® √standalonee √ √

2 Forcefield-Based Simulations/October 1997

1. Introduction

Who should use this documentation

This guide is written mainly for the typical scientist-user of MSI’s simulation engines. Although these programs are written to run with reasonable default values for basic simulations, you should read this guide if you want to make efficient use of the programs, obtain the best results possible, and understand the results.

Prerequisites You should already be familiar with:

♦ The system and windowing software on your workstation.

♦ How to use the particular MSI molecular modeling program that contains the desired simulation engine (Cerius2, Insight, and/or QUANTA).

Your workstation should have:

♦ A licensed copy of Cerius2, Insight, and/or QUANTA as well as of the appropriate simulation engine.

♦ A directory in which you have write permission.

What can simulation engines do?

Energy minimization

Typical uses of energy minimization include:

♦ Optimizing initial geometries of models constructed in a molecular modeling program such as Cerius2™ or Insight®.

♦ Repairing poor geometries occurring at splice points during homology building of protein structures.

♦ Mapping the energy barriers for geometric distortions and con-formational transitions using “torsion forcing” to obtain Ram-achandran-type contour plots for proteins or RIS statistical weights for polymers.

What can simulation engines do?


♦ Evaluating whether a molecule can adopt a template conforma-tion consistent with a pharmacophoric or catalytic site model (“template forcing”).

Molecular dynamics

Typical uses of molecular dynamics include:

♦ Searching the conformational space of alternative amino acid sidechains in site-specific mutation studies.

♦ Identifying likely conformational states for highly flexible polymers or for flexible regions of macromolecules such as pro-tein loops.

♦ Producing sets of 3D structures consistent with distance and torsion constraints deduced from NMR experiments (simu-lated annealing).

♦ Calculating free energies of binding, including solvation and entropy effects.

♦ Probing the locations, conformations, and motions of mole-cules on catalyst surfaces.

♦ Running diffusion calculations.

Other forcefield-based calculations

In addition, simulation engines can be routinely used for:

♦ Calculating normal modes of vibration and vibrational fre-quencies.

♦ Analyzing intramolecular and intermolecular interactions in terms of residue–residue or molecule–molecule interactions, energy per residue, or interactions within a radius.

♦ Calculating diffusion coefficients of small molecules in a poly-mer matrix.

♦ Calculating thermal expansion coefficients of amorphous poly-mers.

♦ Calculating the radial distribution of liquids and amorphous polymers.


1. Introduction

♦ Performing rigid-body rms comparisons between minimized conformations of the same or similar structures or between sim-ulated and experimentally observed structures.

What are forcefields and simulation engines?

The fundamental computation at the core of a forcefield-based simulation is the calculation of the potential energy for a given configuration of atoms (and cells, if requested and possible). The calculation of this energy, along with its first and second deriva-tives with respect to the atomic coordinates (and cell coordinates), yields the information necessary for minimization, harmonic vibrational analysis, and dynamics simulations. This calculation is actually performed by the simulation engine, or forcefield-based program.

Simulation “engine” defined

Simulation engines are the computational packages that handle the application of forcefields in minimization, dynamics, and other molecular mechanics simulations. Currently, MSI-supplied simu-lation engines include CHARMm, Discover, and OFF. (CHARMm is the name for both a simulation engine and for the forcefield included in that engine.)

“Forcefield” and “energy expression” defined

The functional form of the potential energy expression and the entire set of parameters needed to fit the potential energy surface constitute the forcefield (Ermer 1976). The energy expression is the specific equation set up for a particular model and including (or not) any optional terms.

For example, a forcefield would contain bond-stretching parame-ters for all combinations of atoms for which it was parameterized, as well as a defined, summed functional form for the bond-stretch-ing term. The corresponding energy expression would contain bond-stretching parameters for only those combinations of bonded atoms actually found in the model being studied, as well as the specific bond-stretching terms for the number and types of bonds in that model (see example under Example energy expression for water).

Importance of the force-field in simulations

It is important to understand that the forcefield—both the func-tional form and the parameters themselves—represents the single largest approximation in molecular modeling. The quality of the

Using this guide


forcefield, its applicability to the model at hand, and its ability to predict the particular properties measured in the simulation directly determine the validity of the results.

Moledular modeling pro-grams

Molecular modeling programs are the graphical user interfaces (UIFs or GUIs) that can be used to prepare models, set up forcefields, and access the simulation engines. Some simulation engines can also be run in standalone mode, that is, outside the graphical molecular modeling program. Molecular modeling programs currently sup-plied by MSI include Cerius2, Insight, and QUANTA.

Using this guide

This guide contains background information on forcefields, the theories involved in their use, and how they are implemented in MSI’s simulation engines, as well as general methodology and strategies for performing the various types of calculation most commonly done with these programs:

♦ Forcefields presents the concept of an energy surface and com-pares the forcefields available to MSI’s simulation engines, including their functional forms and atom types. Different forcefields have been developed specifically for different types of models or computational experiments.

♦ Preparing the Energy Expression and the Model concerns concepts such as periodic boundary conditions, nonbond interactions, restraints, and constraints. You typically need to refine both the model and the energy expression that you intend to set up, in order to optimize your calculation conditions.

♦ Minimization includes information on the minimization algo-rithms that these programs can use; how, in general, to carry out minimization calculations; and the applicability of minimi-zation in energy and vibration calculations.

♦ Molecular Dynamics covers the dynamics algorithms in these programs, thermodynamic ensembles, control of temperature and pressure, and constraints during dynamics.

♦ Free Energy presents relative and absolute free energy calcula-tions.


1. Introduction

References contains the scientific references cited in this guide. Atom types and forcefield terms are listed under Forcefield Terms and Atom Types.

Additional information

Available documentation Guides are available for every simulation engine and modeling program that MSI provides, including these:

♦ MSI Forcefield Engines: CDiscover.

♦ Cerius2 Forcefield Engines: OFF.

♦ MSI Forcefield Engines: FDiscover.

♦ MSI Forcefield Engines: CHARMm.

♦ Cerius2 Modeling Environment.

♦ Insight II.

♦ QUANTA documentation set.

On-screen help In addition to the task-oriented documentation for each simula-tion engine, on-screen help is available within the Cerius2, Insight, and QUANTA environments. Please see the documentation for the specific program for how to access the help.

Supplemental documen-tation

Additional information on using the Cerius2, Insight, and QUANTA interfaces, including building models and writing scripts for automated running, is contained in their respective guides. Technical information that is mainly of use to program-mers and system administrators is contained in installation/administration guides. Supplemental information that may be of general interest (including additional information on the elec-tronic documentation) is contained in release notes.

MSI’s website The URL for the documentation and customer support areas of MSI’s website are :

http://www.msi.com/doc/http://www.msi.com/support/

Information relevant to forcefields, simulation engines, and mod-eling programs can be found.

Typographical conventions


Typographical conventions

Unless otherwise noted in the text, Forcefield-Based Simulations uses these typographical conventions:

♦ Terms introduced for the first time are presented in italic type. For example:

Instructions are given to the software via control panels.

♦ Keywords in the interface are presented in bold type. In addi-tion, slashes (/) are used to separate a menu item from a sub-menu item. For example:

Select the View/Colors… menu item means to click the View menu item, drag the cursor down the pulldown menu that appears, and release the mouse button over the Colors… item.

♦ Words you type or enter are presented in bold type. For exam-ple:

Enter 0.001 in the Convergence entry box.

♦ UNIX command dialog and file samples are represented in a typewriter font. For example, the following illustrates a line in a .grf file:

CERIUS Grapher File

♦ Words in italic represent variables. For example:

discovery input_file

In this example, the name of the file from which data are read in replaces input_file.


1. Introduction


2 Forcefields

This chapter focuses specifically on the forcefields supported by MSI’s simulation engines.

Who should read this chapter

You should read this chapter if you want to know:

♦ What a forcefield is.

♦ What a potential energy surface is.

♦ How to choose the best forcefield for your system.

This chapter explains The potential energy surface

Empirical fit to the potential energy surface

Forcefields supported by MSI forcefield engines

Second-generation forcefields accurate for many properties

Rule-based forcefields broadly applicable to the periodic table

Classical forcefields

Special-purpose forcefields

Archived and untested forcefields

Related information Preparing the Energy Expression and the Model presents information on how the functional forms of forcefields are used for real simu-lations. You need to read it to optimize how you set up your simu-lation. The general procedure for forcefield-based calculations is outlined under Using forcefields.

The atom types defined for each forcefield are listed under Force-field Terms and Atom Types. Illustrations of various types of cross terms are also included.

The files that specify the forcefields are described in the separate documentation for each simulation engine.


2. Forcefields

The potential energy surface

The complete mathematical description of a molecule, including both quantum mechanical and relativistic effects, is a formidable problem, due to the small scales and large velocities. However, for this discussion, these intricacies are ignored and the focus is on general concepts, because molecular mechanics and dynamics are based on empirical data that implicitly incorporate all the relativ-istic and quantum effects. Since no complete relativistic quantum mechanical theory is suitable for the description of molecules, this discussion starts with the nonrelativistic, time-independent form of the Schrödinger description:

The Schrödinger equation Eq. 1

where H is the Hamiltonian for the system, Ψ is the wavefunction, and E is the energy. In general, Ψ is a function of the coordinates of the nuclei (R) and of the electrons (r).

The Born–Oppenheimer approximation

Although this equation is quite general, it is too complex for any practical use, so approximations are made. Noting that the elec-trons are several thousands of times lighter than the nuclei and

Table 2. Finding information in Forcefields section

If you want to know about: Read:

The theory behind forcefields. The potential energy surface; Empirical fit to the potential energy surface.

What a forcefield is. The forcefield; The energy expression.Characteristics of forcefields. Main types of forcefields.What forcefields are available in which

MSI modeling programs.Table 3. Primary uses of forcefields provided in MSI prod-

ucts.Choosing the best forcefield for your

calculation.Table 3. Primary uses of forcefields provided in MSI prod-

ucts. followed by one or more of the descriptive subsec-tions starting under Second-generation forcefields accurate for many properties.

HΨ R r,( ) EΨ R r,( )=



therefore move much faster, Born and Oppenheimer (1927) pro-posed what is known as the Born–Oppenheimer approximation: the motion of the electrons can be decoupled from that of the nuclei, giving two separate equations. The first equation describes the electronic motion:

Equation for electronic motion, or the potential energy surface

Eq. 2

and depends only parametrically on the positions of the nuclei. Note that this equation defines an energy E(R), which is a function of only the coordinates of the nuclei. This energy is usually called the potential energy surface.

Equation for nuclear motion on the potential energy surface

The second equation then describes the motion of the nuclei on this potential energy surface E(R):

Eq. 3

The direct solution of Eq. 2 is the province of ab initio quantum chemical codes such as Gaussian, CADPAC, Hondo, GAMESS, DMol, and Turbomole. Semiempirical codes such as ZINDO, MNDO, MINDO, MOPAC, and AMPAC also solve Eq. 2, but they approximate many of the integrals needed with empirically fit functions. The common feature of these programs, though, is that they solve for the electronic wavefunction and energy as a function of nuclear coordinates. In contrast, simulation engines provide an empirical fit to the potential energy surface.


Solving Eq. 3 is important if you are interested in the structure or time evolution of a model. As written, Eq. 3 is the Schrödinger equation for the motion of the nuclei on the potential energy sur-face. In principle, Eq. 2 could be solved for the potential energy E, and then Eq. 3 could be solved. However, the effort required to solve Eq. 2 is extremely large, so usually an empirical fit to the potential energy surface, commonly called a forcefield (V), is used. Since the nuclei are relatively heavy objects, quantum mechanical effects are often insignificant, in which case Eq. 3 can be replaced by Newton’s equation of motion:

Hψ r R;( ) Eψ r R;( )=

HΦ R( ) EΦ R( )=


2. Forcefields

Eq. 4

Molecular dynamics and mechanics

The solution of Eq. 4 using an empirical fit to the potential energy surface E(R) is called molecular dynamics. Molecular mechanics ignores the time evolution of the system and instead focuses on finding particular geometries and their associated energies or other static properties. This includes finding equilibrium struc-tures, transition states, relative energies, and harmonic vibrational frequencies.

The forcefield

Components of a force-field

The forcefield contains the necessary building blocks for the calcu-lations of energy and force:

♦ A list of atom types.

♦ A list of atomic charges (if not included in the atom-type infor-mation).

♦ Atom-typing rules.

♦ Functional forms for the components of the energy expression.

♦ Parameters for the function terms.

♦ For some forcefields, rules for generating parameters that have not been explicitly defined.

♦ For some forcefields, a defined way of assigning functional forms and parameters.

This total “package” for the empirical fit to the potential energy surface is the forcefield.

Coordinates, terms, func-tional forms

The forcefields commonly used for describing molecules employ a combination of internal coordinates and terms (bond distances, bond angles, torsions, etc.), to describe part of the potential energy sur-face due to interactions between bonded atoms, and nonbond terms to describe the van der Waals and electrostatic (etc.) interactions between atoms. The functional forms range from simple quadratic forms to Morse functions, Fourier expansions, Lennard–Jones potentials, etc.

RddV

– mt2

2

d

d R=



Purpose of forcefields The goal of a forcefield is to describe entire classes of molecules with reasonable accuracy. In a sense, the forcefield interpolates and extrapolates from the empirical data of the small set of models used to parameterize the forcefield to a larger set of related mod-els. Some forcefields aim for high accuracy for a limited set of ele-ment types, thus enabling good prediction of many molecular properties. Other forcefields aim for the broadest possible cover-age of the periodic table, with necessarily lower accuracy.

Physical significance The physical significance of most of the types of interactions in a forcefield is easily understood, since describing a model’s internal degrees of freedom in terms of bonds, angles, and torsions seems natural. The analogy of vibrating balls connected by springs to describe molecular motion is equally familiar. However, it must be remembered that such models have limitations. Consider for example the difference between such a mechanical model and a quantum mechanical “bond”.

Quantum and mechani-cal descriptions of bonds

Covalent bonds can, to a first approximation, be described by a harmonic oscillator, both in quantum and classical mechanical the-ory. Consider the classic oscillator in Figure 1. A ball poised at the intersection of the pale horizontal line with the parabolic energy surface (thick line) would begin to roll down, converting its poten-tial energy to kinetic energy and achieving a maximum velocity as it passes the minimum. Its velocity (kinetic energy) is then con-verted back into potential energy until, at the exact same height as it had started, it would pause momentarily before rolling back. The interchange of kinetic and potential energy in such a mechan-ical system is familiar and intuitive.

The probability of finding the ball at any point along its trajectory is inversely proportional to its velocity at that point (which is opposite to the probability for a real atom). This probability is plot-ted above the parabolic curve (thin line, Figure 1). The probability is greatest near the high-energy limits of its trajectory (where it is moving slowly) and lowest at the energy minimum (where it is moving quickly). Because the total energy cannot exceed the initial potential energy defined by the starting point, the probability drops to zero outside the limit defined by the intersection of the total energy (pale horizontal line) with the parabola.

Describing a quantum mechanical “trajectory” is impossible, because the uncertainty principle prevents an exact, simultaneous specification of both position and momentum. However, the prob-


2. Forcefields

ability that the quantum mechanical ball will be at a given point on the parabola can be quantified. The quantum mechanical probabil-ity function plotted in the right panel of Figure 1 is very different from the mechanical system. First, the highest probability is at the energy minimum, which is the opposite of the mechanical case. Second, the quantum mechanical ball can actually be found beyond the classical limits imposed by the total energy of the sys-tem (tunneling). Both these properties can be attributed to the uncertainty principle.

Utility of the forcefield approach

With such a different qualitative picture of fundamental physical principles, is it reasonable to use a mechanical approach for obvi-ously quantum mechanical entities like bonds? In practice, many experimental properties such as vibrational frequencies, sublima-

Figure 1. Energy and probability of a mechanical and quantum particle in a harmonic energy well

The energy is indicated by the heavy lines and probability by the thin lines. The total energy of the system is indicated by the pale horizontal line. The classical (mechanical) probability is highest when the particle reaches it maximum potential energy (zero velocity) and drops to zero between these points. The quantum mechanical probability is highest where the potential energy is low-est, and there is a finite probability that the particle can be found outside the classical limits (pale vertical lines).

classical harmonic oscil-lator

quantum harmonic os-cillator

3.0

0.0

3.0

0.0-2 2 -2 2



tion energies, and crystal structures can be reproduced with a forcefield, not because the systems behave mechanically, but because the forcefield is fit to reproduce relevant observables and therefore includes most of the quantum effects empirically. Never-theless, it is important to appreciate the fundamental limitations of a mechanical approach.

Limitations of the force-field approach

Applications beyond the capability of most forcefield methods include:

♦ Electronic transitions (photon absorption).

♦ Electron transport phenomena.

♦ Proton transfer (acid/base reactions).

The power of the force-field approach

The true power of the atomistic description of a model embodied in the energy expression lies in three major areas:

♦ The first is that forcefield-based simulations can handle large systems, since these simulations are several orders of magni-tude faster (and cheaper) than quantum-based calculations. Forcefield-based simulations can be used for studying con-densed-phase molecules, macromolecules, crystal morphology, inorganic and organic interphases, etc., where the properties of interest are not sensitive to quantum effects (e.g., phase behav-ior, equations of state, bond energies, etc.).

♦ The second is the analysis of the energy contributions at the level of individual or classes of interactions. For instance, you can decompose the energy into bond energies, angle energies, nonbond energies, etc. or even to the level of a specific hydro-gen bond or van der Waals contact, in order to understand a physical observable or to make a prediction.

♦ The third area, which is described under Applying constraints and restraints, lies in the modification of the energy expression to bias the calculation. You can impose constraints (absolute conditions), such as fixing an atom in space and not allowing it to move. You can also add extra terms to the energy expression to restrain or force the system in certain ways. For instance, by adding an extra torsion potential to a particular bond, you can force the torsion angle toward a desired value. (You can apply constraints also for quantum-based energy calculations.)


2. Forcefields

The energy expression

The actual coordinates of a model combined with the forcefield data create the energy expression (or target function) for the model. This energy expression is the equation that describes the potential energy surface of a particular model as a function of its atomic coordinates.

The potential energy of a system can be expressed as a sum of valence (or bond), crossterm, and nonbond interactions:

Valence interactions The energy of valence interactions is generally accounted for by diagonal terms, namely, bond stretching (Ebond), valence angle bending (Eangle), dihedral angle torsion (Etorsion), and inversion (also called out-of-plane interactions) (Einversion or Eoop) terms, which are part of nearly all forcefields for covalent systems. A Urey–Bradley term (EUB) may be used to account for interactions between atom pairs involved in 1–3 configurations (i.e., atoms bound to a common atom):

Evalence = Ebond + Eangle + Etorsion + Eoop + EUB Eq. 5

Valence crossterms Modern (second-generation) forcefields generally achieve higher accuracy by including cross terms to account for such factors as bond or angle distortions caused by nearby atoms. Crossterms can include the following terms: stretch–stretch, stretch–bend–stretch, bend–bend, torsion–stretch, torsion–bend–bend, bend–torsion–bend, stretch–torsion–stretch. (These are illustrated under Force-field Terms and Atom Types.)

Nonbond interactions The energy of interactions between nonbonded atoms is accounted for by van der Waals (EvdW), electrostatic (ECoulomb), and (in some older forcefields) hydrogen bond (Ehbond) terms:

Enonbond = EvdW + ECoulomb + Ehbond Eq. 6

Restraints Restraints that can be added to an energy expression include dis-tance, angle, torsion, and inversion restraints. Restraints are useful if you, for example, are interested in the structure of only part of a model. For information on restraints and their implementation and use, see Preparing the Energy Expression and the Model in this

Etotal Evalence Ecrossterm Enonbond+ +=



documentation set and also the documentation for the particular simulation engine.

Example energy expres-sion for water

As a simple example of a complete energy expression, consider the following equation, which might be used to describe the potential energy surface of a water model:

Eq. 7

where Koh, b0oh, Khoh, and θ0

hoh are parameters of the forcefield, b is the current bond length of one O–H bond, b′ is the length of the other O–H bond, and θ is the H–O–H angle.

In this example, the forcefield defines:

♦ The coordinates to be used (bond lengths and angles).

♦ The functional form (a simple quadratic in both types of coor-dinates).

♦ The parameters (the force constants Koh and Khoh, as well as the reference values b0

oh and θ0hoh).

The reference O–H bond length and reference H–O–H angle are the values for an ideal O–H bond and H–O–H angle at zero energy, which is not necessarily the same as their equilibrium values in a real water molecule.

Example forcefield func-tion

Eq. 7 is an example of an energy expression as set up for a simple molecule. Eq. 8 is an example of the corresponding general, summed forcefield function:

V R( ) Koh b boh0

–( )2

Koh b′ boh0

–( )2

Khoh θ θhoh0

–( )2

+ +=


2. Forcefields

Eq. 8

The first four terms in this equation are sums that reflect the energy needed to stretch bonds (b), bend angles (θ) away from their reference values, rotate torsion angles (φ) by twisting atoms about the bond axis that determines the torsion angle, and distort planar atoms out of the plane formed by the atoms they are bonded to (χ). The next five terms are cross terms that account for interactions between the four types of internal coordinates. The final term represents the nonbond interactions as a sum of repul-sive and attractive Lennard–Jones terms as well as Coulombic terms, all of which are a function of the distance rij between atom pairs. The forcefield defines the functional form of each term in this equation as well as the parameters such as Db, α, and b0. The forcefield also defines internal coordinates such as b, θ, φ, and χ as a function of the Cartesian atomic coordinates, although this is not explicit in Eq. 8.

We should note that the energy expression in Eq. 8 is cast in a gen-eral form. The true energy expression for a specific model includes information about the coordinates that are included in each sum. For example, it is common to exclude interactions between bonded and 1–3 atoms in the summation representing the nonbond inter-actions. Thus, a true energy expression might actually use a list of allowed interactions rather than the full summation implied in Eq. 8.

V R( ) Db 1 exp a b b0–( )–( )–[ ] 2

b

∑ Hθ θ θ0–( )2

θ

∑ Hφ 1 s nφ( )cos+[ ]

φ

∑+ +=

Hχχ2

χ

∑ Fbb ′ b b0–( ) b′ b ′0–( )

b ′

∑b

∑ Fθθ′ θ θ0–( ) θ′ θ′0–( )

θ′

∑θ

∑+ + +

Fbθ b b0–( ) θ θ0–( )

θ

∑b

∑ Fθθ′φ

θ′

∑ θ θ0–( ) θ′ θ′0–( ) φcos

θ

∑+ +

Fχχ′ χχ′

χ′

∑χ

∑ Aij

rij12

-------Bij

rij6

------–qiqj

rij---------+

j i>

∑i

∑+ +




The results of any mechanics or dynamics calculation depend cru-cially on the forcefield. The quality of the description of both the system and the particular properties being analyzed is of para-mount importance. Accurate, specific parameters generally give better results than automatic, generic parameters. Choosing the correct forcefield is vitally important in getting reasonable results from energy calculations.

Contents of this section This section gives a general comparison of the forcefields that are available in MSI products and presents the reasoning behind mak-ing a wide variety of forcefields available to our customers. It should enable you to make at least a preliminary choice of which forcefield to use.

Forcefield descriptions Complete descriptions of each forcefield follow in subsequent sec-tions:

Second-generation force-fields

CFF91, PCFF, CFF —consistent forcefields

MMFF93, the Merck molecular forcefield

Broadly applicable force-fields

ESFF, extensible systematic forcefield

UFF, universal forcefield

VALBOND

Dreiding forcefield

Classical forcefields Standard AMBER forcefield

Homans’ carbohydrate forcefield

CHARMm forcefield

CVFF, consistent valence forcefield

Special-purpose force-fields

Glass forcefield

MSXX forcefield for polyvinylidene fluoride

Zeolite forcefields

Forcefields for sorption on zeolites

Forcefields for Cerius2•Morphology module


2. Forcefields

Other forcefields Archived and untested forcefields

Related information The atom types defined by each forcefield are listed under Force-field Terms and Atom Types, and the types of parameters used in the forcefields are described in the documentation for each simulation engine.

Main types of forcefields

MSI provides four main types of forcefields:

♦ Second-generation forcefields capable of predicting many properties.

♦ Rule-based forcefields applicable to a broad range of the peri-odic table.

♦ Classical, first-generation forcefields applicable mainly to bio-chemistry.

♦ Special-purpose forcefields that are narrowly applicable to par-ticular applications or types of models.

A complete list of these forcefields, their main uses, and the simu-lation engine that handles them is given in Table 3.

In addition, we supply (but do not support) several older or untested forcefields.

Second-generation force-fields

♦ The CFF family of forcefields (CFF91, PCFF, CFF) are closely related second-generation forcefields (Maple et al. 1988, 1994a, b, Dinur and Hagler 1991, Waldman and Hagler 1993, Hill and Sauer 1994, Hwang et al. 1994, Hagler and Ewig 1994, Sun et al. 1994, Sun 1994, 1995).

The CFF family of forcefields were parameterized against a wide range of experimental observables for organic com-pounds containing H, C, N, O, S, P, halogen atoms and ions, alkali metal cations, and several biochemically important diva-lent metal cations.

CFF has slightly more atom types than CFF91 (Forcefield Terms and Atom Types).

PCFF is based on CFF91, extended so as to have a broad cover-age of organic polymers, (inorganic) metals, and zeolites.



The CFF family of forcefields have been shown to reproduce experimental results more accurately than classical forcefields such as CVFF and AMBER.

♦ COMPASS stands for Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies. COMPASS is MSI's breakthrough forcefield technology for materials science. It is the first ab initio forcefield that enables accurate and simulta-neous prediction of structural, conformational, vibrational, and thermophysical properties for a broad range of molecules in isolation and in condensed phases. It is also the first high qual-ity forcefield that consolidates parameters for organic and inor-ganic materials previously found in different forcefields.

COMPASS is developed from PCFF, a CFF type forcefield for organic materials and polymers. In addition to the high-quality prediction of molecular properties that PCFF offers, COMPASS has been specifically optimized to yield high accuracy predic-tions of condensed-phase properties of many organic liquids and polymers, making possible, for example, detailed studies of the temperature dependence of solubility parameters of polymers and small molecule liquids. In addition to broad cov-erage in organics, small inorganic molecules, and polymers, this forcefield has been augmented with new functional forms, to include coverage of inorganic materials - metal oxides, met-als, and metal halides. Detailed information about the COM-PASS forcefield can be found in the COMPASS user guide.

♦ The Merck molecular forcefield (MMFF93), developed by T. A. Halgren at the Merck Research Laboratories (1992, Halgren & Nachbar, 1996) is designed to be used with a large variety of chemical models.

The main application of MMFF93 is to the study of receptor–ligand interactions involving proteins or nucleic acids as recep-tors and a wide range of chemical structures as ligands. The forcefield can describe ligands and receptors in isolation as well as in the bound state.

Rule-based forcefields ♦ The ESFF forcefield (extensible systematic forcefield) is a rule-based forcefield that was developed at MSI.

The goal of this forcefield is to provide the widest possible cov-erage of the periodic table, enabling both the structures of iso-lated molecules and crystals to be reproduced. Its scope does


2. Forcefields

not extend to highly accurate vibrational frequencies or other properties such as conformational energies.

♦ The Universal forcefield (Rappé et al. 1992) is an excellent gen-eral-purpose forcefield. All the Universal forcefield parameters are generated from a set of rules based on element, hybridiza-tion, and connectivity.

The Universal forcefield was parametrized for the full periodic table and has been carefully validated for main-group com-pounds (Casewit et al. 1992b), organic molecules (Casewit et al. 1992a), and metal complexes (Rappé et al. 1993).

♦ VALBOND is a combination of the UFF, universal forcefield, and the VALBOND method for the angle energy.

This forcefield combines the advantages of a general forcefield with the strengths of the VALBOND method and may give bet-ter results for non-hypervalent structures where the geometry of ligands around a central atom is unknown.

♦ The Dreiding forcefield (Mayo et al. 1990) is a good, robust, all-purpose forcefield. While a specialized forcefield is more accu-rate for predicting a limited number of structures, the Dreiding forcefield allows reasonable predictions for a very much larger number of structures, including those with novel combinations of elements and those for which there is little or no experimen-tal data.

It can be used for structure prediction and dynamics calcula-tions on organic, biological, and main-group inorganic mole-cules.

Classical forcefields ♦ The AMBER forcefield Weiner et al. 1984, 1986) was parameter-ized against a limited number of organic models. It has been widely used for proteins, DNA, and other classes of molecules and may be considered well characterized.

The standard AMBER forcefield is mainly useful for proteins and nucleic acids. The Homans (1990) carbohydrate forcefield is based on AMBER, but extended to polysaccharides. It is not generally recommended for use in materials science studies.

♦ The CHARMm forcefield (Chemistry at HARvard Macromolecu-lar mechanics) is packaged in a highly flexible molecular mechanics and dynamics engine originally developed in the



laboratory of Dr. Martin Karplus at Harvard University. It has been widely used and can be considered well tested and char-acterized (e.g., Brooks et al. 1983, Momany and Rone 1992).

A variety of systems, from isolated small molecules to solvated complexes of large biological macromolecules, can be simu-lated using CHARMm.

♦ The CVFF forcefield is a classic forcefield having some anhar-monic and cross term enhancements. As the traditional default forcefield in the Discover program, it has been used extensively and can be considered well tested and characterized.

CVFF was parameterized to reproduce peptide and protein properties.

Special-purpose force-fields

♦ In addition to some standard forcefields, the Cerius2•Open Force Field module provides several smaller forcefield param-eter files for more specialized work.

These include separate forcefields for glasses, zeolites, and polyvinylidene fluoride, as well as some forcefields that are intended only for use in the Cerius2•Morphology module.

Advantages of having several forcefields

The ability to choose among several forcefields has several advan-tages:

1. A broader range of systems can be treated:

Some classical forcefields were originally created for modeling proteins and peptides, others for DNA and RNA. Some have been extended to handle more general systems having similar functional groups.

The rule-based forcefields have extended the range of forcefield simulations to a broader range of elements.

The second-generation forcefields currently include parame-ters for all functional groups appropriate for protein simula-tions.

2. Identical calculations with two or more independent forcefields can be compared to assess the dependence of the results on the forcefield:


2. Forcefields

For example, amino acid parameters are defined in the AMBER, CHARMm, CVFF, CFF, and MMFF93 forcefields, so peptide and protein calculations with these forcefields can be compared to assess the effect of the forcefields.

3. The different functional forms used in the various energy expressions increase the flexibility of the Discover program and the Open Force Field module:

You can balance the requirements of high accuracy vs. available computational resources. (Highly accurate forcefields are gen-erally more complex and therefore require more resources.)

Different energy terms can be compared. For example, approx-imations such as a distance-dependent dielectric constant or scaling of 1–4 nonbond interactions can be assessed.

Harmonic bond terms are accurate only at bond lengths close to the reference bond length, but the Morse term can be used to model bond breaking.

4. The development of new forcefields at MSI and elsewhere con-tinues to provide more accurate and more broadly applicable forcefields. As experience is gained in parameterizing force-fields and as new experimental data become available, the range of both properties and systems fit by these newer force-fields will increase.

Primary uses of each MSI forcefield

Table 3 summarizes the forcefields best suited for various types of work and lists the simulation engines that handle each one:



Table 3. Primary uses of forcefields provided in MSI products (Page 1 of 2)

Type and use of forcefieldForcefield

name Simulation engine Forcefield filename(s)

Second-generation, general-purpose

CFF91 Discover; OFFa cff91.frc; cff91_950_1.01

CFF95b Discover; OFF cff95.frc; cff95_950_1.01

CFF Discover, OFF cff.frc; cff1.01MMFF93 CHARMmc mmff_setup.STRPCFF Discover; OFF pcff.frc; pcff_300_1.01 COMPASS98

dDiscover; OFF compass.frc;

COMPASS1.0, compass98.frc; Compass98.01

Rule-based, broadly applicable, general-purpose

ESFF Discovere esff.frcUniversal OFF UNIVERSAL1.02UFF-VAL-

BONDOFF UFF_VALBOND1.01

Dreiding OFF DREIDING2.21

Classical, general-pur-pose (biochemistry)

AMBER Discover, OFFf amber.frcCHARMm CHARMmg

CVFF Discover; OFF cvff.frc; cvff_950_1.01h

Special-purpose:Inorganic oxide glasses Glass OFF glassff_1.01, glassff_

2.01Morphology module of

Cerius2 Lifson OFF morph_lifson1.11Momany OFF morph_momany1.1Scheraga OFF morph_scheraga1.1Williams OFF morph_williams1.01

Polyvinylidene fluoride polymers

MSXX OFF msxx_1.01


2. Forcefields

aOFF = the Open Force Field module of Cerius2.bMarketed as an add-on forcefield, not present in Discover or OFF by default.cCHARMm as run through the Cerius2•MMFF module, not in QUANTA or standard CHARMm.dMarketed as an add-on forcefield, not present in Discover or OFF by default.eIn CDiscover, not FDiscover; in other words, in the Cerius2•Discover and the Insight•Discover_3 modules, not the Insight•Discover module.fIn the Insight•Discover_3 and the Insight•Discover modules but not the Cerius2•Discover mod-ule. An older version of AMBER is accessible through the Cerius2•OFF module.gCHARMm is both the name of a forcefield and the name of a simulation engine that handles the CHARMm forcefield.hCVFF differs slightly in versions 3.0.0 and 95.0 of Insight II—both versions are included in Cerius2•OFF.

Additional information Additional information about forcefields included with Cerius2 is printed to the text window when you load a forcefield. Alterna-tively, you can click the Show information action button in the Load Force Field control panel.

Additional information about forcefields included with Insight 4.0.0 can be obtained with the Forcefield/FF_Info parameter block, which is accessed through the Builder and other modules. (It is not included in Insight 97.0.)

Zeolites BKS OFF bks1.01Burchart OFF burchart1.01Burchart–

DreidingOFF burchart1.01-

DREIDING2.21Burchart–

UniversalOFF burchart1.01-

UNIVERSAL1.02CVFF_aug Discover; OFF cvff_300_1.01

Zeolite sorption Yashonath OFF sor_yashonath1.01Demontis OFF sor_demontis1.01Pickett OFF sor_pickett1.01Watanabe–

AustinOFF watanabe-austin1.01

Older, archived, misc. several Discover; OFF gifts/*, archive/*

Untested, misc. several OFF untested/

Table 3. Primary uses of forcefields provided in MSI products (Page 2 of 2)

Type and use of forcefieldForcefield

name Simulation engine Forcefield filename(s)



Second-generation forcefields accurate formany properties

Availability Second-generation forcefields provided or developed by MSI include CFF91, CFF, PCFF, COMPASS, and MMFF93:

♦ The CFF family of forcefields (CFF91, PCFF, CFF —consistent forcefields) are run via the Discover program, which is available in the Insight•Discover_3, Insight•Discover, and Cerius2•Dis-cover modules. They can also be used by the Cerius2•Open Force Field module. Discover is also used implicitly by other modules in Insight II (such as some in the Polymer suite of products). The CFF and COMPASS forcefields are separately licensed (that is, not present by default within Discover).

♦ MMFF93 (MMFF93, the Merck molecular forcefield) is run via a version of CHARMm that supports the Cerius2•MMFF mod-ule.

Characteristics The topography of an energy surface is usually very complex, especially for large and/or complex models, with many energy minima and barriers and regions of greatly varying energy and curvature. Nevertheless, the forcefield expression must be as accu-rate and complete as possible, to avoid spurious or misleading results. The newer, second-generation forcefields meet this requirement through their greater complexity than the classical forcefields, having expanded analytic energy expressions that include additional terms.

Parameterization The complexity of second-generation forcefields requires the use of a large number of forcefield parameters. There are almost always far more parameters than can be inferred from experiment, such as by microwave or infrared spectroscopy. However, modern quantum mechanical methods can generate enough quantum observables so that all the necessary parameters can be accurately determined by fitting the energy expression to these observables.

Quantum calculations of the energy surfaces of a series of model compounds (equilibrium structures, models at conformational energy barriers, and distorted structures) yield energies as well as their derivatives with respect to atomic coordinates (i.e., the sur-


2. Forcefields

face gradients and curvatures) (Maple et al. 1994a, b). Many atomic partial charges are also determined quantum mechanically.

Intermolecular or nonbond parameters are computed by fitting to experimental crystal lattice constants and sublimation energies of crystals (Hagler et al. 1979a, b).

Since quantum mechanics (Hartree–Fock approximation with the 6-31G* basis set) yields results that differ consistently from exper-iment, the parameterized forcefield is then fit to experimental data by parameterizing a small number of scaling factors (Hwang et al. 1994).

The CFF family of forcefields (within Discover) can use automatic parameters (Automatic assignment of values for missing parameters) when no explicit parameters are present. These are noted in the output file from the calculation.

Advantages of deriving forcefields from quantum calculations

The use of quantum calculations in the development of second-generation forcefields has the advantages that:

♦ Sufficient data are available for accurately determining all the forcefield parameters.

♦ The resulting forcefield may be broad in terms of the types of molecules and molecular environments that may be modeled, since no recourse to experiment is required, even for unusual or transient species. Properties can be modeled for:

Isolated small molecules (structure, thermodynamics, spectros-copy).

Condensed phases (crystal structure, sublimation energies, heats of vaporization).

Macromolecular systems.

♦ The resulting forcefield is consistent, since all parameters, func-tional groups, and molecular species are modeled in the same way. This is in contrast to forcefields whose parameters are derived empirically, since the experimental data for different molecules necessarily come from greatly differing sources and types of measurements, and are sometimes of questionable accuracy.

♦ Fewer atom types are necessary.



CFF91, PCFF, CFF —consistent forcefields

Functional form

All the CFF forcefields (CFF91, CFF, PCFF) have the same func-tional form, differing mainly in the range of functional groups to which they were parameterized (and therefore, having slightly dif-ferent parameter values). These differences can be examined by using the forcefield editing capabilities of Cerius2 and Insight or in the forcefield files. Atom equivalences for assignment of parame-ters to atom types may also differ, as may some combination rules for nonbond terms (see Preparing the Energy Expression and the Model for explanation of these processes, which occur during forcefield setup).

The analytic expressions used to represent the energy surface are shown in Eq. 9. Both anharmonic diagonal terms and many cross-terms are necessary for a good fit to a variety of structures and rel-ative energies, as well as to vibrational frequencies.

The CFF forcefields employ quartic polynomials for bond stretch-ing (Term 1) and angle bending (Term 2) and a three-term Fourier expansion for torsions (Term 3). The out-of-plane (also called inversion) coordinate (Term 4) is defined according to Wilson et al. (1980). All the crossterms up through third order that have been found to be important (Terms 5–11) are also included—this gives a forcefield equivalent to the best used in a formate anion test case (Maple et al. 1990). Term 12 is the Coulombic interaction between the atomic charges, and Term 13 represents the van der Waals interactions, using an inverse 9th-power term for the repulsive part rather than the more customary 12th-power term.

No explicit special atom types are used for carbons in strained three- and four-membered rings. The quartic angle potential, com-bined with crossterms, enables accurate description of normal alkanes, cyclobutane, and cyclopropane with one set of parame-ters.

NoteBecause the Wilson out-of-plane definition is used in the CFF family of forcefields, results calculated with CDiscover, FDiscover, and Cerius2•OFF should agree exactly.


2. Forcefields

Eq. 9



K2 b b0–( )2 K3 b b0–( )3 K4 b b0–( )4+ +[ ]

b

∑

2 θ θ0–( )2 H3 θ θ0–( )3 H4 θ θ0–( )4+ +

V1 1 φ φ10

–( )cos–[ ] V2 1 2φ φ20

–( )cos–[ ] V3 1 3φ φ30

–(cos–[+ +[∑

χχ2 Fbb ′ b b0–( ) b ′ b′0–( )

b ′

∑b

∑ Fθθ′ θ θ0–( ) θ′ θ–(

θ′

∑θ

∑+ +

Fbθ b b0–( ) θ θ0–( )

θ

∑ b b0–( ) V1 φcos V2 2φcos V+ +[

φ

∑b

∑+

b ′ b′0–( ) V1 φcos V2 2φcos V3 3φcos+ +[ ]

φ

∑

θ θ0–( ) V1 φcos V2 2φcos V3 3φcos+ +[ ]

φ

∑

Kφθθ′ φcos θ θ0–( ) θ′ θ′0–( )

θ′

∑θ

∑ qiqj

εrij---------

i j>

∑ Aij

rij9------

Bij

rij6------–

i j>

∑+ +

(1)

(2)

(3)

(5) (6)

(7) (8)

(9)

(10)

(11) (12) (13)


2. Forcefields

CFF91 forcefield

Applicability CFF91 is useful for hydrocarbons, proteins, protein–ligand interac-tions. For small models it can be used to predict: gas-phase geom-etries, vibrational frequencies, conformational energies, torsion barriers, crystal structures; for liquids: cohesive energy densities; for crystals: lattice parameters, rms atomic coordinates, sublima-tion energies; for macromolecules: protein crystal structures.

It has been parameterized explicitly (based on quantum mechan-ics calculations and molecular simulations, see Parameterization) for acetals, acids, alcohols, alkanes, alkenes, amides, amines, aro-matics, esters, and ethers (Maple 1994a, Hwang 1994).

The functional form of CFF91 is exactly as shown in Eq. 9.

Atom types CFF91 has parameters for functional groups that consist of H, Na, Ca, C, Si, N, P, O, S, F, Cl, Br, I, and/or Ar. The atom types of the CFF91 forcefield are listed in Table 27.

Partial charges The bond increment section of the .frc file for CFF91 enables partial charges to be determined whenever the Discover program or the Cerius2•OFF module is able to assign automatic atom types.

CFF forcefield

Applicability CFF (formerly CFF95) was parameterized for additional func-tional groups beyond CFF91 (Maple et al. 1994a, b, Hwang et al. 1994, Hagler & Ewig 1994). It is recommended for all life sciences applications and for organic polymers such as polycarbonates and polysaccharides.

Almost all types of computations within Insight or Cerius2 life sci-ence modules may be performed using CFF. These include inter-molecular and intramolecular energies and forces, optimization of model structures, and molecular dynamics simulations. CFF is not currently implemented for relative free-energy perturbations or for applications in the Docking module of Insight.

Atom types The atom types of the CFF forcefield are listed in the separate doc-umentation for CFF (below).

Additional information Additional information on CFF, which is sold as a separately licensed product, is contained in the MSI Forcefields:CFF book (published separately by MSI).



PCFF forcefield for polymers and other materials

Applicability PCFF was developed based on CFF91 and is intended for applica-tion to polymers and organic materials. It is useful for polycarbon-ates, melamine resins, polysaccharides, other polymers, organic and inorganic materials, about 20 inorganic metals, as well as for carbohydrates, lipids, and nucleic acids and also cohesive ener-gies, mechanical properties, compressibilities, heat capacities, elastic constants. It handles electron delocalization in aromatic rings by means of a charge library rather than bond increments.

Validation Parameterization, testing, and validation of PCFF included the compounds listed for CFF91 and these functional groups: carbon-ates, carbamates, phosphazene, urethanes, siloxanes, silanes, ureas (Sun et al. 1994, Sun 1994, 1995), and zeolites (Hill and Sauer 1994). Metal parameters (listed below) were derived by fitting to crystal structures and elastic constants.

Atom types PCFF has parameters for functional groups that consist of those listed for CFF91 and also He, Ne, Kr, Xe. In addition, it includes Lennard–Jones parameters for the metals Li, K, Cr, Mo, W, Fe, Ni, Pd, Pt, Cu, Ag, Au, Al, Sn, Pb. Atom type coverage in PCFF includes those listed for CFF91 (Table 27) and the atoms listed here.

MMFF93, the Merck molecular forcefield

The Merck molecular forcefield is derived largely from ab initio calculations and can be accurately applied to a variety of con-densed-phase and aqueous systems. It uses a unique functional form for describing the van der Waals interactions (Halgren 1992) and employs novel combination rules that systematically correlate van der Waals parameters with those that describe experimentally characterized interactions involving rare-gas atoms. Electrostatic interactions are scaled to mimic solution effects.

Applicability Conformational energies, geometries, and vibrational frequencies of small organic molecules.

Functional form The MMFF93 energy expression is similar to that of MM2 and MM3:


2. Forcefields

Eq. 10

Where:

EbijQuartic bond stretching term.

EaijkCubic angle bending term (cosine when the refer-

ence angle is ≅ 180°).

EbaijkStretch–bend crossterm.

Eoopijk;lTerm for out-of-plane motion at tri-coordinate cen-

ters, using the Wilson definition of the out-of-plane angle.

EtijklTorsion twisting term.

EvdWijBuffered 14–7 van der Waals interaction term.

EqijBuffered Coulombic term for electrostatic interac-

tions. Use of a distance-dependent dielectric “constant” is supported.

To allow straightforward application to condensed-phase simula-tions employing implicit solvent molecules, MMFF93 includes a dielectric constant in its electrostatic interaction terms.

Charges are implemented via bond increments (similar to CVFF or the CFF family of forcefields) that are included as part of the force-field.

Missing parameters are supplied via a generic step-down and equivalency typing scheme (see Preparing the Energy Expression and the Model).

The terms of the energy expression are calculated in kcal mol-1. They are described in detail by Halgren (1992, 1996a–d, Halgren & Nachbar, 1996).

EMMFF Ebi jEai jk

Ebaijk

Eoop Et EvdW Eq∑+∑+∑+∑+

∑+∑+∑=



Rule-based forcefields broadly applicable to theperiodic table

Availability Rule-based forcefields provided by MSI with generally broad applicability across the periodic table include ESFF, the Universial forcefield, and the Dreiding forcefield:

♦ The ESFF forcefield (ESFF, extensible systematic forcefield) is run via the CDiscover program, which is available in the Insight•Discover_3 module.

♦ UFF-VALBOND (VALBOND) is available via the Cerius2 Open Force Field module.

♦ The Universal (UFF, universal forcefield) and Dreiding (Dreiding forcefield) forcefields are accessible through the Cerius2•Open Force Field module.

Parameterization Although the second-generation (Second-generation forcefields accu-rate for many properties) and classical (Classical forcefields) forcefields derive the forcefield parameters by fitting ab initio and/or exper-imental data sets, these rule-based forcefields rely on atomic param-eters coupled with theoretically and empirically derived rules for generating explicit forcefield parameters. The rules embody phys-ical reality (electronegativity, hardness, atomic radii for UFF and ESFF, simple hybridization for Dreiding) and therefore tend to break redundancies and guarantee transferability. As much as pos-sible, the atomic parameters are directly determined from experi-ment or calculated rather than fit.

Characteristics ESFF can be used for structure prediction of organic, inorganic, and organometallic systems in gas or condensed phases. It covers all elements in the periodic table up to Rn. Its scope does not extend to highly accurate vibrational frequencies or conforma-tional energies (Shi et al., no date).

UFF covers all elements in the periodic table and is the default forcefield in Cerius2. It is recommended for any system that is not covered by the more accurate special-purpose forcefields. It gives better structures than the Dreiding forcefield but may not be as accurate for properties that depend on intermolecular interactions.


2. Forcefields

In the VALBOND formalism, hybrid orbital strength functions are used as the basis for a molecular expression of molecular shapes. These functions are suitable for accurately describing the energet-ics of distorted bond angles not only around the energy minimum, but also for very large distortions.

The Dreiding forcefield predicts bulk material properties that depend on intermolecular interactions better than does UFF, but it is not as widely applicable to the periodic table. It is not as accurate as the special-purpose forcefields for the materials for which they are applicable.

ESFF, extensible systematic forcefield

Derivation

ESFF was derived using a mixture of DFT calculations on dressed atoms to obtain polarizabilities, gas-phase and crystal structures, etc. The training set included primarily organic and organometal-lic compounds and a few inorganic compounds. The focus was on crystal structures and sublimation energies. The training set included models containing each element in the first 6 periods up to lead (Z = 82) (except for the inert gases), Sr, Y, Tc, La, and the lan-thinides (except for Yb).

Parameters and charges are generated on-the-fly, based on the model configuration, the local environment, and the derived rules.

Functional form

Valence energy The analytic energy expressions for the ESFF forcefield are pro-vided in Eq. 11. Only diagonal terms are included.

Bond energy The bond energy is represented by a Morse functional form, where the bond dissociation energy D, the reference bond length r0, and the anharmonicity parameters are needed. In constructing these parameters from atomic parameters, the forcefield utilizes not only the atom types and bond orders, but also considers whether the bond is endo or exo to 3-, 4-, or 5-membered rings.

The rules themselves depend on the electronegativity, hardness, and ionization of the atoms as well as atomic anharmonicities and



the covalent radii and well depths. The latter quantities are fit parameters, and the former three are calculated.

Eq. 11

Angle types The ESFF angle types are classified according to ring, symmetry, and π-bonding information into five groups:

♦ The normal class includes unconstrained angles as well as those associated with 3-, 4-, and 5-membered rings. The ring angles are further classified based on whether one (exo) or both bonds (endo) are in the ring. Additionally, angles with only central atoms in a ring are also differentiated.

(5)(4)

(3)

(2)

(1)

Epot Db 1 eα– rb rb

0–( )( )

–2

b

∑=

Ka

θa0

sin2---------------- θacos θa

0cos–( )2

a

∑

2Ka θacos 1+( )

a

∑

Kaθa θacos2

a

∑2Ka

n2--------- 1 nθa( )cos–( ) 2Ka

β r13 ρa–( )–( )+

a

∑

+

(normal)

(linear)

(perpendicular)

(equatorial)

Dτθ1sin2 θ2sin2

θ10

sin2 θ20

sin2-------------------------------- sign

θ1sinn θ2sin2

θ10

sinn θ20

sinn-------------------------------- nτ[ ]cos+

τ

∑+

Doχ2

o

∑ AiBj AjBi+

rnb9

---------------------------- 3BiBj

rnb6

----------–

nb

∑ qiqj

rnb---------

nb

∑+ + +

(6)


2. Forcefields

♦ The linear class includes angles with central atoms having sp hybridization, as well as angles between two axial ligands in a metal complex.

♦ The perpendicular class is restricted to metal centers and includes angles between axial and equatorial ligands around a metal center.

♦ The equatorial class includes angles between equatorial ligands of square planar (sqp), trigonal bipyramidal (tbp), octahedral (oct), pentagonal bipyramidal (pbp), and hexagonal bipyrami-dal (hbp) systems.

♦ The π system class includes angles between pseudoatoms. This class is further differentiated in terms of normal, linear, perpen-dicular, and equatorial types.

The rules that determine the parameters in the functional forms depend on the ionization potential and, for equatorial angles, the periodicity. In addition to these calculated quantities, the parame-ters are functions of the atomic radii and well depths of the central and end atoms of the angle, and, for planar angles, two overlap quantities and the 1–3 equilibrium distances.

Torsions To avoid the discontinuities that occur in the commonly used cosine torsional potential when one of the valence angles approaches 180°, ESFF uses a functional form that includes the sine of the valence angles in the torsion. These terms ensure that the function goes smoothly to zero as either valence angle approaches 0° or 180°, as it should. The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central atom which is fit.

Out-of-plane centers The functional form of the out-of-plane energy is the same as in CFF91, where the coordinate (φ) is an average of the three possible angles associated with the out-of-plane center. The single parame-ter that is associated with the central atom is a fit quantity.

Nonbond energy

Partial charges The charges are determined by minimizing the electrostatic energy with respect to the charges under the constraint that the sum of the charges is equal to the net charge on the molecule. This is equiva-lent to equalization of electronegativities.



Derivation The derivation of the rule begins with the following equation for the electrostatic energy:

Eq. 12

where χ is the electronegativity and η the hardness. The first term is just a Taylor series expansion of the energy of each atom as a function of charge, and the second is the Coulomb interaction law between charges. The Coulomb law term introduces a geometry dependence that ESFF for the time being ignores, by considering only topological neighbors at effectively idealized geometries.

Atomic charges Minimizing the energy with respect to the charges leads to the fol-lowing expression for the charge on atom i:

Eq. 13

where λ is the Lagrange multiplier for the constraint on the total charge, which physically is the equalized electronegativity of all the atoms. The ∆χ term contains the geometry-independent rem-nant of the full Coulomb summation.

Adjustment to chemical reality

Eqns. 12 and 13 give a totally delocalized picture of the charges in a relatively severe approximation. To obtain reasonable charges as judged by, for example, crystal packing calculations, some modifi-cations to the above picture have been made. Metals and their immediate ligands are treated with the above prescription, sum-ming their formal charges to get a net fragment charge. Delocal-ized π systems are treated in an analogous fashion. And σ systems are treated using a localized approach in which the charges of an atom depend simply on its neighbors. Note that this approach, unlike the straightforward implementations based on the equal-ization of electronegativity, does include some resonance effects in the π system.

Electronegativity and hardness obtained by DFT

The electronegativity and hardness in the above equations must be determined. In earlier forcefields they were often determined from experimental ionization potentials and electron affinities; how-ever, these spectroscopic states do not correspond to the valence

E Ei0 χiqi

12---η iqi

2+ +

i

∑ Bqiqj

Rij---------

i j>

∑+=

qi

λ χ i– χi∆–

η i----------------------------=


2. Forcefields

states involved in molecules. For this reason, ESFF is based on elec-tronegativities and hardnesses, calculated using density func-tional theory as implemented in DMol. The orbitals are (fractionally) occupied in ratios appropriate for the desired hybridization state, and calculations are performed on the neutral atom as well as on positive and negative ions.

van der Waals interactions ESFF uses the 6–9 potential for the van der Waals interactions. Since the van der Waals parameters must be consistent with the charges, they are derived using rules that are consistent with the charges.

Derivation Starting with the London formula:

Eq. 14

where α is the polarizability and IP the ionization potential of the atoms, the polarizability, in a simple harmonic approximation, is proportional to n / IP where n is the number of electrons. Across any one row of the periodic table, the core electrons remain unchanged, so that the following form is reasonable:

Eq. 15

where a′ and b′ are adjustable parameters that should depend on just the period, and neff is the effective number of (valence) elec-trons. Further assuming that α is proportional to R3 and that another equivalent expression to that in Eq. 14 is:

Eq. 16

where ε is a well depth, the following forms are deduced for the rules for van der Waals parameters:

Rules for van der Waals parameters Eq. 17

The van der Waals parameters are affected by the charge of the atom.

Modification for metal atoms

In ESFF we found it sufficient to modify the ionization potential (IP) of metal atoms according to their formal charge and hardness:

Bi α i2 IP⋅∼( )

α a ′IP------

b′neff

IP-------------+=

Bi εR6∼

Ria

IP( )1 3/------------------

b neff1 3/⋅

IP( )1 3/------------------+= and εi c IP( )=



Eq. 18

Treatment of nonmetals and for nonmetals to account for the partial charges when calculat-ing the effective number of electrons.

ESFF atom types

ESFF atom types (Table 31) are determined by hybridization, for-mal charge, and symmetry rules (Atom-typing rules in ESFF). In addition, the rules may involve bond order, ring size, and whether bonds are endo or exo to rings. For metal ligands the cis–trans and axial–equatorial positionings are also considered. The addition of these latter types affects only certain parameters (for example, bond order influences only bond parameters) and thus are not as powerful as complete atom types. In one sense they provide a fur-ther refinement of typing beyond atom types.

Coverage of the periodic table

The ESFF forcefield has been parameterized to handle all elements in the periodic table up to radon. It is recommended for organome-tallic systems and other systems for which other forcefields do not have parameters. ESFF is designed primarily for predicting rea-sonable structures (both intra- and intermolecular structures and crystals) and should give reasonable structures for organic, biolog-ical, organometallic and some ceramic and silicate models. It has been used with some success for studying interactions of mole-cules with metal surfaces. Predicted intermolecular binding ener-gies should be considered approximate.

UFF, universal forcefield

Cerius2 contains a full implementation of the Universal forcefield, including bond order assignment. The Cerius2 implementation has been rigorously tested and results are in agreement with pub-lished work on this forcefield (Rappé et al. 1992, Casewit et al. 1992a, b, Rappé et al. 1993).

Parameter generation is based on physically realistic rules.

Functional form UFF is a purely diagonal, harmonic forcefield. Bond stretching is described by a harmonic term, angle bending by a three-term Fou-rier cosine expansion, and torsions and inversions by cosine–Fou-rier expansion terms. The van der Waals interactions are described

IP IP( )0 qη i+=


2. Forcefields

by the Lennard–Jones potential. Electrostatic interactions are described by atomic monopoles and a screened (distance-depen-dent) Coulombic term.

Atom types The Universal forcefield’s atom types are denoted by an element name of one or two characters followed by up to three other char-acters:

♦ The first two characters are the element symbol (for example, N_ for nitrogen or Ti for titanium).

♦ The third character (if present) represents the hybridization state or geometry (for example, 1 = linear, 2 = trigonal, R = an atom involved in resonance, 3 = tetrahedral, 4 = square planar, 5 = trigonal bipyramidal, 6 = octahedral).

♦ The fourth and fifth characters (if present) indicate characteris-tics such as the oxidation state (for example, Rh6+3 represents octahedral rhodium in the +3 formal oxidation state; H_ _ _b indicates a diborane bridging hydrogen type; and O_3_z is a framework oxygen type suitable for zeolites).


UFF has full coverage of the periodic table. UFF is moderately accurate for predicting geometries and conformational energy dif-ferences of organic molecules, main-group inorganics, and metal complexes. It is recommended for organometallic systems and other systems for which other forcefields do not have parameters.

Parameterization The Universal forcefield includes a parameter generator that calcu-lates forcefield parameters by combining atomic parameters. Thus, forcefield parameters for any combination of atom types can be generated as required.

The atomic parameters are combined using a prescribed set of equations (rules) that generate forcefield parameters for bond, angle, torsion, inversion (i.e., out-of-plane), and van der Waals and Coulombic energy terms. For further details, including the gener-ator equations, see Rappé et al. (1992).

Dummy atoms are used in π-complexation and are associated with explicit parameters.

Important



Charges in the Universal forcefield

The Universal forcefield was developed in conjunction with the charge equilibration (Rappé & Goddard 1991) method. Therefore this method of electrostatic charge calculation is highly recom-mended for use with the Universal forcefield. For more on the charge equilibration calculation, see the documentation supplied with Cerius2•OFF).

Versions UNIVERSAL1.02 is the most up-to-date, and recommended, ver-sion of the UFF. It includes full bond-order correction. (UFF 1.02 differs from UFF 1.01 in that some explicit torsion parameters were corrected and one of the oxygen atom-typing rules was modified.)

UFF-VALBOND is UFF with a different function to calculate the angle energy, so most things which are true for UFF, are true for UFF-VALBOND.

The burchart1.01–UNIVERSAL1.02 forcefield combines UFF with the Burchart forcefield. See burchart1.01-UNIVERSAL1.02 for more information.

VALBOND

Introduction

Most molecular mechanics methods attempt to describe accurate potential energy surfaces by using a variant of the general valence forcefield, and a large number of parameters. These simple force-fields are not accurate outside the proximity of the energetic min-ima and often are difficult to apply to the different shapes and higher coordination numbers of transition metal complexes.

In the VALBOND formalism, hybrid orbital strength functions are used as the basis for a molecular expression of molecular shapes. These functions are suitable for accurately describing the energet-ics of distorted bond angles not only around the energy minimum, but also for very large distortions.

To obtain correct results when using UFF, calculate fractional bond orders after atom typing the structure and before setting up the energy expression. Cerius2 does this correctly by default, and you need not worry about it unless you change the default behavior.


2. Forcefields

The combination of these functions with simple valence bond ideas leads to a simple scheme for predicting molecular shapes.

Structures and vibrational frequencies calculated by the VAL-BOND method agree well with experimental data for a variety of molecules from the main group of the periodic table.

UFF-VALBOND is a combination of the original VALBOND method described by Root et al. (1993), augmented with non-orthogonal strength functions taken from Root (1997) and the Uni-versal Forcefield of Rappé et al. (1992).

Validation

Although the original VALBOND was developed for use with the CHARMM forcefield (Brooks et al., 1983), the table below shows that the quality of the new UFF-VALBOND forcefield is compara-ble to the original, and similar to popular forcefields.

Table 4

Molecule Item Calc Ref Diff ExpRef-Exp

Cal-Exp

Ethane H-C-H 108.7 108.3 0.4 107.5 0.8 1.2Ethane C-C-H 110.2 110.6 -0.4 111.2 -0.6 -1.0Propane C-C-C 112.9 112.0 0.9 112.0 0.0 0.9Propane H-C-H 107.6 107.2 0.4 107.8 -0.6 -0.2Butane C-C-C 112.9 112.0 0.9 113.3 -1.3 -0.4Isobutane C-C-C 111.3 110.7 0.6 110.8 -0.1 0.5Cyclopentane C2-C1-C5 104.4 103.7 0.7 103.0 0.7 1.4Cyclopentane C1-C2-C3 106.0 104.9 1.1 104.2 0.7 1.8Cyclopentane C2-C3-C4 106.8 106.4 0.4 105.9 0.5 0.9Cyclohexane C-C-C 111.9 110.7 1.2 111.4 -0.7 0.5Cyclohexane H-C-H 107.4 107.7 -0.3 107.5 0.2 -0.1Methyl-Cyclo-

hexaneC-C-C(exo) 111.5 110.9 0.6 112.1 -1.2 -0.6

Calc: calculated with UFF-VALBONDRef: taken fromRoot et al. (1993)Exp: experimental values



Norbornane C1-C2-C3 102.5 102.4 0.1 102.7 -0.3 -0.2Norbornane C2-C1-C6 110.6 109.0 1.6 109.0 0.0 1.6Norbornane C1-C6-C4 92.7 91.8 0.9 93.4 -1.6 -0.7Ethylene C=C-H 120.6 120.9 -0.3 121.4 -0.5 -0.8Propene C=C-C 122.9 122.8 0.1 124.3 -1.5 -1.4Propene C=C-H 118.6 118.7 -0.1 121.3 -2.6 -2.7Propene =C-C-H 110.3 110.4 -0.1 116.7 -6.3 -6.4Cis-2-butene C-C=C 125.3 124.9 0.4 125.4 -0.5 -0.1Cyclopentene C3-C2=C1 112.0 111.9 0.1 111.0 0.9 1.0Cyclopentene C2-C3-C4 104.9 104.9 0.0 103.0 1.9 1.9Cyclopentene C3-C4-C5 106.2 106.3 -0.1 104.0 2.3 2.2Cyclohexene C-C=C 123.0 123.0 0.0 124.0 -1.0 -1.0Cyclohexadi-

eneC-C=C 122.8 123.2 -0.4 122.7 0.5 0.1

Cyclohexadi-ene

C-C-C 114.4 113.6 0.8 113.3 0.3 1.1

Norbornene C1-C7-C4 91.8 93.4 -1.6 95.3 -1.9 -3.5Norbornene C2-C2=C3 106.2 106.7 -0.5 107.7 -1.0 -1.5Methanol O-C-H(trans) 111.6 112.8 -1.2 107.2 5.6 4.4Methanol H-C-H 106.3 105.4 0.9 108.5 -3.1 -2.21,4-Dioxane C-C-O 112.8 112.5 0.3 109.2 3.3 3.61,4-Dioxane C-O-C 114.6 111.7 2.9 112.6 -0.9 2.0Formaldehyde O-C-H 122.3 122.5 -0.2 121.8 0.7 0.5Formaldehyde H-C-H 115.4 114.9 0.5 116.5 -1.6 -0.8Acetaldehyde C-C-H 115.5 115.4 0.1 113.9 1.5 1.6Acetone C-C-C 117.1 116.0 1.1 116.0 0.0 1.1Acetone C-C=O 121.4 122.0 -0.6 122.0 0.0 -0.6Methyl-For-

mateO-C=O 123.8 125.9 -2.1 125.9 0.0 -2.1

Methyl-For-mate

C-O-C 115.0 111.9 3.1 114.8 -2.9 0.2

Acetic Acid O-C=O 116.8 117.6 -0.8 126.6 -9.0 -9.8Acetic Acid O-C-O 117.3 117.5 -0.2 110.6 6.9 6.7

Table 4


Cal-Exp



2. Forcefields

Acetic Acid O-C=O 126.0 124.6 1.4 123.0 1.6 3.0Methyl Ace-

tateC-O-C 116.1 115.1 1.0 114.8 0.3 1.3

Piperazine C-C-N 110.9 110.8 0.1 109.8 1.0 1.1Piperazine C-N-C 115.6 113.8 1.8 112.6 1.2 3.0Nitromethane [N-C-H] 110.2 109.8 0.4 107.2 2.6 3.0Succinamide N-C=O 119.6 121.4 -1.8 122.0 -0.6 -2.4Succinamide C-C-N 116.9 114.0 2.9 116.0 -2.0 0.9Succinamide C-C=O 123.6 124.5 -0.9 122.0 2.5 1.6Acetamide C-C-N 117.6 117.5 0.1 115.1 2.4 2.5BHF2 F-B-F 118.4 118.1 0.3 118.3 -0.2 0.1BHCl2 Cl-B-Cl 121.1 119.6 1.5 119.7 -0.1 1.4BF2NH2 F-B-F 119.6 119.9 -0.3 117.9 2.0 1.7BF2NH2 H-N-H 114.4 115.0 -0.6 116.9 -1.9 -2.5NH3 H-N-H 106.8 106.8 0.0 106.7 0.1 0.1NCl3 Cl-N-Cl 106.5 106.5 0.0 107.1 -0.6 -0.6NHCl2 H-N-Cl 106.8 106.7 0.1 102.0 4.7 4.8NHCl2 Cl-N-Cl 106.1 106.1 0.0 106.0 0.1 0.1NH2Cl H-N-H 107.1 107.1 0.0 106.8 0.3 0.3NH2Cl H-N-Cl 106.4 106.3 0.1 102.0 4.3 4.4N3- N-N-N 179.9 180.0 -0.1 180.0 0.0 -0.1NClO Cl-N-O 114.6 114.2 0.4 113.3 0.9 1.3PH3 H-P-H 93.8 93.8 0.0 93.3 0.5 0.5PCl3 Cl-P-Cl 100.1 100.1 0.0 100.1 0.0 0.0CH3PH2 C-P-H 93.1 97.1 -4.0 96.5 0.6 -3.4CH3PH2 H-P-H 91.3 91.3 0.0 93.4 -2.1 -2.1(CH3)2PH C-P-C 99.7 98.5 1.2 99.7 -1.2 0.0(CH3)2PH C-P-H 91.2 95.6 -4.4 97.0 -1.4 -5.8(CH3)3P C-P-C 97.0 95.6 1.4 98.6 -3.0 -1.6AsH3 H-As-H 91.7 91.7 0.0 92.1 -0.4 -0.4AsF3 F-As-F 96.0 96.0 0.0 96.0 0.0 0.0AsCl3 Cl-As-Cl 98.7 98.7 0.0 98.6 0.1 0.1

Table 4


Cal-Exp




AsBr3 Br-As-Br 99.6 99.6 0.0 99.7 -0.1 -0.1AsI3 I-As-I 100.2 100.2 0.0 100.2 0.0 0.0O3 O-O-O 116.8 116.8 0.0 116.8 0.0 0.0(CH3)2O C-O-C 114.7 111.6 3.1 111.7 -0.1 3.0(CH3)2S C-S-C 100.1 99.3 0.8 98.9 0.4 1.2(CH3)2Se C-Se-C 96.9 96.2 0.7 96.0 0.2 0.9(SiH3)2O Si-O-Si 142.6 142.7 -0.1 144.1 -1.4 -1.5(SiH3)2S Si-S-Si 99.4 99.6 -0.2 97.4 2.2 2.0(SiH3)2Se Si-Se-Si 97.9 98.0 -0.1 96.6 1.4 1.3(GeH3)2O Ge-O-Ge 126.1 126.2 -0.1 126.5 -0.3 -0.4(GeH3)2Se Ge-Se-Ge 93.9 94.1 -0.2 94.6 -0.5 -0.7Si4O4C8H24 Si-O-Si 143.4 143.4 0.0 142.5 0.9 0.9Si4O4C8H24 O-Si-O 112.1 112.4 -0.3 109.0 3.4 3.1Si4O4C8H24 C-Si-C 107.4 107.3 0.1 106.0 1.3 1.4Ge4S6(CF3)4 [S-Ge-S] 114.6 113.8 0.8 113.8 0.0 0.8Ge4S6(CF3)4 [Ge-S-Ge] 97.9 99.9 -2.0 99.9 0.0 -2.0Sn6Ph12 [Sn-Sn-Sn] 112.8 112.4 0.4 112.5 -0.1 0.3Sn6Ph12 C-Sn-C 107.0 105.5 1.5 106.7 -1.2 0.3B3Ph3O3 [B-O-B] 121.8 121.8 0.0 121.7 0.1 0.1B3Ph3O3 [O-B-O] 118.2 118.2 0.0 118.0 0.2 0.2PO4P3O3 O1-P1-O2 114.1 114.1 0.0 115.0 -0.9 -0.9PO4P3O3 O2-P1-O4 104.5 104.5 0.0 103.0 1.5 1.5PO4P3O3 O2-P2-O3 99.4 99.3 0.1 99.0 0.3 0.4PO4P3O3 P2-O2-P1 121.7 122.9 -1.2 124.0 -1.1 -2.3PO4P3O3 P2-O3-P3 128.8 127.3 1.5 128.0 -0.7 0.8Ga2Pyr2Br4 Br-Ga-Br 107.2 107.0 0.2 105.8 1.2 1.4

Table 4


Cal-Exp



2. Forcefields

Applicability

UFF-VALBOND can be used for compounds containing elements from across the periodic table.

Because the angular energy function is based on hybridization considerations, improved results are expected for non-hypervalent complexes for which the molecular shape is not a priori known.

Hypervalent molecules are molecules that contain atoms with more occupied orbitals than there are valence orbitals. Thus, for a nor-mal valent atom of the p-block, only the valence s and p orbitals are occupied, and molecules are hypervalent if the electron count around such an atom exceeds eight.

For transition state elements engaged in covalent bonding, only the valence s and the five d orbitals participate in bonding. For these compounds the molecule is hypervalent if the electron count around any central atoms exceeds 12. Most common transition ele-ment complexes are hypervalent.

For compounds containing hypervalent atoms, the forcefield anal-ysis yields similar results as the Universal Forcefield, provided that the hypervalent atoms are declared as a non-VALBOND cen-ter.

The current version of Cerius2 cannot differentiate between hyper-valent and non-hypervalent species, and the onus is thus on the

Ga2Pyr2Br4 Ga-Ga-Br 114.3 115.1 -0.8 116.3 -1.2 -2.0As3(CH3)6In6(C

H3)6

C-In-C 101.7 101.9 -0.2 99.0 2.9 2.7

As3(CH3)6In6(CH3)6

C-As-C 125.6 124.7 0.9 126.0 -1.3 -0.4

[|x|] 0.68 1.28 1.52

Table 4


Cal-Exp




user to correctly assign VALBOND centers for hypervalent mod-els. By default the UFF-VALBOND forcefield will assign all atoms bonded to two or more atoms to be a VALBOND center. This will give correct results for the majority of organic compounds. Depending on the topology Cerius2 may assign gross hybridiza-tion (see Assigning gross hybridization) to hypervalent atoms, but the energy calculated by OFF may be incorrect.

Assigning VALBOND centers

If the model contains hypervalent atoms, VALBOND centers should be assigned by hand. This may be accomplished as follows:

1. Load UFF_VALBOND1.01 via the Open Force Field card

2. From the Open Force Field card select Energy Expression - Automate Setup

3. Disable the Perform Valbond Initialization option

4. From the Open Force Field card select Typing - VALBOND centers

5. Select atoms from the model window and set the appropriate VALBOND centers.

Note

Assigning gross hybridization

The gross hybridization of an atom is equal to the number of occu-pied orbitals minus one. For example, in a sp3 carbon it is three.

Cerius2 will assign values for the gross hybridization to VAL-BOND centers based on the hybridization of the atoms as stored in the data model.

For p-block elements the gross hybridization is of the form spn. For d-block elements VALBOND assumes a hybridization of the form sdm.

It is possible to assign VALBOND centers with any forcefield loaded. However for the angle energy function to be used, the forcefield must be specifically defined to use the VALBOND bend energy function. Currently only the UFF_VALBOND1.01 uses this function.


2. Forcefields

In certain cases, i.e., some hypervalent atoms, Cerius2 may not be able to assign a gross hybridization to all atoms, or occasionally, the user may wish to override the assigned values. Gross hybrid-izations may be assigned manually as follows:

1. Disable automatic VALBOND initialization as described in Assigning VALBOND centers.

2. From the Open Force Field card select Typing - VALBOND centers.

3. Click on the Gross Hybridizations button.

4. Gross hybridizations may be assigned from the panel that pops up. For each VALBOND center a hybridization of the form spndm may be assigned. The values for n and m need not be inte-gers.

Bond Hybridization

Bond hybridization is the net hybridization of an individual bond connected to a VALBOND center. The hybridization is a function of the nature of the VALBOND center, its gross hybridization, and the number and nature of all the ligands connected to the VAL-BOND center.

For example the gross hybridization of the N atom in ammonia, NH3, equals three (sp3).

The bond hybridization of the N-H bonds is calculated as sp3.47. The gross hybridization of O in water is three, and the bond hybridization of the O-H bond is sp3.87. Thus VALBOND calculates these bonds to have more p character than the C-H bond in say methane, where both gross and bond hybridizations are sp3, exactly.

Bond hybridizations are calculated when VALBOND centers are assigned.

Examples

Standard Minimization ♦ From OFF Setup load UFF_VALBOND1.01 via the Open Force Field card.

♦ From OFF Methods select Minimizer and Run.



Minimization of hyperval-ent compound

♦ From OFF Setup load UFF_VALBOND1.01 via the Open Force Field card.

♦ From the Open Force Field card select Energy Expression - Automate Setup.

♦ Disable the Perform Valbond Initialization option.

♦ From the Open Force Field card select Typing - VALBOND centers.

♦ Select the appropriate atoms from the model window and set as VALBOND centers

OR

set all atoms as VALBOND centers and unset selected atoms.

♦ From OFF Methods select Minimizer and Run.

Dreiding forcefield

General force constants and geometry parameters for the Dreiding forcefield are based on simple hybridization rules rather than on specific combinations of atoms. The Dreiding forcefield does not generate parameters automatically the way that UFF and ESFF do; however, its explicit parameters were derived by a rule-based approach.

Functional form The Dreiding forcefield is a purely diagonal forcefield with har-monic valence terms and a cosine–Fourier expansion torsion term. The umbrella functional form is used for inversions, which are defined according to the Wilson out-of-plane definition (see Table 24). The van der Waals interactions are described by the Len-nard–Jones potential. Electrostatic interactions are described by atomic monopoles and a screened (distance-dependent) Coulom-bic term. Hydrogen bonding is described by an explicit Lennard–Jones 12–10 potential (Mayo et al. 1990).


The Dreiding forcefield has good coverage for organic, biological and main-group inorganic molecules. It is only moderately accu-rate for geometries, conformational energies, intermolecular bind-ing energies, and crystal packing.

Atom types Atom typing in the Dreiding forcefield is straightforward. An atom type is denoted by a name of up to five characters:


2. Forcefields

♦ The first two characters are the elemental symbol (for example, C_ for carbon, Sn for tin).

♦ The third character (if present) represents the hybridization state (for example, 1 = linear, sp1; 2 = trigonal, sp2; 3 = tetrahe-dral, sp3; and R = an sp2 atom involved in resonance).

♦ The fourth character (if present) indicates the number of implicit hydrogen atoms (for example, C_R2 is a resonant car-bon with two implicit hydrogens).

♦ The fifth character (if present) is reserved to indicate other spe-cial characteristics (for example, H_ _ _A denotes a hydrogen atom that is capable of forming a hydrogen bond).

Versions The Dreiding II forcefield is an extension and improvement over the Dreiding I forcefield. DREIDING2.21 is the recommended, up-to-date version of the Dreiding II forcefield.

See the archive directory (archive directory) for older versions of the Dreiding forcefield.


Availability Classical forcefields provided by MSI include AMBER, CHARMm, and CVFF:

♦ The standard AMBER forcefield (Standard AMBER forcefield) has been supplemented (Homans’ carbohydrate forcefield), so as to cover oligosaccharides. This enhanced AMBER forcefield is the version that is run via the Discover program, as available in the Insight•Discover and Insight•Discover_3 modules. (Nonvali-dated versions of AMBER are also available for the Cerius2•Open Force Field module, in the directory named “untested”, see untested directory.)

♦ The CHARMm forcefield (CHARMm forcefield) is run through the CHARMm program as implemented in the QUANTA molecular modeling interface.

♦ The CVFF forcefield (CVFF, consistent valence forcefield) is run via the Discover program, as available in the Insight•Discover, Insight•Discover_3, and Cerius2•Discover modules, and via the Cerius2•Open Force Field module.



Characteristics The parameters of classical forcefields were derived by fitting experimental data sets. They were generally designed for biologi-cal macromolecules, although they have been used or adapted for other classes of models. Since they are relatively old, they are well characterized and many research studies have used them.

AMBER forcefield

The standard AMBER forcefield (Weiner et al. 1984, 1986) is parameterized to small organic constituents of proteins and nucleic acids. Only experimental data were used in parameteriza-tion.

However, AMBER has been widely used not only for proteins and DNA, but also for many other classes of models, such as polymers and small molecules. For the latter classes of models, various authors have added parameters and extended AMBER in other ways to suit their calculations. The AMBER forcefield has also been made specifically applicable to polysaccharides (Homans 1990, and see Homans’ carbohydrate forcefield).

AMBER is used mainly for modeling proteins and nucleic acids. It is generally lower in accuracy and has a limited range of applica-bility. The use of AMBER is recommended mainly for those cus-tomers who are familiar with AMBER and have developed their own AMBER-specific parameters. It generally gives reasonable results for gas-phase model geometries, conformational energies, vibrational frequencies, and solvation free energies.

Within Discover, AMBER cannot automatically replace missing parameters with default or generic parameters (see Automatic assignment of values for missing parameters).

Standard AMBER forcefield

Functional form The AMBER energy expression contains a minimal number of terms. No cross terms are included. The functional forms of the energy terms used by AMBER are given in Eq. 19.


2. Forcefields

Eq. 19

The first three terms in Eq. 19 handle the internal coordinates of bonds, angles, and dihedrals. Term 3 is also used to maintain the correct chirality and tetrahedral nature of sp3 centers in the united-atom representation. (In the united-atom representation, nonpolar hydrogen atoms are not represented explicitly, but are coalesced into the description of the heavy atoms to which they are bonded.) Terms 4 and 5 account for the van der Waals and electrostatic inter-actions.

The final term, 6, is an optional hydrogen-bond term that aug-ments the electrostatic description of the hydrogen bond. This term adds only about 0.5 kcal mol-1 to the hydrogen-bond energy in AMBER, so the bulk of the hydrogen-bond energy still arises from the dipole–dipole interaction of the donor and acceptor groups.

Atom types The atom types in AMBER are quite specific to amino acids and DNA bases. In the original publications, the atom types and charges are defined by means of diagrams of the amino acids and nucleotide bases. In the Insight environment, this information has been placed in a residue library. Descriptions of the atom types, from the original papers defining the AMBER forcefield, are shown in Table 25.

Homans’ carbohydrate forcefield

Extension of AMBER to car-bohydrates

Homans’ forcefield for oligosaccharides (Homans 1990) has been incorporated into the AMBER forcefield available in the Discover program. It uses the same functional form as AMBER and extends its applicability to polysaccharides and glycoproteins.

Parameterization Homans’ approach in developing the carbohydrate forcefield was to combine the parameters for monosaccharides (Ha et al. 1988) with the results of ab initio calculations on model compounds rel-

Epot K2 b b0–( )2

b

∑ Hθ θ θ0–( )2

θ

∑ Vn

2------ 1 nφ φ0–( )cos+[ ]

φ

∑+ +=

ε r*/r( )12 2 r*/r( )6–[ ]∑ qiqj/εi jrij∑ Cij

rij12

-------Dij

rij10

-------–∑+ + +

(1) (2) (3)

(4) (5) (6)



evant to the glycosidic linkage (Wiberg and Murcko 1989), to gen-erate an AMBER-compatible forcefield. The bond, angle, and torsion parameters for each monosaccharide residue were in gen-eral taken directly from Ha et al. (1988). However, certain parame-ters required adjustment and others were added, to account for the glycosidic linkage between contiguous monosaccharide residues. The torsion parameters were adjusted to fit the quantum mechan-ical data (6-31G*) of Wiberg and Murcko (1989) for dimethoxymethane.

In addition, the carbohydrate forcefield utilizes charges and van der Waals parameters derived for monosaccharides by Ha et al. (1988). Since the latter parameters were derived without an explicit hydrogen-bonding term, the carbohydrate forcefield also does not contain hydrogen-bonding parameters.

Homans-specific atom types

To account for the anomeric effect associated with carbohydrates, the linking atoms were defined as different atom types. Table 26 lists these atom types, as well as the types corresponding to the ring atoms of sugars.

CHARMm forcefield

CHARMm, which derives from CHARMM (Chemistry at HAR-vard Macromolecular Mechanics), is a highly flexible molecular mechanics and dynamics program originally developed in the lab-oratory of Dr. Martin Karplus at Harvard University. It was param-eterized on the basis of ab initio energies and geometries of small organic models.

Applicability CHARMm performs well over a broad range of calculations and simulations, including calculation of geometries, interaction and conformation energies, local minima, barriers to rotation, time-dependent dynamic behavior, free energy, and vibrational fre-quencies (Momany & Rone, 1992). CHARMm is designed to give good (but not necessarily “the best”) results for a wide variety of modelled systems, from isolated small molecules to solvated complexes of large biological macromolecules; however, it is not applicable to organometallic complexes.

Functional form CHARMm uses a flexible and comprehensive empirical energy function that is a summation of many individual energy terms. The energy function is based on separable internal coordinate


2. Forcefields

terms and pairwise nonbond interaction terms (Brooks et al. 1983). The total energy is expressed by the following equation:

Eq. 20

where the oop (out-of-plane) angle is defined as an improper tor-sion. A more specific (Brooks et al. 1983) statement of the function is:

Eq. 21

The electrostatic term can be scaled to mimic solvent effects. The van der Waals combination rules and functional form are derived from rare-gas potentials. The function optionally used in CHARMm to calculate the hydrogen bond energy is:

Eq. 22

Hydrogen bond energy is not included as a default energy term. The current CHARMm parameter set has been derived in such a way that hydrogen bond effects are described by the combination of electrostatic and van der Waals forces.

Constraint terms are described under Applying constraints and restraints. The use of user terms to customize the CHARMm force-field is described in the CHARMm documentation.

Epot Ebond Eangle Etorsion Eoop+ + += (internal terms)

Eelec EvdW+ + (external terms)

Econstraint Euser+ + (special)

Epot kb r r0–( )2∑ Kθ θ θ0–( )2∑ kφ kφ nφ( )cos–∑ kχ χ χ0–( )2∑qiqj

4πε0rif------------------∑ Aij

r12-------

Bij

r6------–

sw rij2 ron

2 roff2, ,( )∑ Econstraint Euser

+ + +

+ + + +

=

EA

rAD6---------

B

rA4------

–

φA H– D–( )cosm φAA A– H–( )cosn=

:switch rAD2

ron2

roff2, ,( )

:switch φAHD( )cos2 φon( )cos2 φoff( )cos2, ,( )



CVFF, consistent valence forcefield

The consistent-valence forcefield (CVFF), the original forcefield provided with the Discover program (and now available also in Cerius2•OFF), is a generalized valence forcefield (Dauber-Osguthorpe 1988). Parameters are provided for amino acids, water, and a variety of other functional groups.

The augmented CVFF was developed for materials science appli-cations and is provided with Discover in the Insight 4.0.0 program. It includes additional atom types for aluminosilicates and alumi-nophosphates. (The default CVFF that is included with Insight 4.0.0 and Cerius2•Discover also has a few more atom types than the CVFF included with Insight 95.0, since the former Insight series, as well as Cerius2, is intended for use in materials science.)

CVFF also has the ability to use automatic parameters (Automatic assignment of values for missing parameters) when no explicit param-eters are present. These are noted in the output file from the calcu-lation.

Applicability CVFF was fit to small organic (amides, carboxylic acids, etc.) crys-tals and gas phase structures. It handles peptides, proteins, and a wide range of organic systems. As the default forcefield in Dis-cover, it has been used extensively for many years. It is primarily intended for studies of structures and binding energies, although it predicts vibrational frequencies and conformational energies reasonably well.

Out-of-planes and the two versions of Discover

The CDiscover program has undergone extensive validation tests comparing it with FDiscover. These tests have indicated that the two programs provide exactly the same results for all components of the energy expression with one exception: the out-of-plane energy for the CVFF forcefield.

The out-of-plane energy for the CVFF forcefield is calculated as an improper torsion. An improper torsion views three connected atoms and a central atom as if it were a torsion (see Table 24). There are three possible improper torsions that can be generated for a particular out-of-plane, based on permutations of the connected atoms.

For CVFF, only one of these improper torsions is used. The rules that FDiscover employs to select the particular improper torsion


2. Forcefields

are somewhat arbitrary, and it was not possible to replicate them in CDiscover. However, the changes in energy are very small (on the order of 0.01 kcal mol-1). (A more rigorously defined out-of-plane, the Wilson out-of-plane, is used in the CFF forcefield. This energy term provides exact agreement between CDiscover and FDiscover.)

Functional form

The analytic form of the energy expression used in CVFF is shown in Eq. 23.

Eq. 23

Types of terms and com-putational costs

Terms 1–4 are commonly referred to as the diagonal terms of the valence forcefield and represent the energy of deformation of bond lengths, bond angles, torsion angles, and out-of-plane interactions, respectively.

A Morse potential (Term 1) is used for the bond-stretching term. The Discover program also supports a simple harmonic potential for this term. The Morse form is computationally more expensive than the harmonic form. Since the number of bond interactions is usually negligible relative to the number of nonbond interactions, the additional cost of using the more accurate Morse potential is insignificant, so this is the default option.

(10) (11)

Epot Db 1 eα b b0–( )–

–[ ]

b

∑ Hθ θ θ0–( )2

θ

∑ Hφ 1 s nφ( )cos+[ ]

φ

∑+ +=

Hχχ2

χ

∑ Fbb ′ b b0–( ) b ′ b′0–( )

b ′

∑b

∑ Fθθ′ θ θ0–( ) θ′ θ′0–( )

θ′

∑θ

∑+ + +

Fbθ b b0–( ) θ θ0–( )

θ

∑b

∑ Fφθθ′ φ θ θ0–( ) θ′ θ′0–( )cos

φ

∑ Fχχ′ χχ′

χ′

∑χ

∑+ + +

(1) (2) (3)

(5)(4) (6)

(7) (8) (9)

ε r*/r( )12 2 r*/r( )6–[ ]∑ qiqi/εrij∑+ +



When not to use the Morse term

When the model being simulated is high in energy (caused, for example, by overlapping atoms or a high target temperature), a Morse-style function might allow bonded atoms to drift unrealis-tically far apart (see Figure 2). This would not be desirable unless you were intending to study bond breakage.

Use of crossterms Terms 5–9 are off-diagonal (or cross) terms and represent cou-plings between deformations of internal coordinates. For example, Term 5 describes the coupling between stretching of adjacent bonds.

These terms are required to accurately reproduce experimental vibrational frequencies and, therefore, the dynamic properties of molecules. In some cases, research has also shown them to be important in accounting for structural deformations. However, crossterms can become unstable when the structure is far from a minimum. Therefore, although both Cerius2•OFF and the standa-lone Discover program include crossterms by default when using CVFF, the Insight program and Cerius2•Discover explicitly omit the crossterms by default.

Nonbond interactions Terms 10–11 describe the nonbond interactions. Term 10 represents the van der Waals interactions with a Lennard–Jones function. Term 11 is the Coulombic representation of electrostatic interac-tions. The dielectric constant ε can be made distance dependent (i.e., a function of rij).

Figure 2. Morse vs. harmonic potentialsA: Morse potential for a C–H bond; B: harmonic potential for a C–H bond. The Morse potential allows a bond to stretch to an unrealistic length.

A B


2. Forcefields

In the CVFF forcefield, hydrogen bonds are a natural consequence of the standard van der Waals and electrostatic parameters, and special hydrogen bond functions do not improve the fit of CVFF to experimental data (Hagler 1979a, b).

Additional information on the forcefields and how they can be augmented is contained in the documentation for the individual simulation engines, where the file formats are described.

CVFF atom types

The CVFF forcefield supplied by MSI defines atom types for the 20 commonly occurring amino acids, most hydrocarbons, and many other organic models (Table 29).

It automatically supplies generic parameters when specific param-eters are not found (Automatic assignment of values for missing parameters).

Augmented CVFF The augmented version of CVFF (available in version 4.0.0 of the Insight program) includes nonbond parameters (Born model) for additional atom types (Table 30) that are useful for simulations of silicates, aluminosilicates, clays, and aluminophosphates. These added parameters were derived using Ewald summation for non-bond interactions between the additional atom types.

Partial charges The bond increment section of the .frc file for CVFF has been expanded so that partial charges can be determined whenever the Cerius2•OFF module or the Discover program is able to assign automatic atom types.


Availability Several forcefields are provided by MSI that are specialized for cer-tain uses:

♦ Forcefields optimized for glasses (Glass forcefield), for polyvi-nylidene fluoride (MSXX forcefield for polyvinylidene fluoride), and for zeolites (Zeolite forcefields), as well as forcefields intended for use only in the Cerius2•Morphology module (Forcefields for Cerius2•Morphology module), are accessed through the Cerius2•Open Force Field module.



♦ The PCFF (PCFF forcefield for polymers and other materials), COM-PASS, and augmented CVFF (Augmented CVFF) forcefields, which are run via both Cerius2•OFF and the Discover program, are versions of standard forcefields that have been extended for use with polymers and zeolites, respectively.

Characteristics The specialized forcefields described in the following sections have been developed for the purpose of simulating certain sys-tems or performing limited types of calculationfss well. They should not be used for other purposes, since they were not designed to be accurate outside their limited areas of applicability.

Glass forcefield

The Glass forcefield (Glassff) exists in versions that include two- and three-body nonbond terms (glassff_1.01 and glassff_2.01, respectively) and is used for studying a range of inorganic oxide glasses (and other ionic systems) under periodic boundary condi-tions. The newer, three-body, glass forcefield is recommended and documented here.

The glass forcefield is applicable to systems containing Si, O, Al, Li, Na, K, Mg, Ca, B, and Ti. Predicted properties include structure, radial distribution functions, angular distributions, and short-range order.

The form of interaction and parameterization for this forcefield is based mainly on the work of Soules, Garofalini, and co-workers (Soules 1979, Soules & Varshneya 1981, Garofalini 1984, Tesar & Varshneya 1987, Rosenthal & Garofalini 1987, 1988, Zirl & Garo-falini 1989, Garofalini & Zirl 1990, Kohler & Garofalini 1994).

Functional form All ion pairs are subjected to an interaction containing a repulsive van der Waals term and a screened Coulombic term.

The screeening of the Coulombic potential with the complemen-tary error function represents an approximate Ewald summation in real-space only (Woodcock 1975). The reciprocal-space Ewald sum is omitted, to be able to model a bulk amorphous system with finite computational resources. This also reduces the effects of the imposed periodicity of the simulation model.

The potential equation is:


2. Forcefields

Eq. 24

Where:

Aij A pre-exponential constant specific to each element-pair van der Waals interaction.

rij Interatomic distance.

p 0.29 Å.

C Constant that converts the units to kcal mol-1.

q Charge on each species.

erfc Complementary error function.

βij Depends on species.

The RSL2 water potential can be used as part of the off-diagonal van der Waals term.

If you want to customize the glass forcefield, the parameter A for each element combination ij can be computed approximately, using the formula:

Eq. 25

Where:

qi Formal charge on ion i.

ni Number of valence-shell electrons in ion i.

b 3.38E-20 J.

ri Radius repulsion constant of the element ion i, related to the ionic radius.

ρ 0.29 Å.

For the glass forcefield, A parameters were computed for the full set of ion-pair interactions using quoted values of r. Where explicit values of Aij existed in the literature, these were preferred. This produced a parameter set able to tackle a range of ions:

O2-, Si4+, Al3+, Li+, Na+, K+, Mg2+, Ca2+, Zn2+, Ti4+, B3+

Vij r( ) Aijexp rij p⁄–( ) Cqiqj

εrij---------erfc rij βi j⁄( )+=

Aij 1qi

ni----

qj

nj----+ +

bri rj+

ρ-------------- exp=



Automated setup For models within the scope of the glass forcefield, the default Open Force Field settings included in the file can be used without adjustment. Atom typing and charging also can be done automat-ically.

MSXX forcefield for polyvinylidene fluoride

The MSXX forcefield for polyvinylidene fluoride (PVDF) and related polymers and small organic models (Karasawa & Goddard 1992) is based on a combination of first-principles quantum mechanical calculations. It includes cross terms (for accurate vibrational frequencies), and charges are associated with each atom type.

MXFF is designed for modelling and predicting a wide range of properties of PVDF, including geometry, cell parameters, elastic constants, dielectric constants, mechanical stability, and vibra-tional frequencies.

Parameterization Four aspects were addressed to generate an accurate forcefield for PVDF: charges, van der Waals interactions, torsion terms, and valence terms.

Partial atomic charges for the various atoms were obtained using the PS-Q program to calculate potential-derived charges from the Hartree–Fock wavefunction for CF3–CH2–CF2–CH3.

For carbon and hydrogen, van der Waals parameters that were determined from fitting lattice parameters, elastic constants, lattice phonons, and cohesive energies of crystalline polyethylene and crystalline graphite were used. Fluorine parameters were derived for CF4 and crystalline polytetrafluoroethelene. Hartree–Fock cal-culations were used to obtain the torsion potential curve for CF3–CH2–CF2–CH3.

The valence parameters were optimized using Hessian-biased forcefield methodology and the vibrational frequencies of the form I crystal. Van der Waals parameters and charges were fixed during optimization of the valence parameters.

Note


2. Forcefields

See also documentation of PCFF (PCFF forcefield for polymers and other materials) and COMPASS.

Zeolite forcefields

See also documentation on the augmented CVFF (Augmented CVFF).

bks1.01 The BKS forcefield was developed by van Beest et al. (1990) to describe the geometries, vibrational frequencies, and mechanical properties of silicas and aluminophosphates. The parametrization is based on both ab initio and experimental data. The forcefield contains four atom types: Si, O, Al, and P.

Interatomic interactions are considered to be ionic (nonbond) rather than covalent. The van der Waals interactions are described with a Buckingham potential, and the electrostatic interactions are described by atomic monopoles and a Coulombic term.

burchart1.01 The Burchart forcefield was developed by Burchart (1992) to describe the geometry, heats of formation, transitions under pres-sure, crystal morphology, and vibrational frequencies of silicas and aluminophosphates. The parametrization is based mainly on experimental data and includes both valence and nonbond terms. It contains four atoms types: Si, O, Al, P.

This forcefield treats the zeolite framework as largely covalent. Bond-stretching is described by the Morse term, 1–3 interactions by the Urey–Bradley term, van der Waals interactions by an expo-nential-6 term, and electrostatic interactions by partial atomic charges and a screened Coulombic term.

The Cerius2 implementation of the Burchart forcefield differs from the published version in three respects:

The MSXX forcefield uses the Morse functional form for bond stretching, which means that the force goes to zero at large bond distances (see Figure 2). Therefore this forcefield should not be used when starting a run with bad initial geometries. Instead, use an alternative forcefield (e.g., the Universal forcefield) when starting from a bad geometry and then switch to the MSXX when close to the energy minimum.



♦ The Cerius2 implementation of the forcefield ignores the Cou-lombic interaction if the atom pair interacts via a bond interac-tion, but Burchart allows both bond and Coulombic interactions for a pair of atoms.

♦ The Cerius2 implementation of the forcefield does not allow Coulombic or van der Waals interactions between two atoms when they interact via a Urey–Bradley interaction, but Burchart does.

♦ The Cerius2 implementation applies a 0–15 Å spline to both types of nonbond interaction, but Burchart applies a 0–15 Å spline function to the Coulombic term and not to the van der Waals term.

burchart1.01-DREIDING2.21

The Burchart–Dreiding forcefield combines the Burchart and Dre-iding II forcefields. It is useful for the study of properties of (mainly) organic molecules inside silica/aluminophosphate frameworks. There are four distinct types of interaction to con-sider:

♦ Framework—All interactions within the silica/aluminophos-phate framework.

♦ Intramolecular—The interactions within each molecule.

♦ Intermolecular—The interactions between molecules.

♦ Framework–molecule—All nonbond interactions between the framework and the molecules.

The Burchart forcefield treats the framework, and the Dreiding II forcefield treats the intra- and inter-molecular interactions The parameters for the framework–molecule interactions are derived from parameters from both forcefields, combined by the arith-metic combination rule (Combination rules for van der Waals terms).

burchart1.01-UNIVERSAL1.02

The Burchart–Universal forcefield combines the Burchart and Uni-versal forcefields. Similar to the Burchart–Dreiding forcefield described above, the Burchart forcefield treats the framework; the Universal forcefield treats the intra- and inter-molecular interac-tions; and the parameters for the framework–molecule interac-tions are derived from parameters from both forcefields, combined by the geometric combination rule.


2. Forcefields

Forcefields for sorption on zeolites

Several forcefields have been implemented especially for studies of sorption of rigid small molecules onto zeolite structures, using the Cerius2•Sorption module. They can be used in studies of bind-ing sites, interaction energies, Henry’s constants, adsorption iso-therms, and relative selectivity.

sor_yashonath1.01 The parameters for C, H, O, and Na in the Sorption Yashonath forcefield are taken from Yashonath et al. (1988). Mainly for satu-rated hydrocarbons in zeolites

sor_demontis1.01 The parameters for C, H, O, and Na in the Sorption Demontis forcefield are taken from Demontis et al. (1989). Mainly for ben-zene in zeolites.

sor_pickett1.01 The parameters for O and Xe in the Sorption Pickett forcefield are taken from Pickett et al. (1990). Mainly for xenon in zeolites.

watanabe-austin1.01 The Watanabe–Austin forcefield (Watanabe et al. 1995) was fit to experimental adsorption isotherms (Miller et al. 1987) for argon, oxygen, and nitrogen adsorption in zeolite types A, X, and Y (alu-monosilicates with Ca+2, Na+, and Li+ counterions). It contains parameters for argon, oxygen and nitrogen sorbates, Ca, Na, K, and Li cations, and zeolite framework atoms (Si, Al, O).

van der Waals interactions are described by a Lennard–Jones potential. Electrostatic interactions are described by off-center monopoles and a Coulombic term. The change in framework oxy-gen polarizabilities as the aluminium content increases was taken into account during parameterization of the forcefield.

Forcefields for Cerius2•Morphology module

These forcefields include only nonbond interaction terms and are intended only for determining crystal morphology in the Cerius2•Morphology module.

morph_lifson1.11 In the Morphology/Lifson forcefield (Lifson et al. 1979), you need to assign the charges on the C, C_O, and H_N. Charges should be assigned by assuming electroneutrality of the CH3, CH2, CH, amide, CO, NH, NH2, and COOH groups. This forcefield is recom-mended for studies of the morphology of crystals or carboxylic



acids and amides and is limited to regid-body calculations involv-ing C, H, N, and O.

The van der Waals term is Lennard–Jones; electrostatic interactions are described by partial atomic charges and a Coulombic term; and hydrogen bonding is described by an explicit Lennard–Jones 10–12 term.

morph_williams1.01 The Morphology/Williams (Williams 1966) forcefield is applicable only to hydrocarbon crystal morphology, because it contains only parameters for carbon and hydrogen.

morph_momany1.1 morph_scheraga1.1

The Morphology/Momany and Morphology/Scheraga force-fields (Momany et al. 1974, Némethy et al. 1983) were developed for polypeptides and are suited for predicting the packing config-urations and lattice energies in crystals of hydrocarbons, carboxy-lic acids, amines, and amides (also amino acids and polypeptides for Morphology/Scheraga). They are limited to rigid-body calcu-lations.

The van der Waals term is Lennard–Jones; electrostatic interactions are described by atomic monopoles and a screened Coulombic term; and hydrogen bonding is described by an explicit 10–12 term.

The Morphology/Scheraga forcefield contains an updated version of the Momany parameter set. Because these parameter sets con-tain parameters only for van der Waals energy calculations, they are appropriate for use within the Morphology module but not in other applications such as energy minimization.


Cerius2 Several old and nonvalidated forcefields for the Cerius2•Open Force Field module are included in the untested and archive sub-directories of the forcefield directory (Cerius2-Resources/FORCE-FIELD directory). These forcefields are documented below.

Another forcefield, CHEAT95, which is an addendum to the CHARMm forcefield, for polysaccharides, is available free at MSI’s website (http://www.msi.com) and is not documented here.


2. Forcefields

Insight II Several of old, nonvalidated, unsupported forcefields for some Insight modules are included in subdirectories of the $BIOSYM/gifts directory. These forcefields are documented (minimally) by means of README files in those directories.

In Insight 4.0.0, several additional unsupported forcefields are present in an archive subdirectory in the $BIOSYM_LIBRARY directory. Additional information is available through the Force-field/FF_Info parameter block, which is accessed through the Builder and other modules (but which is not included in Insight 97.0).

archive directory The Cerius2 archive directory contains copies of forcefield files that were previously released with CERIUS, as well as a copy of the Dreiding forcefield previously released with POLYGRAF. Newer versions of most of these forcefields are available in the top-level forcefield directory, as discussed above. These archived forcefields should be used only when necessary for compatibility with work carried out using CERIUS 3.0–3.2 or POLYGRAF.

The “CFF93” forcefield that is included in Cerius2•OFF is actually CFF89 (see CFF91, PCFF, CFF —consistent forcefields), which was parameterized only for the alkyl functional group and alkane models (Hwang et al. 1994).

Most of these are CERIUS forcefields and produce the same results when used in Cerius2 as in CERIUS 3.2. However, GRAFDREIDING1.00 is the Dreiding II forcefield from POLYGRAF3.2.1.

Notes on the archive/GRAFDREIDING1.00 force-field

The GRAFDREIDING1.00 forcefield is the direct result of a format conversion (via the pf converter, as discussed in the documenta-tion for C2•OFF) of the dreidii321.par forcefield of POLYGRAF3.2.1. Nevertheless, discrepancies may result between the energies calculated in Cerius2 and in POLYGRAF.

This is because only the first-found inversions and torsions con-tribute to the respective energy terms, and these are not necessar-ily picked in the same order by Cerius2 and POLYGRAF. However, you will obtain identical energies if the options to find all torsions and all inversions are chosen in Cerius2 and POLYGRAF (see Scaled torsion terms).

This version of the Dreiding II forcefield also differs from both the published one (Mayo et al. 1990) and the other Dreiding forcefield



files in Cerius2 in that this version uses the geometric combination rule for nonbond interactions (see Combination rules for van der Waals terms).

We strongly recommend that you use the UNIVERSAL1.02 or DREIDING2.21 forcefields in preference to the GRAFDREIDING1.00 field, unless you want to match the results of POLYGRAF calculations.

Note

untested directory The untested directory contains several forcefields that have not been validated by Molecular Simulations. Some are incomplete and intended only as examples of how to incorporate parameters from other forcefields. We supply these forcefields as a conve-nience to our customers, but do not support them.

♦ amber1.01 — This forcefield file is based on the AMBER (Weiner et al. 1984, 1986) forcefield developed for the simula-tion of proteins and nucleic acids. Its atom types are limited to those needed for these structures: H, C, O, S, N, and P.

♦ amber2.01 — This forcefield is an extension of the amber1.01 forcefield. In addition to the atom types of amber1.01, it con-tains atom types for several biologically important cations: Br, Na, Cl, and Ca.

♦ mm2_77_1.01 and mm2_85_1.01 — These forcefields are based on the MM2 and MMP2 forcefields developed by Allinger ’s group (Kao & Allinger 1977, Liljefors et al. 1987, Sprague et al. 1987). They are particularly suitable for small organic models. The mm2_77 forcefield applies only to saturated hydrocarbons, and the mm2_85 forcefield is a significant extension of the MM2 and MMP1 forcefields. The mm2_85 parameters are compatible with those of mm2_77, but calculations on conjugated systems can be done with the mm2_85 forcefield. The mm2_85 force-field also has more atom types for a much larger range of ele-ments.

When using the GRAFDREIDING1.00 forcefield, if you load in a Biodesign (bgf) format structure file that already has the correct atom types assigned (for example, a structure that was atom typed using the default Dreiding forcefield rules and saved in a POLYGRAF Biodesign file with its default atom types), do not atom type the structure again.


2. Forcefields

♦ The “CFF93” forcefield for hydrocarbons (cff93_1.01), although not in the untested directory, is no longer supported, since it has been superceded by the CFF91 and CFF forcefields, which should be used instead.


2. Forcefields


3 Preparing the Energy Expression and the Model

Many forcefields allow you a great deal of flexibility with respect to how atom types are assigned to atoms, which terms of the energy function are used, and how the simulation engine applies the forcefield. You can also perform computational experiments by using alternative functional terms and applying constraints and restraints to your model.


Although largely automated model preparation and energy expression setup is possible for simple systems, you should read this chapter if you want to perform the best simulations possible. You need to read this chapter if you want or need to know about:

♦ The general procedures for using and selecting forcefields.

♦ Using nondefault atom typing.

♦ Customizing an energy expression’s parameters.

♦ Using nondefault functional forms of the energy expression (e.g., to avoid difficulties in converging).

♦ Applying constraints or restraints to your model to perform computational experiments.

♦ Working with periodic systems.

♦ Working with large models.

This chapter explains Using forcefields

Selecting forcefields

Assigning forcefield atom types and charges

Parameter assignment

Using alternative forms of energy terms


3. Preparing the Energy Expression and the Model

Applying constraints and restraints

Modeling periodic systems

Handling nonbond interactions

Related information in this book

Forcefields presents the functional forms of energy expressions and describes the forcefields that are available in Molecular Simula-tions products.

The atom types defined for each forcefield are listed under Force-field Terms and Atom Types.

The files that specify the forcefields are described in detail in sep-arate documentation.

Table 5. Finding information in Preparing the Energy Expression and the Model section


Charge assignment. Assigning charges.Missing parameters. Automatic assignment of values for missing parameters.Custom parameters. Manual parameter assignment.Editing forcefields. Editing a forcefield.Hydrogen bond terms. Hydrogen bonds and hydrogen-bond terms.Criteria for defining hydrogen bonds. Hydrogen bonds and hydrogen-bond terms.Optimizing or simulating only part of a

model.Applying constraints and restraints.

Reducing computational costs. Applying constraints and restraints.Forcing a model towards a desired

conformation.Applying constraints and restraints.

Images of models in periodic systems. Modeling periodic systems.Relation between Cartesian and crys-

tal axes.Figure 7. Relationship between Cartesian coordinate sys-

tem (xyz) and periodic system (abc) in Discover and CHARMm.

Solvent molecules. Minimum-image model; Explicit-image model.Combination rules for van der Waals

terms.Combination rules for van der Waals terms.

Dielectric constants. The dielectric constant and the Coulombic term.Neighbor lists. Neighbor lists and buffer widths.

Using forcefields


Specific information For specific information on procedures, please see the manual for the molecular modeling program and/or simulation engine that you are using (see Available documentation):

Using forcefields

Graphic interface mode All MSI’s simulation engines and forcefields can be used through at least one graphical molecular modeling interface (Cerius2, Insight II, QUANTA, see Table 1).

Standalone mode The Discover and CHARMm programs can also be run in a com-mand-based, standalone mode with input from a text interface and/or a script and other input files.

Mixed-mode use For Discover and CHARMm, you can optionally perform some tasks through the appropriate graphical interface and others in standalone mode. For example, you might want to prepare model structure and command input files with the graphical interface, save both types of files, edit the command input file with a text edi-tor so as to perform some complex simulation, start the run in stan-dalone mode, and then analyze the results with the aid of one of the graphical interfaces. In addition, facilities exist for directly entering specific BTCL or CHARMm commands from the Insight or QUANTA interface and for reading a BTCL or CHARMm com-mand input file into the Insight, Cerius2, or QUANTA interface.

Additional information How to run Discover and CHARMm in standalone mode is docu-mented separately (see Available documentation).

General procedure Regardless of whether simulation engines are run through the graphical interface, in standalone mode, or in some combination of both modes, the general sequence of activities for doing forcefield-based calculations is as follows.

Charge groups, functional groups. Charge groups and group-based cutoffs.Cell multipoles, nonperiodic systems. Cell multipole method.Ewald sums, periodic systems. Ewald sums for periodic systems.

Table 5. Finding information in Preparing the Energy Expression and the Model section




Important

1. Read in the forcefield—Based on the type of model and the sci-entific problem that you want to simulate, decide which force-field to use (see Forcefields). If it is not the default forcefield for the molecular modeling program you are using, you need to specify the desired forcefield.

The forcefield parameter files (which contain parameters that specify force constants, equilibrium geometries, van der Waals radii, and other data needed for calculating energies) are read in as part of this process.

2. Prepare the model:

a. If necessary, read in monomer/residue definitions—In most cases, the molecular modeling program automatically does this for you, or (depending on the type of model) it is not necessary.

Information about residues or monomers, the basic chemical units that comprise many models, is stored in topology (or library, dictionary, or template) files. The atoms, atomic properties, bonds, bond angles, torsion angles, improper torsion angles, hydrogen bond donors, acceptors, and ante-cedents, nonbond exclusions, and charge increments are all specified on a per residue basis.

If this information is not included in a structure or model file you intend to read in (for example, a Brookhaven Protein Database file), then you may have to specify the appropriate topology file.

b. Read in or construct a model—Read in a model from an appropriate file or construct it using the builder functional-ity in the molecular modeling program. Make sure your final model is correct with respect to atom connectivity, hybridization, bond orders, valences, etc.

c. Assign forcefield atom types and charges to each atom in your model (see also Assigning forcefield atom types and charges). If you are using UFF, calculate fractional bond orders. These are largely automated processes.

For specific instructions, see the documentation for the appropriate simulation engine and/or molecular modeling program (see Available documentation).

Using forcefields


d. If necessary, define charge groups—The need for charge groups depends on the type of model and the scientific problem that you want to simulate. Some calculations are impossible without this kind of information, others can be significantly speeded up by supplying it (see also Charge groups and group-based cutoffs).

e. Read in or generate Cartesian coordinates—In most cases, the molecular modeling program automatically does this for you when you save a model that you have built or read in from a non-MSI type of file.

3. Set up the energy expression—You may want to use alternative terms in the energy expression, use or avoid using certain default terms, specify how nonbond interactions are handled, apply constraints or restraints (biases) to your model, etc. (see also this chapter).

4. Set up the calculation—Unless you want your calculation to run under the default conditions, you need to specify items such as which minimization algorithm(s) and what termination criteria to use (see also Minimization, Molecular Dynamics, and Free Energy).

Most programs also allow you to control your calculation by specifying various nondefault conditions. You may want to use a more robust minimizer for a highly strained model, then switch to a more accurate one for the final stages of computa-tion. Some forcefield engines allow sophisticated if-tests and conditional branching. You can determine what will happen if certain parameters are not found (see also Parameter assign-ment). You can also send jobs that are time consuming to some other (faster) computer or run them in the background.

5. Specify output—If desired, specify nondefault kinds or amounts of output.

6. Run the calculation.

7. Analyze the results—The molecular modeling programs pro-vide analysis functionality that allows you to view your results in the form of graphs and tables, as well as by graphicallly dis-playing the final (and/or intermediate) conformations of the model.



Selecting forcefields

For details on how to select a forcefield in the molecular modeling program that you are using, please see the appropriate specific documentation (see Available documentation). A brief summary:

♦ Cerius2•Discover—Use the Run Discover control panel (which is accessed by selecting Run on the DISCOVER menu card) or the Select Discover Forcefield control panel (accessed by select-ing Forcefield/Select on the DISCOVER card).

♦ Cerius2•OFF—Use the Load Force Field control panel, which is accessed by selecting Load on the OPEN FORCE FIELD menu card.

♦ Cerius2•MMFF, go to the MMFF card deck and select Run to access the MMFF Energy Minimization control panel (check the Use MMFF For Energetic Calculations check box if you want MMFF93 to be used for calculations in other relevant modules).

♦ Insight II—Use the Forcefield/Select parameter block. This parameter block is found in several modules, including the Builder.

♦ QUANTA—Use the CHARMm/CHARMm Mode menu item on the main QUANTA menu bar.


Who should read this sec-tion

You should read this section if you want to understand what atom types are, how types and charges are assigned automatically, and how you can make your own atom type or charge assignments.

In addition, if you want to understand parameter assignment (Determination of which parameters are used with which atom types) and/or edit the forcefield parameters (Manual parameter assign-ment), you need to understand something about atom type assign-ment first.

Availability All molecular modeling programs supplied by MSI perform auto-matic and/or semi-automatic atom-type and charge assignment



(which needs to be re-done if you switch to a different forcefield). Please see the guidebook for the appropriate molecular modeling program for details on how to assign atom types and charges for the simulation engine you are using (see Available documentation).

What are atom types in forcefields?

The simulation engine needs the forcefield atom type of each atom in the model in order to determine which forcefield parameters to use. Forcefield parameters apply to particular combinations of atom types as specified by the forcefield.

Relation between force-field atom types and chemical atoms

The forcefield atom types are related to the microchemical envi-ronment of the atoms in a way defined by the particular forcefield. For example, a methane model has only two atom types, one for the carbon and one for the hydrogens, even though each of the atoms may have a distinct atom name for labeling purposes. The hydrogen atoms are equivalent by symmetry; therefore, they would all have the same atom type in any forcefield.

As a more complicated example, consider propane, which has four distinct types of atoms: methyl carbon atoms, methyl hydrogen atoms, a methylene carbon atom, and the methylene hydrogens. In principle, a forcefield could consider these to be four distinct atom types, but in practice, the chemical difference between the carbon atoms or between the hydrogen atoms is very small, so in most forcefields the carbon atoms are all assigned the same atom type, and all the hydrogens are assigned a second atom type.

Assigning atom types to a model

Atom types and charges supplied by the structure file

Atom types are assigned by the simulation engine or the molecular modeling program. Atom types are automatically assigned by using a set of rules that link the type of an atom to its element type and its chemical microenvironment (for example, the number and nature of connected atoms). Different forcefields use different atom types and atom-typing rules, which are contained in a resi-due library or the forcefield file.

The atom type information can also be supplied by a molecular data file such as an .msi file (OFF), an .mdf file (Discover and OFF), or an RTF (or PSF) file (CHARMm). These structure files are typi-



cally created in the Cerius2, Insight, or QUANTA molecular mod-eling program.

To make sure that atom types are assigned:

♦ Cerius2 warns you when you try to perform some action for which atom types need to have been assigned, if they have not been assigned.

♦ Insight checks whether types (and charges) have been assigned when you exit the model-building module and gives you an opportunity to assign them.

♦ QUANTA takes care of assigning the atom types when the model is saved in a structure file.

Charge information also is saved in the structure file.

To assure that you use the most appropriate atom types in your studies, you should always check the assigned atom types against the appropriate table under Forcefield Terms and Atom Types. In most cases, Cerius2 and Insight automatically assign the atom types. However, these assignment engines of course require the models to be correctly built. One of the most critical types of infor-mation is the bond order, which should be set before the forcefield is assigned.

Atom typing in different modeling programs

In Cerius2 and Insight, atoms with unassigned atom types are labelled with question marks when you label the model according to the atom type (FFTYPE or potential type).

♦ In Cerius2•Discover, use the Discover Atom Typing control panel (accessed by selecting the Forcefield/Typing item on the DISCOVER card) to assign all atom types and charges (if you do not want to do this automatically). You can also select indi-vidual atoms and manually assign an atom type different from the one assigned automatically.

♦ In Cerius2•OFF, use the Force Field Atom Typing control panel (accessed by going to the OFF SETUP deck of cards and select-ing the Typing/Atoms card menu item) to assign all atom types. Charges are also assigned if the forcefield being used contains charge information. You can also select individual atoms and manually assign an atom type different from the one assigned automatically.



♦ In Insight II, assignment of potential types (and charges) to each atom is done with the Forcefield/Potentials parameter block, which appears automatically when appropriate or can be accessed from the Biopolymer, Builder, and other modules of the Insight program. You can re-type individual atoms by using the Atom/Potential parameter block in the Biopolymer or Builder module.

♦ QUANTA automatically assigns atom types when you con-struct or modify a model. Library (or “dictionary”) files for commonly used chemical units (amino acids, nucleic acids, etc.) are supplied with CHARMm. You can also manually assign atom types different from what were assigned automatically, by using the Molecular Editor (accessed from Applications/Build-ers/3-D Builder on the main QUANTA menu bar).

Important

Important

Assigning charges

Charges (when available) are generally assigned at the same time as the forcefield atom type (see Assigning atom types to a model).

An atom-type charge is simply a fixed value associated with the atom types. Overall neutrality of a model is not necessarily achieved by assigning forcefield atom types. You may prefer to assign charges specifically. (The exact method depends on the molecular modeling program being used.)

Importance of correct charge assignment

Electrostatic interactions play a critical role in determining the structures of inorganic systems and in defining the packing of organic molecules.

A newly assigned atom type (including associated parameters such as mass and charge) replaces any previously assigned or calculated value.

Currently, the automatic atom-typing engines cannot distinguish a metal atom from a metallic ion. Hence, by default, “metal” atom types are assigned by the automatic typing engines. So for metal ions, you need to assign the formal charges and atom types by hand.



Many forcefields already include charges

Forcefields that include Coulombic terms generally already include standard charges (or “bond increments”) associated with the atom types. These forcefields have been parameterized with nonzero atom type charges or charge increments (Table 6) and therefore you usually just assign charges automatically when you do atom typing, instead of having to assign specific charges:

aCHARMm as run through the Cerius2•MMFF module, not in QUANTA or stan-dard CHARMm.

Important

Finding charges, if needed

If you need to specifically assign charges, most relevant modules allow you to set atomic charges directly or specify an overall net charge for the whole structure using charge editing functions.

For small models, you can obtain values for charges by using an ab initio or semiempirical quantum chemistry module (for example, MOPAC).

Table 6. Forcefields parameterized with nonzero atom charges or bond increments

Forcefield Engine

CFF91–95, CFF, PCFF OFF, Dis-cover

COMPASS OFF, Dis-cover

CVFF OFF, Dis-cover

bks1.01 OFFburchart1.01 OFFburchart1.01-DREIDING2.21 (not all atom types) OFFburchart1.01-UNIVERSAL1.02 (Burchart atom types only) OFFglassff_1.01 OFFMMFF93 CHARMma

msxx_1.01 OFFCHARMm CHARMm

If you want to assign charges different from those in the forcefield, you need to assign the charges after atoms typing (and automatic charge assignment) is done.



For larger, distorted, models and when charge assignment is done by the charge equilibration method (in Cerius2), you usually need to perform a short minimization before assigning charges. This is because charge equilibration calculations on distorted models can lead to assignment of unrealistic charges.

Charge assignment in dif-ferent modeling programs

♦ In Cerius2•Discover, charges are automatically assigned when atoms are typed.

♦ In Cerius2•OFF, if you are using UFF or the Dreiding forcefield, charges should be assigned to the model by using the Charges module, accessed from the OFF SETUP deck of cards (see Cerius2 Forcefield Engines: OFF). The Charges module uses the charge equilibration approach developed by Rappé & Goddard (1991) to predict the charges from the model geometry and the atomic electronegativities. If the model geometry changes much during minimization, you should iterate the procedure of reassigning charges and reminimizing until the energy reaches a constant. The Charges control panel also allows you to edit or assign charges manually.

♦ In Insight II for all forcefields except CFF, assignment of charges (and atom types) to each atom is done with the Forcefield/Potentials parameter block, which appears automatically when appropriate or can be accessed from the Biopolymer, Builder, and other modules. Potential function atom types must be (and are) assigned before charges or partial charges are assigned. The Insight program assigns atom types and partial charges to each atom in the structure based on information in a residue library file or (if not found in a residue library file) on the bond increments found in the forcefield file. You can edit the charges on individual atoms with the Atom/Charge parameter block in the Biopolymer or Builder module.

♦ QUANTA automatically assigns charges when you construct or modify a model. Library (or “dictionary”) files for commonly used chemical units (amino acids, nucleic acids, etc.) are sup-plied with CHARMm. You can also manually assign charges different from what were assigned automatically, by using the Molecular Editor (accessed from Applications/Builders/3-D Builder on the main QUANTA menu bar).





If you are a novice user or routinely run relatively simple calcula-tions on relatively simple or standard models, you do not need to read this section. However, if, for example, an error message informs you of missing parameters or you want to customize your energy expression for an atypical model, then you do need to understand how the simulation engines determine what parame-ters are used for which atoms, bonds, angles, etc.

You should also understand something about atom type and charge assignment (Assigning forcefield atom types and charges) to make effective use of this section.

Determination of which parameters are used with whichatom types

Before calculating the energy of a model, the simulation engine must construct the complete energy expression for the model by associating the correct forcefield parameters with the appropriate atoms and other coordinates. For example, methane has one type of bond (C–H) and one type of bond angle (H–C–H). The program must create a list of the four actual bonds and then associate the C–H bond parameters with each. Similarly, there are six H–C–H angles, but they are characterized by the same set of parameters.

It is important to understand how the parameters from the force-field are associated with individual internal coordinates, because the energy, derivatives, structures, and almost all other properties calculated by the program depend on these forcefield parameters and the way in which they are associated with the internal coordi-nates. The following sections describe two facets of this process: atom type equivalences and wildcards in parameter definitions.

Atom type equivalences

Chemically distinct atoms often differ in some, but not all, of their forcefield parameters. For example, the bond parameters for the C–C bonds in ethene and in benzene are quite different, but the nonbond parameters for the carbon atoms are essentially the same.



In Discover, rather than duplicating the nonbond parameters in the forcefield parameter file, atom type equivalences are used to simplify the problem. In the example, the phenyl carbon atom type is equivalent to the pure sp2 carbons of ethene insofar as the non-bond parameters are concerned.

The Discover program recognizes five types of equivalences for each atom type: nonbond, bond, angle, torsion, and out-of-plane. Crossterms such as bond–bond terms have the same equivalences (insofar as atom types are concerned) as the diagonal term of the topology of all the atoms defining the internal coordinates. For the bond–bond term, this means that the atom type equivalences for angles would be used.

The actual format of the equivalence data in the forcefield param-eter file is detailed in the File Formats documentation. For the equivalences used in any particular forcefield, you should exam-ine the actual forcefield parameter file for current information.

CHARMm PRM files handle equivalences for nonbond parame-ters by using partial wildcards, for example, N* means that the associated nonbond parameters apply to any nitrogen type that is not specifically parameterized.

For forcefields in Cerius2•OFF, wildcards are usually used.

Wildcard atom types in the parameter file

For some internal coordinates, the parameters do not depend strongly on the specific atom types of one or more atoms. For example, the parameters of torsion terms may not be strongly affected by the end atoms. This means that the torsion parameters are essentially defined by the central bond rather than its substitu-ents.

The forcefield engines allow wildcard atom types to conveniently handle this type of situation. This special atom type, indicated by an X in CHARMm .PRM files and in relevant Cerius2 forcefield files and by an asterisk (*) in Discover forcefield files, matches any atom type when the forcefield engine is searching for the parame-ters to associate with a particular internal coordinate. (In CHARMm, this applies only to bond, angle, torsion, and improper-torsion parameters.)



Automatic assignment of values for missing parameters

Availability

What happens if parame-ters are not found

Some classic and second-generation forcefields are not completely parameterized for all their atom types. (For rules-based forcefields (Rule-based forcefields broadly applicable to the periodic table), all parameters are generated according to rules rather than read from the forcefield file.) When parameters for classic and second-gener-ation forcefields are not available, one of several things can hap-pen, with varying consequences:

♦ The missing parameters are simply ignored (i.e., set to zero in the energy expression). The simulation runs and yields results, but they may be very inaccurate.

♦ Setup of the energy expression is interrupted, the simulation run is not started, and a message is output to the textport or text window.

♦ Missing parameters are obtained automatically from a simpler generic set of parameters (using many wildcards, see above). The results may be reasonable, but not as accurate as if specific parameters existed.

Temporary patches for missing parameters; pre-cautions

A forcefield may include automatic parameters for use when bet-ter-quality explicit parameters are not defined for a particular bond, angle, torsion, or out-of-plane interaction. These parameters are intended as temporary patches, to allow you to begin calcula-tions immediately. While MSI has made every effort to ensure that the automatic parameters used in CVFF, the CFF family of force-fields, and CHARMm produce reasonable geometries for a wide

Table 7. Automatic parameter assignment in MSI’s molecular modeling programs

Modeling program Automatic parameters Comments

Cerius2 yes not for all forcefieldsInsight II (CDiscover) yes but not for AMBER forcefieldinsight II (FDiscover) yes controllable in standalone mode only, not for AMBER QUANTA yes but only if the PSF Generator is used



variety of models, we cannot guarantee that the automatic param-eters are appropriate in every instance. You therefore should always carefully evaluate any results that you obtain using auto-matic parameters.

How missing parameters are supplied

Discover automatically assigns values for parameters missing from the CFF and CVFF forcefields by switching to an automatic forcefield. This switching is accomplished with an equivalence table that converts the original set of atom types to a smaller set of generic atom types.

Cerius2•OFF behaves similarly.

QUANTA’s parameter chooser looks through the existing CHARMm parameters for similar cases and averages them all to come up with suggested values.

Discover’s automatic forcefield

In the automatic forcefield in the Discover program, the atom types for bonds, angles, torsions, and out-of-plane deformations have different levels of specificity. For example, while bond-stretching parameters are determined by the atom types of both atoms; angle-bending and torsion parameters may be determined by the atom type of only the central atom(s). A wildcard (*), repre-senting any type of atom, is used for the end atoms of torsions and angles.

In some cases, angle-bending parameters are specified by two atoms (rather than only the central atom). This can lead to ambigu-ity—for example, C–C–N (if not explicitly defined in the force-field) can be associated with c_–c_–* or with n_–c_–*. The underscore in this example is used to denote the generic (or auto-matic) atom types. Here, a one-sided wildcard (*#, where # is an integer indicating the precedence), is used for one of the end atoms in an angle.

Cerius2•OFF handles precedence with an additional field (P0, P1, … P9) rather than wildcards.

In interpreting the wildcard, the Discover program and the Cerius2•OFF module use the parameter for which the integer is lower. The parameters for a C–C–N angle would, for example (if not explicitly defined in the forcefield), be taken from those for atom types n_–c_–*6 rather than c_–c_–*7, because 6 is smaller than 7.



An example As an example, the parameters for the angle oh–c"–c" in oxalic acid (Figure 3) are not present in CFF91.

When the automatic parameter assignment process is used in Dis-cover, it looks at the auto-equivalence table in the cff91.frc file to find the generic atom types for this angle (indicated in bold type):

#auto_equivalence cff91_auto

! Equivalences! -----------------------------------------------------------------------------!Ver Ref Type NonB Bond Bond Angle Angle Torsion Torsion OOP OOP Inct End Atom Apex Atom End Atoms Center Atoms End Atom Center Atom!--- --- ---- ---- ---- --- -------- ----------- --------- ------------ -------- ----------- 2.0 2 c" c" c" c’_ c_ c’_ c_ c’_ c_ c’_ 2.0 2 oh o o_ o_ o_ o_ o_ o_ o_ o_ …

Thus, for parameter assignment purposes only, atom type oh is reassigned to o_, c" is reassigned to c’_ for the apex atom, and c" is reassigned to c_ for the end atom. The parameters for the oh–c"–c" angle could be taken from either the o_c’_*7 or the c_c’_*9 lines in the quadratic_angle section of the cff91.frc file—o_c’_*7 is chosen because 7 is lower than 9:

#quadratic_angle cff91_auto

> E = K2 * (Theta - Theta0)^2

!Ver Ref I J K Theta0 K2!---- --- ---- ---- ---- --------- --------- 2.0 2 c_ c’_ *9 120.0000 40.0000

Figure 3. Oxalic acid structure and CFF91 atom types

oh

h*

c"

o’

c"

o’

oh

h*



2.0 2 n_ c’_ *8 120.0000 53.5000 2.0 2 o_ c’_ *7 110.0000 122.0000 2.0 2 o’_ c’_ *6 120.0000 68.0000 2.0 2 h_ c’_ *2 110.0000 55.0000 …

Manual parameter assignment

Notification of missing parameters

If parameter(s) for a potential type are not present in the forcefield file and are not generated when the energy expression is set up, an appropriate error message is written to the textport or text win-dow of your molecular modeling program. (In QUANTA, this occurs only if you use QUANTA’s Applications/Builders tools to construct your model; there is no warning for models that are read in from some other application or database.)

Missing and/or automatic parameters are also listed in an output file after the completion of a simultion run. You can find out if parameters are missing before starting your run:

♦ Cerius2 notifies you if atom types are missing and lists them in the text window. It then quits trying to set up an energy expres-sion.

If terms are missing, in Cerius2•Discover the energy expression is set up unless one or more of the Stop check boxes in the Dis-cover Parameters control panel (accessed by selecting Force-field/Parameters in the DISCOVER card) are checked.

If terms are missing, in Cerius2•OFF the energy expression is set up by only if Ignore undefined terms is checked in the Energy Expression control panel (accessed from the Energy Expres-sion/Setup card menu item on the OPEN FORCE FIELD card).

♦ In Insight 97.0, you can request a list of missing parameters with the Forcefield/Tabulate parameter block in the Builder or Biopolymer module.

♦ In Insight 4.0.0, you can use the Forcefield/FF_Info parameter block (in the Builder or other modules) to check the model for unassigned potentials and charges.

♦ In QUANTA, you can request a parameter report with the CHARMm/Parameters/Set Options menu item.



Obtaining new parameters

If the forcefield you are using is similar in functional form and atom typing to another forcefield which does contain the desired parameters, you may be able to use those parameters in your force-field, at least on a trial basis. You may also be able to obtain new parameters (or help in deriving them yourself) from the scientific literature (see References) or from the developers of the forcefield you are using.

Editing a forcefield

Expert users can edit MSI forcefields in different ways to custom-ize them to their needs or to create new forcefields.

Editing through a graphi-cal interface

MSI’s principal molecular modeling programs include forcefield editors (see Available documentation):

♦ Cerius2—Use the Force Field Editor (in the OFF SETUP card deck), which allows you to edit all existing defined terms in the current forcefield and also lets you create new forcefields. The Cerius2•FFE module also allows you to create, edit, and delete atom types and to change the rules on which automatic atom-typing is based.

You can change the functional form of terms in rules-based forcefields (see Rule-based forcefields broadly applicable to the peri-odic table) by adding an explicit term. Any explicit term is always used in preference to a generated term.

Important

♦ Insight II 4.0.0—Use the Forcefield/Edit_FF parameter block, which is found in several modules and allows you to edit the parameters of the current forcefield for all terms except cross-terms. (This parameter block is not included in Insight 97.0, since well parameterized forcefields exist for life science appli-cations.)

The Cerius2•FFE module cannot be used to modify the CFF or CVFF forcefield files that are included with Cerius2—if you want to use customized versions of these forcefields, you need to modify them with Insight II 4.0.0 or as described in the Discover documentation.



♦ QUANTA—The Edit/Atom Data/Parameters menu item allows you to change a limited set of parameters for any atom type. The PSF Generator allows you to edit automatically sup-plied parameters when you do a CHARMm calculation in PSF mode.

Manual editing Expert users can edit the files that define many forcefields with a text editor:

♦ The classical and second-generation forcefields that are avail-able through the Discover program—How to modify the parameter file is explained in separate documentation. The forcefield files are located in the directory defined by the $BIOSYM_LIBRARY environment variable.

The potential template rule file used by Insight can also be edited. You may add new atom types by making additions to this file (refer to the Insight II documentation for a complete description).

♦ CHARMm—Parameter file contents and formats are explained in the electronic documentation supplied with CHARMM. The forcefield file is located in the directory defined by the $CHM_DATA environment variable—PARM.BIN is the binary version and PARM.PRM the ASCII version.

♦ ESFF—Since this forcefield is rule-based, a parameter file needs to be generated and then edited. Please see the CDiscover book.

♦ CFF—Since this forcefield is encrypted, it is edited indirectly, by means of a special template file, as explained in MSI Force-fields: CFF.

ImportantFor forcefields accessed through Cerius2, we strongly recommend that you never edit these files by hand. Please use the Force Field Editor module. This is important because some forcefield values are linked to others and only the Force Field Editor reliably assures that related values are modified in a coordinated way.




The energy expression is the heart of the forcefield. Potential energy is described in the energy expression as the sum of various terms that indicate the energy costs of bond stretching, angle bend-ing, etc. Not all terms are present in all forcefields, and the func-tional forms of the terms vary among forcefields (see Forcefields).

This section and the following main section (Applying constraints and restraints) describe energy term preferences that you can set and restraint terms that can be optionally included in the energy expression.


If you are a novice user, you should alter the default energy terms and parameters as little as possible. One exception to this recom-mendation is nonbond methods (see Handling nonbond interac-tions), where you should choose the method according to the model type rather than necessarily accept the default settings.

Availability

Table 8. Modifications of energy expression in MSI’s simulation engines

term modification engine (restrictions) details

any removal FFEa, CHARMm hereall of a type scaling FDiscover, CDiscover hereall of a type editing FFE, Insight 4.0.0b herebond stretching Morse vs. harmonic FDiscover, CDiscover (CVFF) herebond stretching scaling OFFc (C–H bonds) heretorsion twisting scaled, averaged,

or first foundOFF here

out-of-plane movement averaged or first found

OFF here

van der Waals interac-tions

Lennard–Jones vs. quartic

FDiscover (standalone) here

hydrogen bond interac-tions

if used, how set up OFF, FDiscover (standalone, AMBER), CDis-cover (standalone, AMBER), CHARMm

here

crossterms removal FDiscover, CDiscover (CVFF) here1–3 or bond stretching–

angle bending interac-tions

Urey–Bradley term vs. crossterm

OFF, CHARMm (standalone only) here



aThe Force Field Editor module in Cerius2.bThe Forcefield/Edit_FF parameter block in the Builder and other modules can be used to edit parameters in all terms except crossterms for AMBER, CVFF, and CFF. (Parameter block not present in Insight 97.0.)cThe Open Force Field module in Cerius2.

Most methods for changing the functional form of the energy expression are available via the graphical UIFs:

♦ Cerius2•FFE, OFF—Controls in the Energy Terms control panel in the Open Force Field module. (You can also use the Energy Terms control panel in the Force Field Editor to modify the forcefield itself.)

♦ CDiscover—Items in the Forcefield menu of the Cerius2•Dis-cover module and commands in the Specify pulldown of the Insight•Discover_3 module.

♦ FDiscover—Commands in the Parameters pulldown of the Insight•Discover module.

♦ QUANTA—Controls in the CHARMm Energy Setup dialog box (accessed via the CHARMm/Energy Terms menu item) and the CHARMm Update Parameters dialog box.

Discover and CHARMm also offer additional functionality when run in standalone mode.

Removing terms from the energy expression

Why remove terms You may, for example, want to save computation time during the early stages of minimization of a model that is far from its equilib-rium conformation by not calculating any cross terms. Or you may have found that certain terms are insignificant with respect to the purposes of your study.

How to remove terms You can effectively remove terms from the energy expression in several ways:

♦ In Cerius2•OFF, you can use the Energy Terms Selection control panel in the Open Force Field module to include or exclude entire classes of terms (e.g., all bond–bond crossterms) when setting up the energy expression.



♦ With the CVFF forcefield in Discover, you can choose to turn off (i.e., not use) all cross terms.

♦ You can accomplish the same end for other terms in CVFF and for any class of terms in the other forcefields supplied with the Discover program by scaling terms with a zero scaling factor (see next section).

♦ In CHARMm, you can omit (“skip”) any type(s) of terms or constraints, for example, all bond terms.

Scaling or editing any selected type of term

Uses The contributions of various terms in the potential energy expres-sion to the total energy can be scaled up or down and/or otherwise edited. This can be useful, for example, in the early stages of min-imizing very “bad” structures, where large contributions by cer-tain terms might interfere with convergence.

How it works The Cerius2•Force Field Editor module allows you to directly change the parameters in any term (e.g., for all C_3–H_ bond terms, but not for all bond terms) in the forcefield. The Energy Terms control panels include entry boxes for all relevant parame-ters.

In the Discover program, scaling applies to an entire class of energy terms (e.g., all bond terms) in the energy expression. The force constants (or some other parameters) for the chosen class of terms are multiplied by some constant factor. For example, all bond interactions can be scaled by one factor and all van der Waals radii by another.

In the Insight 4.0.0 molecular modeling program, you can use the Forcefield/FF_Edit parameter block in the Builder and other mod-ules to directly change the parameters in any term except cross-terms (e.g., for all C–H bond terms, but not for all bond terms) in the forcefield. The *_Par parameter blocks accessed through the editor include entry boxes for all relevant parameters.

Alternative bond terms

With the CVFF forcefield in Discover, you can choose to use qua-dratic bond terms rather than Morse bond terms. The Morse term



can allow bonds to stretch to unrealistic lengths (Figure 2), so you may get quicker convergence from a hightly distorted configura-tion if you replace the Morse term by a harmonic term. You do this by specifying the “no Morse” version of CVFF.

Scaled torsion terms

If all torsions about a common bond were simply summed, the tor-sion energy term could be too large. Cerius2•OFF therefore allows several methods for scaling torsion terms (Discover and CHARMm automatically handle torsions optimally, because of how their forcefields are parameterized):

Behavior in Cerius2•OFF ♦ The usual treatment in Cerius2•OFF is to divide the sum of all the parameterized torsion terms around a common bond by the number of torsions around that bond (as is done in Discover).

♦ An alternative method of scaling torsions is to use the energy of only the first torsion found about a bond. This method is not generally recommended, because the torsion term used (and, therefore, the torsion energy) depends on the order in which atoms are created.

♦ Calculated energies for torsions that are exocyclic to aromatic rings (Figure 4), tend to be too high and may be scaled by an additional factor, usually 0.4.

Figure 4. Torsion exocyclic to an aromatic ring



Inversion terms

The inversion, improper, or out-of-plane torsion term represents the energy involved in inverting a chiral center or otherwise changing this out-of-plane angle.

In Cerius2•OFF, you may use the first inversion term found or the average of all inversion energies, but the first approach is not rec-ommended.

Nonbond functional form

In the Cerius2•Force Field Editor module, you can use the van der Waals Energy Terms control panels to choose among several non-bond functional forms. (These are listed in the online help, which is accessed by right-mouse clicking over the Function popup.) However, you would have to change the relevant parameters as well, if you wanted good results.

You can use the DSL (the FDiscover command language) set com-mand to choose between the usual Lennard–Jones 6–12 or 6–9 potential (e.g., term 10 in Eq. 23) and a quartic form:

Eq. 26

The quartic form is useful when you need to eliminate bad van der Waals contacts, but the second derivatives are not calculated.

Hydrogen bonds and hydrogen-bond terms


Many forcefields, especially the newer ones, fully account for hydrogen bonds by other terms in the forcefield and so do not have or require specific terms for handling hydrogen bond inter-actions.

However, some older forcefields include specific terms for hydro-gen bonding (e.g., older versions of AMBER). Others (e.g., CHARMm) allow you to use a hydrogen bond term if you want (but MSI does not recommend this). If you are using a forcefield with explicit hydrogen bond terms, you should read this section.

k srmin( )2 r2–[ ] 2



Lack of hydrogen bond terms is an asset

If specific hydrogen bonds are required, generation of a list of hydrogen bonds is a major step in evaluating the energy of a sys-tem. This process involves looking at all possible pairs of hydro-gen bond donors and acceptors and selecting those that meet certain criteria (Figure 5):

♦ The hydrogen bond length is less than a defined cutoff.

♦ The deviation of D–H–A from linearity is less than a defined cutoff. Typically, the best hydrogen bond has a D–H–A angle of 180°.

Since hydrogen bond interactions depend on both angle and dis-tance, both angle cutoffs and distance cutoffs must be specified for a switching function (see Nonbond cutoffs). A switching, or spline, function (Figure 15) is needed to conserve energy by smoothing transitions over the cutoffs.

Specifying the criteria In Cerius2•OFF, the hydrogen bond criteria can be changed by using the Hydrogen Bond Preferences control panel (accessed by selecting the Energy Terms/Hydrogen Bond menu item from the OPEN FORCE FIELD card). This control panel also allows speci-fication of switching function (“spline”) parameters.

In Discover, default hydrogen bond criteria are contained in the forcefield file (amber.frc). CDiscover allows you to use the BTCL forcefield scale command to scale hydrogen bond terms (if they exist). In FDiscover, they can be changed by editing the command input file to change the variables HBDIST and HBANGL.

Figure 5. Distance and angle criteria for hydrogen bondsA = hydrogen acceptor; D = hydrogen donor.

D

HA

distance

angle



In CHARMm, you can change the hydrogen bond criteria with the CHARMm Update Parameters dialog box, which is accessed from the CHARMm/Update Parameters menu item. This dialog box also allows specification of switching function parameters for hydrogen bonds. Setting the Update Frequency to 0 (the default) effectively omits the hydrogen bond term from the potential energy expression. You can also omit explicit hydrogen bond terms by using the CHARMm/Energy Terms menu item.

Bond–angle cross terms vs. Urey–Bradley terms

An alternative or supplement to bond–angle interactions is the Urey–Bradley term, which accounts for 1–3 interactions between two atoms that are bonded to a common atom.

In Cerius2•OFF, use the Energy Terms Selection control panel to specify whether to use the Urey–Bradley term (assuming it is available in the current forcefield).

In CHARMm, the Urey–Bradley term, if present, can be omitted from the energy expression (standalone only) or can be specified in the parameter file (ANGLE statement).


Why read this section Constraints and restraints allow you to focus the calculation on a region or conformation of interest and also to set up computational experiments. Such experiments are one of the primary uses of molecular modeling, allowing you control over a model at the atomic level. Several examples are described under When to use constraints/restraints.

Restraints vs. constraints The seminal difference between a constraint and a restraint is that a constraint is an absolute restriction imposed on the calculation, while a restraint is an energetic bias that tends to force the calcula-tion toward a certain restriction (even though many people use these terms as if they were interchangeable).

Availability



Table 9. Constraints and restraints in MSI’s simulation engines

constraint/re-straint type enginea details

atom fixed (constraints) OFFb, FDiscover, CDis-cover, CHARMm

here

template forc-ing

harmonic (Eq. 28) restraint FDiscover here

tethering and template forcing

quadratic (Eq. 29) restraint CDiscoverc here

tethering harmonic (Eq. 28) restraint FDiscover heretethering mass-weighted harmonic (Eq. 30) restraint CHARMm herequartic droplet harmonic (Eq. 31) restraint CHARMm heredistance harmonic (Eq. 32) restraint OFF heredistance quadratic (Eq. 29), flat-bottomed (Eq. 34),

or cosine (Eq. 36) restraintCDiscover3 here

distance harmonic (Eq. 32) or flat-bottomed (Eq. 33) restraint

FDiscover here

distance flat-bottomed (Eq. 35) restraint CHARMm heredynamics RATTLE algorithm (constraints) CDiscover heredynamics SHAKE algorithm (constraints) CHARMm heredynamics consensus dynamics (Eq. 28) (standalone

only)FDiscover, CDiscover here

angle harmonic (Eq. 37) restraint OFF hereangle quadratic (Eq. 29), flat-bottomed (Eq. 34),

or cosine (Eq. 36) restraintCDiscoverc here

torsion harmonic (Eq. 38) restraint OFF heretorsion quadratic (Eq. 29), flat-bottomed or J3 dihe-

dral (Eq. 34), cosine (Eq. 36), cis (Eq. 39), trans (Eq. 40), or cis/trans (Eq. 41) restraint

CDiscoverc here

torsion flat-bottomed (Eq. 33) restraint (standalone only)

FDiscover here

torsion cosine (Eq. 42) or harmonic (Eq. 38) torque (one of these is standalone only)

FDiscover here

torsion harmonic (Eq. 38) restraint CHARMm hereinversion harmonic (Eq. 43) restraint OFF herechiral flat-bottomed (Eq. 34) restraint CDiscover3 hereout-of-plane quadratic (Eq. 29), flat-bottomed or J3 dihe-

dral (Eq. 34), or cosine (Eq. 36) restraintCDiscover3 here

out-of-plane harmonic (Eq. 43) restraint (standalone only)

FDiscover here



aThe standalone modes of running simulation engines may give access to additional constraints and restraints—please see the appropriate documentation.bThe Open Force Field module in Cerius2.cNot available yet in the Cerius2•Discover module; restraints applied with CDiscover (in Insight and stan-dalone modes) can also be scaled.

Most restraints and constraints are available via the graphical UIFs:

♦ Cerius2•OFF—Controls in the Restraints control panel, which is accessed from the Energy Terms/Restraints card menu item and in the Atom Constraints control panel, which is accessed from the Atom Constraints card menu item. The latter also allows you to color-code immovable atoms.

♦ Cerius2•Minimizer—Controls in the Atom Constraints control panel, which is accessed from the Constraints/Atoms card menu item.

♦ CDiscover—Controls in the Atom Constraints control panel of the Cerius2•Discover module and commands in the Specify pulldown of the Insight•Discover_3 module.

♦ FDiscover—Commands in the Constraint pulldown of the Insight•Discover module.

♦ QUANTA—Controls accessed by the CHARMm/Constraints Options and CHARMm/SHAKE Options menu items.

Discover and CHARMm offer additional restraint and constraint functionality when run in standalone mode.

When to use constraints/restraints

Constraints and restraints are often used to control and direct the minimization.

Fixed-atoms example For example, you can fix some atoms in space, not allowing then to move. For example, part of the structure of a molecule may have been well solved experimentally, but the structures of other areas are less clear. Or you might want to keep parts of your model (e.g., solvent molecules) rigid to decrease computational costs.



Torsion-rotation example You can add extra terms to the energy expression to restrain or bias the system in certain ways. For example, if you are investigating the adiabatic energy barrier to rotation about a bond, you would restrain the value of that torsion and minimize the structure. Repeating this procedure for a set of torsion values in the range 0°–360° yields a complete energy profile for rotation about the bond. A similar process is used to generate phi/psi maps and other mul-tidimensional energy surfaces in studies of model conformation.

Docking example If a substrate is being docked onto an enzyme and a specific hydro-gen bond between the enzyme and the ligand is thought to be involved in binding, the donor and acceptor atoms can be pulled together to provide a docking coordinate. In this way, the results are not so dependent on the initial starting configuration, which may have been only a crude graphic alignment. In cases like this, the restraint is turned off at some point to make sure that the biased minimum is close to a true minimum.

Modeling incomplete models

Another example of the use of restraints is in modeling incomplete systems. Often, it is difficult or impossible to construct a realistic environment around parts of a model system. For example, only a partial structure of a large protein complex may be available, and some atoms must be restrained to stay near their initial crystal positions because they do not “feel” interactions with neighboring (missing) amino acids, membrane, or solvent. If the site of interest (for instance, a binding site for a competitive inhibitor) is well characterized but other parts of the enzyme are unknown or would require too much computation time if they were included, a limited study can still be carried out with the ends tethered to their crystal coordinates. Usually, these restraints are permanent parts of the model. The results of such calculations must be criti-cally evaluated but can be valid if the ligand binding does not depend on interactions with missing pieces of the model or on con-formational flexibility in the tethered regions.

Relaxing crystal structures As a final example, tethering can be used to gently relax a crystal structure. Often, crystal coordinates, even if highly refined, have several strained interactions due to intrinsically disordered or poorly defined atomic positions, which, upon minimization, give rise to large initial forces. If these forces are not restrained, they can result in artifactual movement away from the original structure. The general approach is to progressively relax parts of the model in stages, starting with the least well determined atoms, until the



entire system can minimize freely. The restraints are ultimately removed so that the final minimum represents an unperturbed conformation. It is usually not necessary to minimize to conver-gence at each stage—the object is to relax the most-strained parts of the system as quickly as possible without introducing artifacts.

Fixed atom constraints

Cost-saving Fixed atoms are constrained to a given location in space; they can-not move at all. Fixed atoms reduce the expense of a calculation in two ways:

♦ Terms in the energy expression involving only fixed atoms can be eliminated, because they merely add a constant to the total energy. Since the positions of fixed atoms cannot change, nei-ther can the contribution of the terms that depend only on these positions. (Interactions between moving and fixed atoms are calculated.)

♦ Fixing atoms reduces the number of degrees of freedom in the system, so minimizers converge in fewer steps and dynamics requires fewer steps to sweep out the available conformational space.

Important

Uses Use atom constraints when you want to apply minimization or dynamics to part of a model, while keeping the remainder of the model fixed and rigid. For example, use atom constraints to quickly minimize a sorbate in a zeolite by fixing the atom positions of the zeolite frame and allowing only the sorbate atoms to move. Or fix all residues in a protein except for those in the active site.

The energy calculated by simulation engines is correct only to an arbitrary constant, depending on the model as well as the fixed atoms. Thus, only differences in energy between conformations of the same model having the same fixed atoms are meaningful.



Template forcing, tethering, quartic droplet restraints,and consensus conformations

Uses Typical uses of these related types of restraints are to bias the con-formation of one model towards that of another, to bias selected atoms towards their experimentally known positions, to restrain the core of a model while allowing its solvent-exposed constitu-ents more freedom of movement, or to find an identical or close set of conformations that a group of related models can achieve.

Template forcing To force the conformation of one model to be similar to that of a template model, a one-to-one correspondence between atoms in the template and in the moving structure is set up, and (for exam-ple) one of the following restraint terms is added to the energy expression:

Eq. 27

or:

Eq. 28

The term in Eq. 27 is proportional to the root-mean-square (rms) deviation of the analog atoms from the template atoms. (This form cannot be used with the Newton–Raphson minimizer in FDis-cover.) The values obtained for the energy and the rms function depend on the value of the forcing parameter K. Typical values for this constant are in the neighborhood of 5 kcal Å-1. It is often instructive to look at the dependence of the energy and rms func-tions on the forcing parameter by making several determinations with different forcing parameters. If several runs (minimization or dynamics) are made, it may also be helpful to plot the energy as a function of the rms value. For tethered minimizations, a very large

E K

Ri Ritemplate

– 2

N---------------------------------------

pairs

N

∑

1 2/

=

E Ki Ri Ritemplate

– 2

pairs

N

∑=



forcing constant (e.g., 2000.0 kcal Å-1) is often used to prevent sig-nificant movement of any of the tethered atoms.

Eq. 28 represents a conceptually more straightforward restraint, with each atom restrained by an isotropic spring to the position of its template atom. In either form, the summation is over a list of pairs of atoms to restrain: one from the moving model, and one from the template model. FDiscover uses this quadratic form by default.

The Ki in Eq. 28 are determined by the distance of atom i from the atom defining the origin as:

where rmin is the distance at which the tethering turns on, kmin is the initial force constant at that distance, rmax is the distance where the force constant reaches its maximum allowed value, and kmax is the maximum allowed force constant. If rmin and kmin are not given, the default values are zero. If rmax is zero, tethering uses a constant force constant of kmax.

In CDiscover, a simpler quadratic is used:

Eq. 29

where V is any appropriate internal (bond length, angle, etc.—the same functional form is used for several types of restraints).

Advantages of each type of template-forcing restraint

The first form (Eq. 27) gives the best rms fit for the least energetic cost, but individual atoms may remain quite far from their tem-plate position. The second form (Eq. 28) restrains each atom indi-vidually, so each atom is forced toward its template partner. The resulting rms fit is not as good as that from Eq. 27, but no one atom is allowed to deviate as much as is possible with Eq. 27. The form in Eq. 28 also allows for a different force constant for each pair, which means that different atoms or classes of atoms can be treated differently.

Ki

0 r rmin<

kmin kmax kmin– r rmin–( ) rmsx rmin–( )⁄×+ rmin r rmax≤ ≤

kmaxrmax r<

=

E scale factor k V V0–( )2=



Tethering Tethering is the same as template forcing, except that the atoms are restrained to their original positions rather than to positions in a template structure. Both Eq. 27 and quadratic forms are applicable for tethering; however, Eq. 28 is used by FDiscover and Eq. 29 by CDiscover, because tethering is usually used to keep atoms from moving too far from their original positions.

CHARMm allows mass-weighted tethering by calculating an additional energy term for all atoms that are to be restrained. This term has the form:

Eq. 30

Where Econs is the constraint energy, ki is the force constant, mi is the mass of atom i (if mass weighing is used) or 1, ri is the position of atom i, r0 is the reference position about which the atom is to be centered, and n is an exponent.

Quartic droplet restraint The quartic droplet restraint term in CHARMm is designed to put the entire model into a “cage” by constructing a restraining sphere around a model. The potential is scaled so that atom positions fur-thest from the center of mass or the geometric center of the model have the greatest restraining force applied.

The quartic droplet restraint term is based on the center of mass (COM) or the center of geometry of the model. No net force or torque is introduced by the center of mass term. The potential function is:

Eq. 31

FDiscover (standalone only) allows similar restraints within spherical shells.

Consensus dynamics Consensus dynamics is used to find the consensus configuration of a set of analogs. In essence, all models in the set are treated as both moving molecules and templates.

Econs kimi ri r0–( )n

i

∑=

Edroplet k mi ri rCOM–( )n

i

∑=



Standalone FDiscover uses the harmonic template-forcing restraint (Eq. 28).

The database capability of CDiscover is used in settng up consen-sus dynamics calculations, using the restraint in Eq. 29.

General internal-coordinate restraints

In CHARMm, you can apply general internal coordinate restraints by applying restraints to all bonds, angles, and/or dihedral angles that have entries in an internal coordinates table. This facility is global, that is, not applicable to specific internal coordinates.

Distance and NOE restraints

Uses Distance restraints are used to bias the distance between two atoms, bonded or not, toward a given value. Some uses are to cyclize linear models by bringing the ends closer together, dock different models, and fit distance data derived from NOE and other experiments.

Several functional forms for distance restraints: har-monic…

Several commonly used functional forms are supported.

One is a simple harmonic function, which in FDiscover has the form:

Eq. 32

where K is a force constant, Rij is the current distance between the atoms, and Rtarget is the target distance. A large force constant tends to force the distance to be close to the target distance; a smaller force constant results in a correspondingly smaller bias.

In CDiscover, the quadratic form is the same, except that scaling is enabled (Eq. 29).

In Cerius2, the form is the same, except that K is multiplied by 0.5. Rtarget can be defined explicitly or automatically extracted from the model as the current distance between atoms.

…and flat-bottomed… The second form is also harmonic, but it is separated into several piecewise continuous regions, resulting in a flat-bottomed poten-tial (Figure 6). For FDiscover, the form is:

E K Rij Rtarget–( )2=



Eq. 33

For CDiscover, the flat-bottomed form is:

Eq. 34

where V is any appropriate internal (bond length, angle, etc.—the same functional form is used for several types of restraints).

The restraining potential used in CHARMm is:

Eq. 35

Where Rlim is the value of R where the force equals fmax.

E

E1 R1 Rij–( )F+ Rij R1<

K2 Rij R2–( )2 R1 Rij< R2≤

0 R2 Rij< R3≤

K3 Rij R3–( )2 R3 Rij R4≤<

E4 Rij R4–( )F+ R4 Rij<

=

E scale factor

k V V0–( )2

0.0

k V V1–( )2

=

V V0≤

V0 V V1< <

V V1≥

E

kmin

2---------- R Rmin–( )2 R Rmin<

0 Rmin R Rmax< <

kmin

2---------- R Rmax–( )2 Rmax R Rlim< <

fmax RRlim Rmax+

2----------------------------–

Rlim R<

=



…and cosine CDiscover also allows a cosine form of restraint:

Eq. 36

Figure 6. Distance restraint functionE as a function of R, the distance between two atoms or dihedral angles, defined as in Eq. 33.

0R2 R3

E(r)

0R2 R3

0R2 = R3

R1 R4

R1 R4

R1 R4

E scale factor k2--- 1 n V V0–( )( )cos–( )=



where V is any appropriate internal (bond length, angle, etc.) and n is the periodicity.

Advantages of the flat-bottomed functional form

It is not necessary for the flat-bottomed potential (Figure 6) to be symmetric. By appropriate definition of the points R1, R2, etc., any of the regions may be eliminated. For Eq. 33, the important regions are those from R1 to R2 and from R3 to R4, where a harmonic poten-tial is applied, and the flat bottom from R2 to R3.

This form of the restraint allows a range of acceptable distances and is particularly useful for incorporating experimental distance information, such as those from NOE experiments, into a calcula-tion. The flat bottom allows for experimental error in the deter-mined distance. The two outer regions (Figure 6) have a constant gradient, which is useful for avoiding unreasonably large forces if the initial structure is far from the target value.

Distance and angle constraints in dynamics simulations

The RATTLE and SHAKE algorithms effectively remove very-high-frequency vibrations from consideration during dynamics simulations. Use of these algorithms can allow for a larger time step during simulation.

In CDiscover, the BTCL rattle command is used before the dynam-ics command to set up constraints in bonds, angles, or water mol-ecules in a molecular dynamics simulation. It can be used to constrain bonds or any atom pairs to user-defined distances. It can be used to constrain angles spanned by two constrained bonds. In addition, it can be used to fix the geometry of water molecules so that the fixed-geometry water models SPC and TIP3P can be used in a simulation. This functionality is available in a limited way via the Calculate/Dynamics parameter block in the Discover_3 mod-ule of Insight (click More to display the Rattle toggle).

In CHARMm, SHAKE is used to constrain bond lengths and angles spanned by two constrained bonds during dynamics runs. (However, its use is recommended only for constraining all bonds in which one of the bonded atoms is a hydrogen.) The SHAKE algorithm cannot be used with the Newton–Raphson or ABNR minimizers (see Minimization).



Angle restraints

In Cerius2, an angle restraint can be applied to a group of any three atoms. The restraint is implemented such that:

Eq. 37

Where: Ka is the angle force constant; θ is the angle between the selected atoms; and θ0 is the desired restrained angle of the selected atoms. θ0 can be defined explicitly or can be automatically extracted from the model as the current angle connecting selected atoms.

In CDiscover the default form of angle restraints is cosine (Eq. 36). Quadratic (Eq. 29) and flat-bottomed (Eq. 34) angle restraints can also be used.

Torsion restraints

Uses Some uses of torsion restraints are to enforce chiral and prochiral centers, prevent cis–trans conversions, and fit NOE J-coupling con-stants from NMR experiments. Conversely, other uses are to force torsion rotation in order to perform phi/psi mapping, perform conformational searching, and induce conformational changes.

Functional forms Several forms of torsion restraints are used in the literature and implemented in MSI’s simulation engines.

Harmonic restraints, or periodic restraints (Eq. 42 with n = 1), are appropriate for forcing a torsion angle to a particular value. The periodic form with a periodicity greater than one is useful for restraining a torsion to one of several related angles. For instance, a threefold potential could keep a torsion either trans or at one of the two gauche conformations, depending on the starting confor-mation and the strength of the potential applied.

Implementation In Cerius2•OFF, a torsion (dihedral) restraint can be defined among any group of four atoms. The restraint is implemented such that:

Eq. 38

E 0.5Ka θ θ0–( )2=

E Kt φ φ0–( )2=



Where Kt is the torsion force constant; φ is the angle between the i–j–k and j–k–l planes; and φ0 is the desired restrained angle of the selected atoms, which can be defined explicitly or automatically extracted from the model as the current angle connecting selected atoms.

In CDiscover, you can specifically restrain dihedrals to be cis:

Eq. 39

or trans:

Eq. 40

or either cis or trans:

Eq. 41

You can also use the flat-bottomed function (Eq. 34) to apply J3 dihedral restraints to fit the results of NOE experiments. A plain cosine form (Eq. 36) and a quadratic form (Eq. 29) are also avail-able. The torson involving any four atoms can be restrained.

In FDiscover, the functional forms include a simple harmonic form analogous to Eq. 32 and a piecewise continuous form like Eq. 33 with R interpreted as the angle, rather than the distance. Another form is the periodic function of Eq. 42:

Eq. 42

where V gives the strength of the restraint, n is an integer period-icity, and φ0 is the phase angle.

CHARMm uses a harmonic potential to restrict the motion of a dihedral angle to a value close to a reference position or to examine a series of different conformations when making potential energy maps.

E scale factor k2--- 1 φcos–( )=

E scale factor k2--- 1 φcos+( )=

E scale factor k2--- 1 2φ( )cos–( )=

E V 1 nφ φ0–( )cos+[ ]=



Inversion, out-of-plane, and chiral restraints

Uses Typical uses include prevention of changes in chirality or prochirality. (A molecule is chiral if no stable conformation of it can be superimposed on its mirror image—most chiral organic mole-cules can be described in terms of chiral centers, i.e., an atom that has four distinct substituents. Two chemically identical substitu-ents on an otherwise chiral tetrahedral center are prochiral; in addi-tion, sp2 hybridised planar systems with three different substituents are considered prochiral.)

Implementation In Cerius2•OFF, an inversion (improper torsion or out-of-plane angle) restraint can be defined among any four atoms i, j, k, l, where i defines the inversion center. The restraint is implemented such that:

Eq. 43

Where Ki is the force constant for the out-of-plane; χ is the angle between the i–j–l and i–k–l planes; and χ0 is the desired restrained out-of-plane angle of the selected atoms, which can be defined explicitly or automatically extracted from the model as the current angle connecting selected atoms. There must be a real atomic cen-ter for the inversion.

The CDiscover program can impose a flat-bottomed chiral restraint (Eq. 34) to invert the chirality or force it to be R or S.

CDiscover can also impose a cosine (Eq. 36), quadratic (Eq. 29), or flat-bottomed (Eq. 34) out-of-plane restraint.

The FDiscover DSL language can be used to impose chirality and prochirality restraints having the same functional form as Eq. 43, where χ0 is the out-of-plane angle corresponding to R or S.

Plane and other geometrical constraints and restraints

The BTCL language of CDiscover allows sophisticated geometric manipulation of molecular and other objects, including constraints and restraints, by means of the geometry, molGeom, restraint and other commands. A subset of this functionality is accessible in the

E Ki χ χ0–( )2=



Calculate/Geometric parameter block in the Insight•Discover_3 module.


Why read this section Periodic boundary conditions refers to the simulation of models con-sisting of a periodic lattice of identical subunits. By applying peri-odic boundaries to simulations, the influence, for example, of bulk solvent or crystalline environments can be included, thereby improving the rigor and realism of a model.

Availability

aThe Open Force Field module in Cerius2.

Most methods for controlling the treatment of periodic systems are available via the graphical UIFs:

♦ CDiscover—The program detects whether a system is periodic (in Cerius2: fully automatic, depends only on which model is current; in Insight: semi-automatic, you need to execute the Setup/System parameter block) and displays the appropriate controls or parameters in the interface of the Cerius2•Discover or Insight•Discover_3 module.

♦ FDiscover—You can choose the minimum-image or explicit-image convention in the Parameters/Variables parameter block of the Insight•Discover module. You have to specify whether a system is periodic by toggling PBC (periodic boundary condi-tions) in the Run/Run parameter block.

Table 10. Periodic boundary methods in MSI’s simulation engines

periodicity engine details

minimum image OFFa, FDiscover, CHARMm hereexplicit image FDiscover, CDiscover, CHARMm herecrystal simulations OFF, CDiscover, CHARMm herebonds across boundaries OFF, CDiscover, CHARMm here



♦ QUANTA—Use the CHARMm/Periodic Boundaries menu item to turn periodic boundary conditions on and off and to specify where to obtain this information.

Discover and CHARMm offer additional functionality when run in standalone mode.

Models are specified in Cartesian space

Some simulation engines accept only Cartesian coordinates, not crystal coordinates (others are able to convert between the two sys-tems). This is important when using asymmetric space groups, since the symmetry operators assume that the input coordinates correspond to the standard asymmetric unit as defined in the International Tables for Crystallography (Reidl 1983).

For Discover, it is assumed that the x Cartesian axis corresponds to the a crystal axis and that the b axis lies in the x,y plane (see Figure 7).

For Cerius2•OFF, by default the c lattice vector is parallel to the z Cartesian axis and the b lattice vector lies in the y,z plane (Figure 8).

CHARMm can handle models that are defined in either crystal or Cartesian space. In converting from crystal to Cartesian axes, the

Figure 7. Relationship between Cartesian coordinate system (xyz) and periodic system (abc) in Discover and CHARMm

x

y

z

a

b

c



a, b, c crystal axes are aligned with the x, y, z Cartesian axes (Figure 7).

Minimum-image model

Tip

Simulation in bulk solvent The left side of Figure 9 shows a solute molecule surrounded by enough solvent to occupy the volume (and shape) of a cube. A sim-ulation carried out on this isolated cubic system is a poor approx-imation of what would happen in a true bulk solvent environment. For example, the solute can diffuse toward a surface or solvent molecules can evaporate. To remedy this, on the right of Figure 9 the cube is replicated in three dimensions to form a 3 × 3 × 3 lattice of identical cubes. This is a much better representation of bulk sol-vent for the interior cube, because molecules near the surfaces now interact with solvent in adjacent cubes. The imaged atoms are used to calculate energies and forces on the real atoms in the interior cube. The energies and forces on the imaged atoms themselves are

Figure 8. Relationship between Cartesian coordinate system (xyz) and periodic system (abc) in Cerius2•OFF

z

y

x

c

b

a

For periodic systems in which nonbond interactions dominate, the Ewald sum method (Ewald sums for periodic systems) is preferred over the the minimum-image convention.



not calculated because their motions are computed as symmetry operations on the real atoms, for example, by translations along the cubic axes.

Implications of minimum-image model for calculat-ing nonbond interactions

Consider the implications of this model for a specific case. In Figure 10, molecule A1 is located near an edge of the square. (For simplicity, this discussion focuses on a two-dimensional lattice.) In addition, eight images of A1 (A2–A9) are present in the adjacent symmetrically related squares. Consider the interactions of mole-cules A with molecules B. The closest image of B to A1 is actually not B1, but rather B5. If molecules in the interior cell are allowed to interact only with the molecule or molecular image closest to it, this is called a minimum-image model. Each molecule interacts only with those molecules and images within a distance of half the cell size. The advantage of this approach is its simplicity. It is straight-forward to compute energy between a given pair of molecules without explicitly keeping track of the images in neighboring cells. All periodic boundary algorithms imply a cutoff criterion, but the minimum-image convention implies a maximum distance for this cutoff of no more than half the cell dimensions.

For a description of the minimum-image convention, see also Allen and Tildesley (1987).

Figure 9. Solute surrounded by solventA solute surrounded by an isolated cube of solvent is replicated periodically in three dimensions in order to better represent a bulk or crystalline environment.

solute



Explicit-image model

A more general approach—ghost mole-cules

Simulation engines (Discover and CHARMm) can also use a more general approach by generating explicit images of the interior mol-ecules, also called ghost molecules, which interact with the interior molecules. These ghost molecules are replicated to as great a dis-tance as necessary (but no farther than necessary) to satisfy the desired potential energy cutoff criteria.

The left side of Figure 11 shows molecule A1 interacting with sev-eral images of B (B1, B2, B3, B5) within the specified cutoff radius (shown as a shaded circle centered on A1). A1 interacts with sev-eral of its own images as well (A3, A5, A6, A8).

Figure 10. Minimum-image modelMinimum-image model showing that each real molecule interacts with at most only one image of each real molecule.

A2B2

A3 A4

A5

A7 A8 A9

A6A1

B3 B4

B6

B9B8B7

B5 B1



Cutoff distances and non-bond interactions

The right side of Figure 11 shows which molecules in the adjacent unit cells become explicit ghost molecules for a given cutoff dis-tance. Not every molecule in an adjacent cell becomes a ghost. However, if a cutoff distance that is longer than the cell length is used, ghosts from unit cells beyond the nearest neighbor cells may be included. As molecules (effectively, see below) move in and out of the boundaries, the molecules that are ghosts can change. There-fore, the ghost list is regenerated periodically.

Nonbond interactions do not have to be calculated between ghost atoms. This helps to significantly reduce computation time.

When group-based cutoffs (Charge groups and group-based cutoffs) are used, the nonbond potential is cut off on the basis of charge groups (i.e., only if two groups are within the cutoff is the interac-tion calculated), and only those groups in molecular ghosts that are within the cutoff distance of a real group are included in the ghost atom list.

How images and “real” molecules move

Ghost molecules follow their symmetrically related counterparts. However, when it comes time to move the molecules (in a dynam-

Figure 11. Explicit-image modelExplicit-image model showing how a cutoff distance defines which molecules in adjacent unit cells are selected as ghost images. (Different cutoff distances are used in the left and right figures.) Left: explicit-image model—a larger cutoff including interactions with more images is possible than with the minimum-image convention; right: the shaded region identifies which molecules are selected as ghost images within the cutoff distance of any molecules in the unit cell.

cutoff

ghost molecules

real molecules

cuto

ff

A2

A1

A3 A4

A5 A6

A7 A8 A9

B2

B1

B3 B4

B5 B6

B7 B8 B9

A2

A1

A3 A4

A5 A6

A7 A8 A9

B2

B1

B3 B4

B5 B6

B7 B8 B9



ics step or minimization iteration), only the real molecules (A1 and B1) are actually moved according to the accumulated forces each molecule has felt. The ghost molecule positions are simply regen-erated by applying the defined symmetry relations to the new positions of the molecules.

Perfect symmetry is maintained between the primary structure and all its image objects. For many applications, this condition is satisfactory. However, it is not possible to study, for example, coop-erative changes between image objects.

To maintain all molecules in the central cell, image centering is used. Molecules that happen to migrate to an edge of the primary struc-ture and would appear in one of its image objects instead reappear in the primary structure from the opposite direction. Thus a con-stant number of atoms is maintained and no molecules are lost, no matter how far they may diffuse during the calculation.

Crystal simulations

Energies of crystals can be calculated and the lattice parameters a, b, c, α, β, and γ can be optimized with Cerius2, CDiscover, and CHARMm:

♦ In the Cerius2•OFF module, you can choose to optimize cell dimensions and angles in 2D or 3D periodic systems or to con-strain some of these coordinates. From the MINIMIZER card (accessed from the OFF METHODS deck of cards), you can access cell constraints options with the Constraints/Cell menu item.

Crystal simulations are also available in several Cerius2•OFF Instruments modules. For example, you can use the Crystal Packer module to optimize crystals or calculate their energy and can include minimization of periodic structures in a Mechanical Properties run.

♦ In the Cerius2•Discover module, the Optimize Cell checkbox in the Discover Minimize control panel is automatically checked if the current model is periodic.

♦ In the Insight•Discover_3 module, crystal optimization is requested by toggling Optimize_Cell in the Calculate/Mini-mize parameter block. Crystal optimization is also available



within the Structure_Refine, Amorphous_Cell, and other Insight II modules.

♦ In QUANTA, use the CHARMm/Periodic Boundaries menu item to turn periodic boundary conditions on and obtain crystal energies.

Because crystal patching is not available in CHARMm, bonds between crystal images are not handled well. Similarly, hydro-gen bond interactions described by an explicit hydrogen bond function cannot be used. The only forces that can be calculated between primary and image atoms in crystals are nonbond forces.

Bonds across boundaries

Allowing bonds (with additional energy terms including angles, dihedrals, and improper dihedrals) between the primary atoms and image atoms enables you to study polymers such as DNA or industrial polymers.

Cerius2•OFF, CDiscover, and CHARMm can handle bonds across cell boundaries. (However, CHARMm is best used only for linear polymers, since it does not handle 3D lattices or networks well.)


Electrostatic (Coulombic) and van der Waals interactions together are referred to as nonbond interactions.

Why read this section Nonbond terms can involve extensive calculation. To avoid a heavy calculation burden, some approximation scheme is often employed. Choosing the best method for your particular model can save computational expense without sacrificing accuracy.

In addition, you have some direct control over the functional terms for nonbond interactions:

♦ You might be able to improve your simulation by changing the default combination rules for van der Waals interactions between non-identical atom types (OFF, Discover, Combination rules for van der Waals terms).



♦ You can change the dielectric constant to account for nonaque-ous solvents and/or solvent screening or make the dielectric “constant” a function of distance (OFF, Discover, CHARMm, The dielectric constant and the Coulombic term).

Availability

aThe Open Force Field module in Cerius2.bStandalone only for periodic systems, cannot be used for constant-pressure or constant-stress dynam-ics.

Most methods for specifying how to treat nonbond interactions are available via the graphical UIFs:

♦ Cerius2•OFF—The Van Der Waals Preferences and Coulomb preferences control panels (accessed from the Energy Terms card menu item), and similar control panels in several OFF Instruments modules.

♦ Cerius2•MMFF—The MMFF Nonbonded Preferences control panel.

♦ CDiscover—Controls in the Discover Non-Bond control panel of the Cerius2•Discover module and commands in the Specify/Nonbonds parameter block of the Insight•Discover_3 module.

♦ FDiscover—Commands in the Parameters pulldown of the Insight•Discover module.

Table 11. Nonbond methods in MSI’s simulation engines

method type of system engine details

atom-based (single) cutoffs

periodic, nonperiodic OFFa, FDiscover, CDiscover, CHARMm here

group-based cutoffs periodic, nonperiodic FDiscover, FDiscover, CHARMm heredouble cuttoffs periodic, nonperiodic FDiscover heretail corrections disordered periodic CDiscover herecell-based cutoffs periodic CDiscover herecell multipole method nonperiodic, peri-

odicbCDiscover here

Ewald sums periodic OFF, CDiscover here



♦ QUANTA—Controls accessed through the CHARMm/Update Parameters menu item.

Discover and CHARMm also offer additional functionality when run in standalone mode.

You may use different methods for van der Waals and electrostatic interactions

Typically, both van der Waals and Coulombic interactions are cal-culated by the same method and (if by the nonbond cutoff method) with the same nonbond list. However, different methods and parameters may be used for van der Waals and Coulombic terms in CDiscover and (except for some operations) in Cerius2•OFF and CHARMm. This allows you, for instance, to use a large cutoff for electrostatic interactions and a smaller cutoff for van der Waals interactions.

The van der Waals interaction potential is relatively short range and dies out as 1 ⁄ r6. By 8–10 Å, the energy and forces are quite small. Thus, using cutoffs to bring the van der Waals potential to zero at about 10 Å can be a reasonable approximation. The Cou-lombic interactions, on the other hand, die off as 1 ⁄ r, so even at considerable distances the energy of interaction is not negligible. But this depends on the model: except for a few formally charged groups, most molecules are composed of neutral fragments with dipoles and quadrupoles. Thus, in most models the major compo-nent of the electrostatic interaction between molecules or parts of molecules is a dipole–dipole interaction, which falls off as 1 ⁄ r3.

♦ To specify different methods for treating van der Waals and Coulombic interactions in the Cerius2•Discover module, select the Forcefield/Nonbond menu item in the DISCOVER card and check the Treat VDW and Coulomb Separately check box in the Discover Non-Bond control panel.

♦ Cerius2•OFF does not allow you to select different cutoffs for Coulombic and van der Waals interactions if you are using a non-Ewald method for both. However, you may independently select an Ewald or non-Ewald method for either. If you do select Ewald for both, you can independently set the cutoff and convergence parameters for each.

NoteFor models having 2D periodicity (e.g., built using the Cerius2•Surface Builder) the Ewald method is available for the Coulombic terms but not for the van der Waals terms.



♦ To specify different methods for treating van der Waals and Coulombic interactions in the Insight•Discover_3 module, go the Specify/Nonbonds parameter block, click More and then set Define Cutoffs to Separate.

♦ In QUANTA and in Cerius2•MMFF, you can use different switching functions for the van der Waals and electrostatic interactions.

Automatic exclusions Van der Waals and Coulombic interactions are ordinarily calcu-lated between all atom pairs that are not specifically excluded. Most forcefields exclude nonbond terms for atoms connected by bonds (1–2 interactions) and valence angles (1–3). Some forcefields also exclude nonbond terms between end atoms in torsion (1–4) interactions. These interactions are illustrated Figure 12.

1–4 interactions and AMBER

If 1–4 nonbond interactions in torsions are included in the non-bond list, they may be scaled. For example, with the AMBER force-field (as implemented in both Cerius2•OFF and Discover) these nonbond interactions must be scaled by 0.50.

Scaling by 0.5 occurs by default with Cerius2 when AMBER is loaded.

In the Insight•Discover module, you need to toggle the p1_4 parameter in the Parameters/Scale_Terms parameter block on and enter 0.5 in the p14 parameter box,

The standalone version of the FDiscover program handles this scaling with the following DSL command:

> scale 1-4 by 0.5

Figure 12, Types of interactions usually excluded from nonbond calculations

11

1

22 2

3

3

4



Important

The equivalent BTCL command for the CDiscover program is forcefield scale with the vdw_1_4 keyword.

Combination rules for van der Waals terms

van der Waals radius com-bination rules

Any van der Waals interaction parameters that are actually defined for heterogenous atom pairs are called off-diagonal parame-ters. Off-diagonal parameters that are not available for such atom pairs are calculated by averaging those for each of the two atom types, using a geometric, arithmetic, or (in CDiscover and Cerius•OFF) 6th-power combination rule:

Eq. 44

Eq. 45

Eq. 46

Availability In Discover and Cerius2•OFF, a choice of combination rule is available and is specified in the forcefield file (see the File Formats documentation).

Quality of results The arithmetic mean gives marginally better equilibrium distances for van der Waals interactions than the geometric combination rule

This term is not set by default in Discover, even for the AMBER forcefield, so you must remember to set the p1_4 parameter either the first time that you run the Discover program from Insight when using AMBER or in your command input file for each standalone job that uses AMBER.

Aij AiiAjj= Bij BiiBjj=

rij riirjj= εi j εiiεjj=

geometric:

εi j εiiεjj= rij* rii

*rjj

*+

2-----------------=

arithmetic:

εi j

εi iεjj 2rii* 3

rjj* 3

rii* 6

rjj* 6

+----------------------------------------= rij

* rii* 6

rjj* 6

+

2-----------------------

1 6/

=

sixth-power:



(Halgren 1992). The 6th-power rule (not available with all force-fields) yields even better results (Waldman and Hagler 1993).

van der Waals combina-tion rules and Ewald sums

With the Ewald method (Karasawa & Goddard 1989) (Ewald sums for periodic systems), the geometric mean leads to faster conver-gence than the arithmetic mean.

In addition, because the Ewald sum calculation proceeds much faster when only diagonal parameters are used, the Cerius2•OFF Van Der Waals Preferences control panel includes an option to ignore off-diagonals even when they are present (they are not present in any of the Discover forcefields).

The dielectric constant and the Coulombic term

Role of the dielectric con-stant in modeling

The electrostatic potential is computed from the partial atomic charges associated with the model (Assigning charges). Approxi-mate solvent-screening effects can be included by specifying a nondefault value for the dielectric constant ε if it is explicitly included in the forcefield. (The “dielectric constant” used in mod-eling is not the dielectric constant that most experimental chemists would think of— it is instead an empirical, dimensionless scaling factor.)

The dielectric constant reflects the polarizability of the solvent molecules. A polarizable solvent such as water has a greater dielectric constant than less polar liquids. Electrostatic interactions in polarizable solvents with high dielectric constants are greatly attenuated. In closely packed molecules, however, there are fewer solvent molecules to screen the charge interactions.

A relatively large dielectric constant can be used for simulating the aqueous environment of small systems. However, many calcula-tions on models use a smaller dielectric constant. For example, a dielectric constant between 2.0 and 10.0 has been used for simula-tions in the interior of a protein. A typical value for water would be around 4.

Additional information For a helpful review, please see Harvey (1989). A tutorial on dielec-tric constants in forcefields can be found at MSI’s website:

http://www.msi.com/support/insight/insight/dielectric.html



A distance-dependent dielectric “constant”

The dielectric constant can be kept constant, or the Coulombic term can be made a shielded function, where the dielectric “con-stant” is a function of distance (r ·ε). This is useful for electrostatic interactions in closely packed molecules, where the number of sol-vent molecules between two interacting charges is usually fewer than in bulk solvent. A distance-dependent dielectric constant is also useful for models in which explicit solvent molecules are not included.

The distance-dependent dielectric function (also called a shielded dielectric) is generally used with the Dreiding and AMBER force-fields and may be used with others.

A shielded Coulombic term is faster to calculate than a non-shielded term because no square root has to be evaluated.

Important

Availability The dielectric constant can be changed and/or made distance dependent in any of MSI’s simulation engines.

In Cerius2•FFE, the Coulombic control panel allows you to choose the form of the Coulombic term—the term can be distance-depen-dent, not distance-dependent, or corrected by an erfc term (see Glass forcefield).

Special considerations for the AMBER forcefield

With the AMBER forcefield, in most applications a distance-dependent dielectric (ε = f (r)) should be used.

In Cerius2•OFF, the Epsilon value is 1.0 unless you change it in the Coulomb Preferences control panel (which is accessed by selecting the Energy Terms/Coulomb card menu item).

In the Insight•Discover_3 module, in the Specify/Nonbonds parameter block you would click More, then set Dielectric to Dist_Dependent and enter 4.0 for the Dielectric Value.

The equivalent BTCL command for the CDiscover program is forcefield with the distance_dep keyword set to true and dielect set to 4.0.

A distance-dependent dielectric constant cannot be used on a periodic model with the Ewald sum method (Ewald sums for periodic systems).



In the Insight•Discover module, in the Parameters/Set parameter block you would enter 4.0 in the Dielectric parameter box and make sure the Dist_Dependent parameter is toggled on.

The standalone version of the FDiscover program handles this with the following DSL command:

> set dielectric = 4.0*r

Nonbond cutoffs

Why read this section An energy expression such as Eq. 8, which is representative of cur-rent forcefields, is computationally tractable only for systems with relatively small numbers of atoms. The number of internal coordi-nates grows linearly with the size of a model, so the computational work involved in the first nine terms in Eq. 8 also grows linearly.

However, inspection of the final summation, which represents the nonbond interactions, reveals a quadratic dependence on the number of atoms in the system: If the system of interest has 1000 atoms, the nonbond summation has about 500,000 terms. If it has 10,000 atoms, the summation has 50,000,000!

Therefore, it is common to neglect or approximate the nonbond interactions for widely separated pairs of atoms.

Choosing how to treat long-range nonbond interactions is an important factor in determining the accuracy and the calculation time of an energy evaluation.

Several cutoff methods are discussed below, and a review was published by Brooks et al. (1985b). More recently, two other methods—cell multipoles (Cell multipole method) and Ewald sums (Ewald sums for periodic systems)—have also become available. You should read all these sections to decide which method is best for your model and computational problem.

NoteThe same nonbond method(s)a and specifications should generally be used for all energy calculations within a given project.



aHowever, the method and/or specifications used for van der Waals interactions may differ from those used for Coulombic interactions (see You may use different methods for van der Waals and electro-static interactions).

Effect of nonbond cutoff distance on calculation of nonbond interactions

To appreciate the impact of cutoffs on computational efficiency, consider a receptor–ligand–solvent system with 5000 total atoms. An example would be a small protein (100–150 residues) sur-rounded by 1–2 layers of water.

Figure 13 shows how the number of nonbond interactions increases with the cutoff distance. This calculation would run at least 10 times faster with an 8.0 Å cutoff than with no cutoff (assuming that the nonbond term is rate limiting, which it usually is). The trade-off is, of course, that interactions beyond the cutoff distance are not accounted for.

The significance of nonbond interactions beyond the cutoff dis-tance depends on the system being simulated. In modeling an iso-lated molecule or cluster, the use of cutoffs for the van der Waals interactions is quite reasonable. The potential is relatively short range and dies out as 1/r6. Consequently, by 8–10 Å, both the energy and forces are quite small.

In contrast, the situation is slightly different in modeling disor-dered or ordered crystalline systems. For a typical disordered sys-tem, which might consist of a cube of organic material with ~25 Å edges, contributions of the van der Waals interactions at distances greater than 8–10 Å to the energy and pressure can amount to ~50–200 kcal mol-1 and ~500–1000 bar (0.05–0.10 GPa), respectively, while contributions of electrostatic interactions are much smaller. Also, contributions of remote nonbond interactions of all types to the resultant force on atoms is small. Fortunately, it is possible in such systems to apply tail corrections, which permit the use of 8–10 Å cutoffs while simultaneously yielding accurate values of energy and pressure.

Finally, in periodic crystalline systems, both van der Waals interac-tions and electrostatic interactions can be significant up to 15 Å or more. For example, in a calculation of the energy as a function of cutoff distance in the [Ala–Pro–D-Phe]2 crystal, Kitson and Hagler showed that the nonbond energy accounted for changes from 63% to 97% of the asymptotic value as the cutoff distance was increased from 8 to 15 Å (Kitson and Hagler 1988).



Figure 14 shows how the van der Waals component of the non-bond energy varies as a function of cutoff distance for an [Ala–Pro–D-Phe]2 crystal. The van der Waals energy changes by 40% as the cutoff distance is increased from 8 to 15 Å. The exact depen-dence of the energy on the cutoff distance depends on the system itself and should be calibrated for each new system.

Figure 13. Number of nonbond interactions as a function of cutoff distance

The number of nonbond pairwise interactions (in millions) expected for a 5000-atom system as a function of cutoff distance. The time required to evalu-ate the total energy of this system is approximately proportional to the number of nonbond interactions.

6 8 10 12 14 nonecutoff distance (Å)

0

5

10

15

nonb

ond

inte

rac

tions

(10

6 )



Atom-based cutoffs and nonbond cutoff terms

Direct cutoff not recom-mended

MSI applications make several methods available for calculation of long-range nonbond interactions. Cerius2•OFF offers (among others) the direct method, which is straightforward and can be applied to nonperiodic and periodic models. Nonbond interac-tions are simply calculated to a cutoff distance and interactions beyond this distance are ignored.

However, the direct method can lead to discontinuities in the energy and its derivatives. As an atom pair distance moves in and out of the cutoff range between calculation steps, the energy jumps, since the nonbond energy for that atom pair is included in one step and excluded from the next. (For small models you may, of course, calculate all nonbond interactions by setting a large enough cutoff distance and using the direct method.)

Minimizing discontinuities in the potential energy sur-face

To avoid the discontinuities caused by direct cutoffs, most simula-tion engines use some kind of switching function (Figure 15) to smoothly turn off nonbond interactions over a range of distances.

Figure 14. Van der Waals energy as a function of cutoff distanceThe van der Waals energy for the hexapeptide crystal, [Ala–Pro–D-Phe]2 as a function of cutoff distance. Note that the van der Waals energy does not con-verge until approximately 20 Å. Simulation done with FDiscover.

0 10 20 30 40–60

–50

–40

–30

cutoff distance (Å)

van

de

r Wa

als

ene

rgy

(kc

al m

ol-1

)



(The variable names and definitions differ among MSI simulation engines, as illustrated in the figure).



Figure 15. Application of a switching functionApplication of a switching function; energy = E(r) · S(r). Variable names in MSI simulation engines that relate to cutoffs are also illustrated. Thick dark curve = the unmodified van der Waals potential; dashed curve = the switching function S(r); grey curve = the resulting, switched potential.

nonbond potential, E(r)

switching function, S(r)

E(r) • S(r)

CUTDIS

SWTDIS

CUTOFF

distance (r)

1.0

0.0

ener

gy

(kca

l mo

l-1)

cutoff

spline width

buffer width

FDiscover

CDiscover

spline-on distance

spline-off distance Cerius2•OFF

ron

roffCHARMm

rcutoff

buffer width



Implementation in Cerius2•OFF

Cerius2•OFF allows you to use a cubic spline switching method, by which the energy is multiplied by a spline function. The inter-action cutoff is defined by two parameters: the spline-on and the spline-off distances (Figure 15). Within the spline-on/spline-off range, the nonbond interaction energy is attenuated by the spline function. Beyond the spline-off distance, nonbond interactions are ignored. The current defaults in the Open Force Field module set a narrow on/off range that results in fast calculations. Using a broader spline range gives more accurate results, but slower calcu-lation.

Eq. 47

Implementation in Dis-cover

In the Discover program the switching function is defined by two variables: the point where the function reaches zero and the range over which the function decreases from one to zero (Figure 15).

The Discover program uses a fifth-order polynomial for the switching function. It is formulated so that the first and second derivatives are zero at both the inner and outer ends of the switch-ing region. Thus, the interaction energy and its first and second derivatives are continuous, although higher derivatives are not.

Implementation in CHARMm

CHARMm and Cerius2•MMFF offer two types of switching func-tions, which its documentation refers to as a switching function (not the same as the Discover switching function) and a shifted potential.

Two variables corresponding to the ends of the switching regions (ron and roff, Figure 15) are required to define the switching func-tion. Depending on the value of rij, the following values are used to multiply individual electrostatic or van der Waals energy terms:

Eq. 48

OFF switching function

1.0

r2off rij–( )2 r2

off 2r2i j 3r2

on–+( )

r2off r2

on–( )3-----------------------------------------------------------------------------------

0.0

=

r2i j r2

on<

r2on r2

ij r2 off< <

r2i j r2

off>

CHARMm switching function

1.0

roff rij–( )2 roff 2rij 3ron–+( )

roff ron–( )3------------------------------------------------------------------------

0.0

=

rij ron<

r on rij roff< <

rij roff>



However, energy can be significant at the cutoff distance, which can result in artificially large forces at long ranges. This is espe-cially true for relatively short (that is, less than 12 Å) cutoff dis-tances and small ranges (that is, when roff – ron < 3 Å).

The shifted potential modifies the radial function so that energy and forces go to zero at some cutoff distance (Eq. 49). The individ-ual electrostatic and van der Waals terms in the energy function are simply multiplied by this term:

Eq. 49

One disadvantage to the CHARMm functions is a discontinuity in the second derivatives at the cutoff distance.

Neighbor lists and buffer widths

To maximize the efficiency of nonbond calculations, Cerius2•OFF, Discover, and CHARMm create a neighbor list that contains all pair interactions to be considered during calculation of the nonbond interactions. Atom pairs are not included in the list if they are too far apart or if they are excluded (Automatic exclusions).

Advantages of a neigh-bor list

Neighbor list generation was chosen over other approaches using cutoffs for computational efficiency:

♦ A pairwise search through all atoms at every step of a calcula-tion is computationally expensive.

♦ During minimization or dynamics, the distances between atoms do not change radically between one step and the next.

The buffer region Although a neighbor list requires time to set up, the net result is time saving for models containing more than about 50 atoms, because the list is not recalculated each time the energy expression is evaluated. Since the list is not updated at every step, it includes atoms in a buffer region (the distance between the two right-most lines in Figure 15) that might move close enough together to con-tribute to the energy calculation before the next update of the neighbor list.

Updating the neighbor list To ensure that no atoms outside the buffer region can move close enough to interact during an energy minimization or molecular dynamics simulation, the nonbond list is automatically updated

1rij

rcutoff--------------

2

–

2



whenever any atom moves more than one-half the buffer width. Thus, the width of the buffer region, coupled with the velocity with which atoms move, determines the maximum amount of time before the neighbor list is updated.

Charge groups and group-based cutoffs

Dipoles must not be split by cutoff distances

To understand the implications of the generalization that the stron-gest electrostatic interactions in many molecules are due to dipoles rather than fully charged groups (see You may use different methods for van der Waals and electrostatic interactions), note that the interac-tion energy for two monopoles, each of one e.u. of charge, is about 33 kcal mol-1 at 10 Å, while that for two dipoles formed from unit monopoles is no more than about 0.3 kcal mol-1. It is clear that ignoring monopole–monopole interactions would give grossly misleading results, whereas ignoring dipole–dipole interactions would be only a modest approximation.

If nonbond cutoffs were applied to such a model on an atom-by-atom basis, this could generate spurious monopoles by artificially splitting dipoles (by having one of a dipole’s atoms inside the cut-off and one outside). Instead of ignoring a relatively small dipole–dipole interaction, this would artificially introduce a large mono-pole–monopole interaction. To avoid these artifacts, the Discover and CHARMm simulation engines can apply cutoffs over charge groups. (In CHARMm in PSF mode, every residue is a charge group.)

Functional groups and charge groups

A charge group is a small group of atoms close to one another which has a net charge of zero or almost zero. Often, charge groups are identical to common chemical functional groups. Thus, a carbonyl group, methyl group, or carboxylic group would be an approxi-mately neutral charge group.

Implementation in Discover—switching atoms

The Discover program designates one atom from each charge group as the switching atom and generates the neighbor list by con-sidering the distance between the switching atoms of two charge groups. If the distance is less than the cutoff distance, then the pair-wise interactions between all atoms in the two groups are included. If the distance is greater than the cutoff, they are all excluded. Similarly, when calculating the actual interaction energy, the Discover program switches off the interactions between all atom–atom pairs in the two charge groups based only on the dis-



tance between the two switching atoms. This procedure prevents artifactual splitting of dipoles.

Implementation in CHARMm

If group-based cutoffs are used in CHARMm, the neighbor list is stored in terms of group pairs.

Charge group size and the cutoff distance

The size of a charge group, as defined by the greatest distance from the switching atom to another atom in the same group, must be significantly smaller than the cutoff distances. Otherwise, an inter-action between two atoms close to each other might be ignored because the switching atoms of the two groups are farther apart than the cutoffs. Typical groups are no more than 1–3 Å large, so cutoffs larger than 7–8 Å are reasonable. However, some models contain considerably larger groups.

The Discover program checks the size of the groups against the cutoff distances, then outputs an error message and terminates if the cutoffs are too short relative to the group size. If this happens, you must either increase the cutoffs or define smaller groups.

Charge group neutrality… The Discover program also warns you about significantly non-neutral groups. Some can be expected if the model contains for-mally charged functional groups, such as protonated amines and carboxylates. However, other non-neutral groups usually indicate an error in group definitions.

… and defining or check-ing charge groups

In Cerius2•Discover, you can specify the tolerance with which neutrality is defined when you ask Discover to perform charge grouping.

In Insight, charge groups and switching atoms are defined, edited, and checked with the Forcefield/Groups parameter block, which is found in the Builder, Biopolymer, and other modules. Potentials and charges for the atoms must be fixed or accepted before defining charge groups.

In CHARMm, you can edit charge groups in the RTF files with any text editor. In PSF mode, every residue is a charge group.

Double cutoffs

The FDiscover program also incorporates an improvement over a single cutoff distance called double cutoffs, or, as it is sometimes called in dynamics, multiple timesteps. The nonbond interaction



potential at a distance is a smooth function that does not vary rap-idly.

With double cutoffs, two cutoff distances—an inner and outer one—are assigned. The two distances define an inner spherical region and an outer shell around a given atom.

Whenever the neighbor list is updated, interactions are calculated in exactly the same way as in the single cutoff scheme, but consid-ering the outer cutoff as the cutoff. Then the resulting interaction energy is partitioned into contributions from atoms within the inner cutoff and atoms within a spherical shell from the inner to the outer cutoff. The contribution from the shell is treated as a con-stant in subsequent molecular dynamics steps, until the next neighbor list update occurs. The basic idea behind double cutoffs is to calculate the shell contribution only when the neighbor list needs to be updated, while calculating inner cutoff contributions at every step.

These double cutoffs make the calculation less expensive by allow-ing smaller values for the inner cutoff than could normally be used with a single cutoff. Accuracy is regained at minimal cost by using a large distance for the outer cutoff.

Discontinuities in the potential energy surface

It is important to realize that the effective potential energy surface is not quite continuous when double cutoffs are used. The magni-tude of the discontinuities depends on the cutoff distances and the system that is being studied.

These discontinuities are only a minor problem for dynamics, where they are manifested as small fluctuations in the total energy.

Their effect during minimization depends on the minimizer that is used, because some minimization algorithms, such as conjugate gradient, are quite sensitive to discontinuities in the surface. Other algorithms, such as steepest descents, are relatively robust.

Tail corrections

Long-range van der Waals interactions in disordered periodic systems

For disordered periodic systems, contributions to the potential energy and pressure from van der Waals interactions outside the cutoff can be written as:



Eq. 50

Eq. 51

where Ni and ρi denote the number and number density of atoms of type i, Uαβ (r) denotes the van der Waals nonbond potential describing interactions between atoms of type α and β, and gαβ (r) denotes the pair correlation function describing the probability of finding α and β at separation r relative to the probability of finding the pair at an infinite distance (McQuarrie, 1976, Chapter 13).

Except in rare cases, the function gαβ (r) is short range, reaching its limiting value of unity at distances of ~10 Å. Moreover, gαβ (r) – 1.0 is small even at shorter distances. In consequence, accurate esti-mates of the tail corrections for all normal nonbond cutoff values may be safely made by setting all gαβ (r) = 1.0 in Eq. 50 and Eq. 51.

Computational costs Note also that applying Eqs. 50 and 51 at each step in a simulation contributes negligibly to the overall simulation cost, since for con-stant-volume simulations the full correction may be precomputed, and in simulations where the volume fluctuates it is necessary only to recompute the volume at each step.

Cell-based cutoffs

CDiscover allows cell-based cutoffs for periodic systems. This is another image-based method, in which the neighbor list is based on a specified number of cell layers surrounding the central cell.

Cell multipole method

Mor rigorous, controllable, and efficient than cutoffs

The cell multipole method (CMM, available only in CDiscover) provides a treatment of the nonbond interactions for both nonpe-riodic and periodic systems that is more rigorous and efficient than cutoffs. This method (Greengard and Rokhlin 1987, Schmidt

∆Utail12--- Nα ρβ4π r2gαβ r( )Uαβ r( ) rd

rc

∞

∫β 1=

ν

∑α 1=

ν

∑=

∆Ptail16--- ρα ρβ4π r2gαβ r( )

r Uαβd r( )rd

----------------------- rd

rc

∞

∫β 1=

ν

∑α 1=

ν

∑=



and Lee 1991, Ding et al. 1992) is a hierarchical approach that allows the accuracy of the nonbond calculation to be controlled. Short-range interactions are treated in the usual way, but long-range group–group interactions are treated in terms of multipoles. Computational time scales as N (the number of atoms).

The cell multipole method applies to the general energy expres-sion of the following form:

Eq. 52

where Φi is the potential at atom i, Rij is the distance between atom i and atom j, p is a number (p = 1 for Coulombic and 6 for London dispersion interactions, for example), and the λ’s are general charges. For Coulombic interactions, the λ’s are real charges.

Near- and far-field poten-tials

The general potential Φi may be divided into a near-field potential due to the surrounding atoms (those within a few angstroms) and a far-field potential due to the rest of the atoms that interact with the ith atom.

The number of interactions in the near field is limited, so it is rela-tively easy to calculate the near-field potential exactly. The number of interactions in the far field is of order N 2, making an exact cal-culation of this potential intractable for large models. The cell mul-tipole method calculates the far-field potential accurately and efficiently in the following manner.

Derivation of cell multi-pole method

We begin by placing an arbitrarily shaped molecule in a rectangu-lar box. The box is then cubed into a number of basic cells of length 4–6 Å and containing 2–4 atoms on average. The basic cell level is denoted level A in Figure 16. Starting from a corner of the box, every eight basic cells may be considered to constitute a larger, par-ent cell, termed level B. Every eight parent cells may constitute a grandparent cell, termed level C. This procedure is repeated until only a few large cells fill the box. For example, considering any atom in cell A0 of the three-level cell system (Figure 16) the other atoms in A0 and all atoms in An contribute to the near-field poten-tial, and the atoms in Af, B, and C contribute to the far-field poten-tial.

E λ iΦi

i 1=

N

∑ λ iλ j

Rijp----------

i j>

∑= =



Key steps used in cell mul-tipole method

The cell multipole method involves the following two key steps:

1. Multipole expansion and calculation of general multipole moments.

The potential associated with each basic cell can be represented as a general potential originating at the center of the cell. This

C C C C C

C

B B B B B B

C

BAf Af Af Af Af Af

B BAf An An An Af Af

C

BAf An A0 An Af Af

B B

CAf An An An Af Af

BAf Af Af Af Af Af

B BAf Af Af Af Af Af

C

B B B B B B

C

B B B B B B

C C C C C

Figure 16. Three-level hierarchical cell systemDefinition of hierarchical cells and division of near field and far field for a basic cell A0. Larger cells are formed as cells are farther from cell A0 (this constitutes the hierarchy). Note that the near field is one layer thick.



potential may be expanded into an infinite series of multipole moments. For example, the potential associated with cell Af in Figure 16 centered at rAf

, is expressed as:

Eq. 53

where R = rAf- r; r is any point outside cell Af; α, β = x, y, z; and

Z, D, and Q are monopoles, dipoles, and quadrupoles, respec-tively.

The potentials associated with the higher-level cells can be expanded in an analogous manner, with moments derived from lower-level cell moments.

2. Generation of Taylor coefficients.

Using this expansion to represent the potentials associated with Af-, B-, and C-level cells, the far-field potential of cell A0 may be obtained by summing all the far-cell contributions. The result-ing potential may now be expanded as a Taylor series about the center of cell A0:

Eq. 54

where rA0 is the position vector of the center of cell A0 and ∆rα

= rα - rA0α. The Taylor coefficients in Eq. 54 are due to all the far-cell contributions.

A key point of the cell multipole method is that, once the set of Tay-lor coefficients is calculated at rA0

, the far-field potential of any atom in cell A0 is obtained easily through Eq. 54.

Since the Taylor coefficients must be generated for every basic cell, another key point of the cell multipole method is efficient genera-tion of these coefficients. A hierarchical procedure is used, in which coefficients determined for higher-level cells are propa-gated to the coefficients for lower-level cells. Thus, coefficients for

ΦAfr( ) Z

Rp-------

DαRα

α

∑Rp 2+-------------------------–

QαβRαRβ

αβ

∑Rp 4+----------------------------------- …–+=

ΦA0r( ) T

0( )rAo

( ) Tα1( )

rA0( )∆rα

α

∑ 12--- Tαβ

2( )rA0

( )∆rα∆rβ

αβ

∑ …+ + +=



a child B cell are obtained by adding contributions directly trans-lated from the C-level coefficients at the center of the parent C cell to the coefficients at the center of B, generated by considering only the B-cell contributions.

Improved computational performance and accu-racy

The cell multipole method is an order-N method (Greengard and Rokhlin 1987, Schmidt and Lee 1991, Ding et al. 1992). The time savings with respect to an exact N 2 algorithm, as well as the improved accuracy relative to using cutoffs, can be dramatic. Table 12 shows results from several calculations on hemoglobin. When the conventional method with 9.5-Å cutoffs is used, the computational and setup times are greatly reduced, but at the cost of a disturbingly large error (over 1% of the correct energies). The last 6 lines of the table show results for second-, third-, and fourth-order multipole expansions at two levels of computational accu-racy. The short-range treatment becomes progressively better towards the bottom of the table. However, the overall CPU time increases. It is practical to achieve essentially exact results (within a fraction of a kcal mol-1) in reasonable times.

For systems larger than hemoglobin, the improvement in perfor-mance can be even more dramatic. For a system ten times larger, the cell multipole method would take 3–10 minutes for the energy evaluation, depending on the accuracy desired. The exact N 2 cal-culation, in contrast, would take about three days!

Table 12. Computational efficiency of cell multipole methodThe results shown are for a calculation with CDiscover 3.1 on hemoglobin (32250 atoms) on an SGI Crimson (50 MHz MIPS R4000),

level of calculation

time (s) error (kcal mol-1)a

setupbenergy eval-

uationc van der Waals Coulomb

exact pairwise calculation 468 2809 0.00 0.009.5-Å cutoff 12.6 30.2 1485 1359coarsed, 2nd-order multi-

pole23.9 15.1 275 -26.0



aRelative to the exact pairwise calculation of the energy.bTime required to set up atom lists and multipole expansions (overhead needed at the beginning of a calculation and periodically during dynamics or minimization).cTime required for recurring evaluation of energy and gradients during a calculation.dThat is, a reasonably accurate, fast calculation with low overhead.eThat is, calculation with highest accuracy and greatest overhead.

Nonbond interaction energies

Due to the nature of the cell multipole method, specific nonbond interaction energies cannot be calculated unless you use the ESFF forcefield. When this method is used with other forcefields, the per-atom energy is calculated by using the cell multipole method, and the nonbond interaction energy is calculated using the group-based method. You can specify cutoffs for the group-based method of nonbond analysis. A large cutoff in the group-based method may give reasonably accurate energies compared with the cell multipole method.

Ewald sums for periodic systems

Nonbond energies of periodic systems

Mainly for crystals The Ewald technique (Tosi 1964, Ewald 1921) is a method for com-putation of nonbond energies of periodic systems. Crystalline sol-ids are the most appropriate candidates for Ewald summation, partly because the error associated with using cutoffs (Nonbond

coarse, 3rd-order multi-pole

58.5 15.1 243 -25.0

coarse, 4th-order multi-pole

199 16.1 243 -3.50

finee, 2nd-order multipole 96.7 57.2 8.95 -10.6fine, 3rd-order multipole 219 56.6 6.95 -2.50fine, 4th-order multipole 718 57.7 0.24 -0.18

Table 12. Computational efficiency of cell multipole methodThe results shown are for a calculation with CDiscover 3.1 on hemoglobin (32250 atoms) on an SGI Crimson (50 MHz MIPS R4000),

level of calculation

time (s) error (kcal mol-1)a

setupbenergy eval-

uationc van der Waals Coulomb



cutoffs) is much greater in an infinite system. The technique can also be applied to amorphous solids and solutions.

Ewald method compared with cutoff-based methods—Coulombic energy…

Figure 17 shows the electrostatic energy for quartz as computed by various techniques. One would feel that all the techniques should converge to the same value at high cutoff distances. However, the direct atom-based cutoffs approach yields results that fluctuate wildly as the cutoff increases, even for rather large cutoffs. The problem is that the sum is only conditionally convergent. As the cutoff increases, charges of opposite sign are taken into account and the partial sum is modified significantly. Worse, reordering the terms of a conditionally convergent series can yield arbitrary results. The problem then is to find physically and chemically meaningful orderings of the series.

Figure 17. Electrostatic energy vs. cutoff distance for quartzThe electrostatic energy of quartz was calculated with CDiscover 3.1 by several methods. Medium line with points: using atom-based cutoffs; thin dark line: using cell-based cutoffs; thick line: using group-based cutoffs; same thick line: by the Ewald method with dipole correction; and medium dashed line: by the Ewald method without dipole correction.

-300

-200

-100

0

35 40 45 50 55 60

ele

ctro

sta

tic e

nerg

y (k

ca

l mo

l-1)

cutoff (Å)



The cell-based (Cell-based cutoffs) and group-based (Charge groups and group-based cutoffs) cutoff techniques are natural candidates. However, they yield somewhat different values (Figure 17), due to the different cutoff conventions employed. The group-based tech-nique computes the result for a sphere, but the cell-based tech-nique computes the result for a parallelepiped that preserves the shape of the unit cell.

A standard Ewald calculation that does not take the dipole moment of the unit cell into account yields yet another value. An Ewald calculation that includes the effect of the dipole moment agrees with the group-based calculation (Figure 17).

…and van der Waals energy

For van der Waals energy, the energy sum is absolutely conver-gent, and no chaotic behavior arises from the direct approach. Even so, as Figure 18 indicates, the convergence of the dispersive energy is slower than might be expected. Even with a cutoff dis-tance of 30 Å, the error is a significant fraction of 0.1 kcal mol-1. (The Ewald calculation is less costly for comparable accuracy.) The repulsive energy, on the other hand, converges at a cutoff distance of only 15 Å and needs no special treatment. (Atom-based calcula-tions for much larger systems, however, show that sometimes even the repulsive energy can exhibit a surprisingly high error at a cutoff of 12 Å.)



Theory of Ewald technique

For full details on the Ewald summation method and parameter optimization procedure used in MSI’s simulation engines, please refer to Karasawa and Goddard (1989).

The Ewald approach to improving convergence is to multiply a general lattice sum:

Eq. 55

by a convergence function φ(r), which decreases rapidly with r. Of course, to preserve equality, one must then add a term equal to the product of 1 - φ(r) with the lattice sum:

Figure 18. van der Waals energy vs. cutoff distance for NaClThe graph shows the (solid lines) dispersive and (dashed line) repulsive portions of the van der Waals energy as a function of the cutoff distance, as calculated by the (thin lines) atom-based and (thick line) Ewald methods. The Ewald calcu-lation was performed with CDiscover 3.1 to an accuracy of 1 e-6, which requires a cutoff distance of 9.5 Å.

-33.4

-33.0

-32.6

5 10 15 20 25 30 35

dis

pe

rsiv

e e

nerg

y (k

ca

l mo

l-1)

cutoff (Å)

17.336

17.338

17.340

rep

ulsi

ve e

nerg

y (k

ca

l mo

l-1)

Sm12---

Aij

ri rj– RL– m----------------------------------

L i j, ,

∑=



Eq. 56

Here, the first term converges quickly, because φm(r) decreases rap-idly. Ewald’s insight was that the second term can be Fourier trans-formed to provide a rapidly converging sum over the reciprocal lattice. The sum over L in Eq. 56 runs over all lattice vectors, but the i = j terms must be omitted when L = 0.

The convergence func-tions

The convergence functions are, for the electrostatic energy:

Eq. 57

and for the dispersive energy:

Eq. 58

Optimizing computational effort

The electrostatic convergence function φ1 was also used by Catlow and Norgett (1976) and Karasawa and Goddard (1989). The disper-sive convergence function φ6 was recommended and used by Karasawa and Goddard. The convergence parameter η plays a similar role in both cases. As η increases, the real-space sum con-verges more rapidly and the reciprocal space sum converges more slowly. (That is, a large η implies a heavy computational load for reciprocal space, and a small η implies a heavy computational load for real space.) Cutoffs must be adjusted accordingly, and process-ing time is affected by the cutoffs. A value of η that balances pro-cessing in the real and reciprocal spaces proves to be optimal. The same value of η can be used for both the dispersive and electro-static energy, and thus they can be combined for greater efficiency.

Implementation in Dis-cover

The Discover program automatically chooses η so as to balance the computational loads for real and reciprocal space.

Implementation in Cerius2• OFF

Cerius2•OFF instead uses the inverse of η as input, that is, the ratio of the time required for a real-space calculation to the time for

Sm12---

Aijφm ri rj– RL–( )

ri rj– RL– m-------------------------------------------------

L i j, ,

∑ 12---

Aij 1 φm ri rj– RL–( )–( )

ri rj– RL– m---------------------------------------------------------------

L i j, ,

∑+=

φ1 erfc ηr( ) 1 erf ηr( )–2

π------- exp s2–( ) sd

η r

∞

∫= = =

φ6 r( ) 1 ηr( )2 12--- ηr( )4+ +

exp ηr( )2–( )=



a reciprocal-space calculation. The value chosen for the time ratio does not affect the accuracy of the calculation, only the time taken to perform it. Real-space calculations typically take longer than reciprocal ones, so the value of the time ratio is usually greater than 1.

Electrostatic energy The Ewald expression for the electrostatic energy is (dropping a factor of 1/4πε0):

Eq. 59

where a = η | ri - rj - RL |; ri = Hsi; h = 2π(HT)-1n (reciprocal lattice vectors); Ω = det(H) = cell volume; and b = h ⁄ 2η.

In Eq. 59, the first term corresponds to the real-space sum, the con-tribution (L = 0, i = j) being omitted. The second term in Eq. 59 cor-responds to the reciprocal-space sum. For electrostatic interactions, the double sum over i and j is reduced to a single sum. This provides a substantial performance improvement (N1.5 instead of N2.5, where N is the number of atoms per unit cell). A similar reduction of the reciprocal-space sum occurs for the disper-sive energy only if the geometric combination rule ( ) holds. The third term in Eq. 59 arises from the self-energy of the charge distributions produced by the introduction of the conver-gence function. The final term of Eq. 59 is zero if the unit cell is charge neutral, which is normally the case. Indeed, the electro-static energy in an infinite system of non-neutral cells is formally infinite. However, some applications can involve the use of non-neutral cells. For example, Cation Locator adds cations to the sys-tem one by one, and neutrality is not achieved until the final cation is added. The effect of the final term in the equation is to give con-vergence to a finite value of the energy, which corresponds to a sys-tem where the excess charge is neutralized by a compensating uniform background charge density.

EQ

η2--- qiqj

erfc a( )a

------------------2πΩ------ qi h ri•( )cos

i

∑2

qi h ri•( )sin

i

∑2

+ exp b2–( )

h2-----------------------

h 0≠

∑+

L i j, ,

∑ηπ

------- qi2

i

∑ Π2Ω-------– η qi

i

∑ 2

–

=

Bij BiiBjj=



The Ewald sum as it appears in Eq. 59, with no h = 0 term, strictly represents an infinite crystal. A real macroscopic but finite crystal also includes surface contributions, which can be substantial (Deem et al. 1990) and which depend on the dipole moment of the unit cell and the shape of the crystal. However, in a real environ-ment, physical effects such as surface reconstruction and dielectric effects in the surrounding medium serve, in general, to diminish the surface charge. The Cerius2•OFF and CDiscover programs therefore omit such terms, corresponding to the so-called “tin-foil” boundary conditions.

Accuracy of Ewald calculations

You choose the accuracy before beginning calcula-tion

The Ewald method allows you to select, before running the calcu-lation, a level of accuracy for the calculation. (Estimation of the cutoff and convergence constants is difficult, so a facility to auto-matically calculate these parameters to a certain accuracy (Karasawa & Goddard 1989) is provided instead.)

Depending on the system, an Ewald calculation with accuracy = 1 e-4 can be comparable in performance to an atom-based calcula-tion with a large cutoff (19 Å) over the range that has been tested. In addition, the Ewald results are significantly more accurate.

Computation time vs. accuracy and model size

Ewald processing time grows as N1.5, where N is the number of atoms in the unit cell. Increases in accuracy do not require unrea-sonable increases in the Ewald lattice cutoff. For acetic acid, for example, increasing the accuracy by 2 orders of magnitude (from 1 e-2 to 1 e-4), with constant repulsive cutoff, increased processing time only about 1.5 fold (from 22.08 to 35.34 seconds) and increased the Ewald lattice cutoff less than 20% (from 11.7 to 13.4 Å).

Troubleshooting Although the default Ewald accuracy is acceptable for most single-point energy calculations, tighter accuracy may be required for some minimization and dynamics runs, to assure acceptable gra-dient accuracy. A value as low as 0.00025 may be preferable if your minimization run fails to converge or your dynamics run misbe-haves.

Nonbond interaction energies

Due to the nature of the Ewald sum method, specific nonbond interaction energies cannot be calculated. When this method is used, the per-atom energy is calculated by using the Ewald sum method, and the nonbond interaction energy is calculated using



the group-based method. You can specify cutoffs for the group-based method of nonbond analysis. A large cutoff in the group-based method may give reasonably accurate energies compared with the Ewald sum method.

Ewald sum for models with 2D periodicity

Only for Coulombic terms If the model has 2D periodicity, an Ewald sum may be applied to the Coulombic terms, using the method of F. Harris (communica-tion). A non-Ewald method must be used for the van der Waals terms in this case, and a large cutoff may be required to obtain good accuracy.

Slow but accurate The method is similar in principle to the 3D Ewald sum, but more complex in that one direction must be treated nonperiodically. To avoid loss of accuracy, no cutoff is applied in the nonperiodic direction. However, the method can still be optimized, since fewer terms are required in the periodic sums for those atom pairs hav-ing a large separation in the nonperiodic direction. Although the method is slower than a non-Ewald method (or a 3D Ewald sum), it provides very good accuracy.

Activated automatically The method is available in Cerius2•OFF and is activated automat-ically if you select the Ewald method for Coulomb terms when the current model has 2D periodicity.


4 Minimization

This chapter explains This chapter concentrates on the static information that can be extracted from the potential energy surface, as well as on the algo-rithms used for this purpose. The main areas that are covered—minimization and harmonic vibration calculations—are usually lumped together as molecular mechanics, to differentiate them from molecular dynamics calculations (Molecular Dynamics), in which the time evolution of the system is considered.

This chapter explains how minimization is implemented and pre-sents general procedures for performing minimization calcula-tions. To make the most effective use of minimization, you should read this entire chapter, which includes:

General minimization process

Minimization algorithms

General methodology for minimization

Energy and gradient calculation

Vibrational calculation

Related information Forcefields and Preparing the Energy Expression and the Model focus on the representation of the potential energy surface and the useful ways that it can be biased through the addition of restraints and constraints, as well as other information on preparing the model for calculations.

Specific information For specific information on setting up and running minimizations with the various MSI simulation engines, please see the relevant documentation (see Available documentation).


4. Minimization

Uses of minimization An important method for exploring the potential energy surface is to find configurations that are stable points on the surface. This means finding a point in the configuration space where the net force on each atom vanishes. By adjusting the atomic coordinates and unit cell parameters (for periodic models, if requested) so as to reduce the model potential energy, stable conformations can be identified.

Perhaps more important, the addition of external forces to the model in the form of restraints (Preparing the Energy Expression and the Model) allows for the development of a wide range of modeling strategies using minimization strategies as the foundation for

Table 13. Finding information in Minimization section


Simple 2D illustrations of minimization algorithms. Specific minimization example.Minimization algorithms used by MSI simultion engines. Table 14.The steepest-descents method of minimization. Steepest descentsThe conjugate-gradient method. Conjugate gradient.The full, iterative Newton–Raphson method. Iterative Newton-Raphson method.Quasi (or pseudo) Newton–Raphson methods. Quasi-Newton–Raphson.Flowcharts for Newton–Raphson methods. Figure 24; Figure 25.Truncated Newton–Raphson methods. Truncated Newton–Raphson.Before you begin a minimization. General methodology for minimization.Using MSI’s simulation engines for minimization. Minimizations with MSI simulation engines.Meaning of the minimized structure and its calculated

energy.Significance of minimum-energy structure.

Using the various minimization algorithms. When to use different algorithms.Precautions with large models. When to use different algorithms.Precautions with models having bad conformations. When to use different algorithms.Convergence problems. Starting structures and choice of force-

field; Failure to converge.Disk-space storage requirements. Conjugate gradient vs. Newton–Raphson

and disk space.Using different constraints/restraints at different stages

of minimization.When to use constraints/restraints.

Ending a run. Convergence criteria.Using MSI’s minimization engines for vibrational calcu-

lations.General methodology for vibrational cal-

culations.



answering specific questions. For example, the question “How much energy is required for one molecule to adopt the shape of another?” can be answered by forcing specific atoms to overlap atoms of a template structure during an energy minimization.


Energy evaluatiom Minimization of a model is done in two steps. First, the energy expression (an equation describing the energy of the system as a function of its coordinates) must be defined and evaluated for a given conformation. Energy expressions may be defined that include external restraining terms to bias the minimization, in addition to the energy terms.

Conformation adjustment Next, the conformation is adjusted to lower the value of the energy expression. A minimum may be found after one adjustment or may require many thousands of iterations, depending on the nature of the algorithm, the form of the energy expression, and the size of the model.

The efficiency of the minimization is therefore judged by both the time needed to evaluate the energy expression and the number of structural adjustments (iterations) needed to converge to the min-imum.

Specific minimization example

To introduce the various minimization algorithms, the application of each algorithm to the minimization of a pure quadratic function in two dimensions is discussed. Although the energy surface is most certainly anharmonic in regions away from the minimum, it may be considered to be locally harmonic at the minimum.

A simple illustration in two dimensions

Rather than a complex energy expression, the target function used in this illustration is an elliptical surface in two dimensions, as described by Eq. 60:

Eq. 60E x y,( ) x2 5y2+=


4. Minimization

Begin with the function and an initial guess of its minimum

This simple function illustrates the properties of the minimization algorithms and captures the mathematical essence of the formula-tions. Every minimization begins with some energy expression analogous to Eq. 60. In addition to an energy expression defining the energy surface, a starting set of coordinates—an initial guess—for (x,y) must be provided.

Figure 19 is a contour plot of the energy E in the (x,y) plane. Each ellipse is spaced two energy units apart and represents a locus of points having the same energy. (This is analogous to a contoured topographical map.)

Of course, the minimum of this simple function is trivial and can be deduced by inspection to be (0,0).

The minimizer must find the direction to the minimum and its distance from the initial guess

Given an energy expression that defines the energy surface (such as in Figure 19) and an initial starting point, a minimizer must determine both the direction towards a minimum and the distance to the minimum in that direction. A good initial direction is simply the slope or derivatives of the function at the current point. The derivatives of Eq. 60 are a two-dimensional vector:

Figure 19. Energy contour surface of a simple functionAn energy contour surface for the function x2 + 5y2. Each contour represents an increase of two arbitrary energy units.

4 6 8 10 12 16 202

0.0 5.0-5.0-2.0

0.0

2.0

x

y 14



Eq. 61

Here, the derivatives are proportional to the coordinates, so that, the further you are from the minimum, the larger the derivatives.

Improve efficiency by find-ing how the derivatives change

The derivatives, however, merely point downhill and not neces-sarily towards the minimum (see Figure 20). Thus, as you move in the direction of the initial derivatives, the new derivatives change and point in yet a new direction. In order to improve the efficiency, the more sophisticated algorithms such as conjugate gradients and Newton–Raphson use information on how the derivatives change, to determine the direction.

Line search

Before detailing the different algorithms, the concept of a line search is introduced. Line searches are an implicit component of most minimizers.

Most minimizers use line searches

Minimizers usually have two major components. The more generic part is the so-called line search, which actually changes the coordinates to a new lower-energy structure. As an illustration, consider Figure 20 in which the gradient direction from an arbi-trary starting point has been superimposed on our elliptic func-tion. The starting point (x0, y0) is defined as point a.

E∇ 2x 10y,( )=


4. Minimization

What is a line search? In simple terms, a line search amounts to a one-dimensional mini-mization along a direction vector determined at each iteration. For the path shown in Figure 20, it would be along derivative vector (2x,10y), and the one-dimensional surface can be expressed para-metrically in terms of coordinate α (see also Figure 21):

Eq. 62

where (x′, y′) are coordinates along the line away from the current point (x0, y0) in the direction of the derivative at (x0, y0).

If the energy of these new points is calculated and plotted as a function of α, the curve in Figure 21 is obtained.

Figure 20. Energy surface for Eq. 60The derivative vector from the initial point a (x0, y0) defines the line search direc-tion. Note that the derivative vector does not point directly toward the mini-mum. Compare this representation with that in Figure 21, where the line (b–a–c–d) is searched in one dimension for the minimum. Note that the minimum (point c) occurs precisely at the point where the derivative vector is tangent to the energy contours, which implies that the subsequent derivative vectors are orthogonal to the previous derivatives.

0.0 5.0-5.0-2.0

0.0

2.0

x

y

d

c

a (x0,y0)

b

x ′ y ′,( ) x0 αx∂

∂E

x0 y0,+ y0 α

y∂∂E

x0 y0,+,

=



The minimum along α (that is, c) coincides with the point at which the line is tangential to the energy contour. Because the maximum derivative’s direction is perpendicular to the search line at this point, each line search is orthogonal to the previous one. This is an important property of line searches, which is also included in the discussion of the conjugate gradient algorithm under Conjugate gradient.

Efficiency and cost of extensive line searches

Line searches do not depend on the algorithm that generated the direction vector. The general strategy is to simply bracket the one-dimensional minimum between two points higher in energy, for example points b and d in Figure 21. Then, by a set of successive iterations, the actual minimum is approached (e.g., starting at point a, the first step might lead to b, then the direction might reverse to lead to d and finally to c). Extensive line searches are attractive because they extract all the energy from one direction before moving on to the next. Also, the fact that the new deriva-tives are always perpendicular to the previous directions produces an efficient path to the minimum for surfaces that are approxi-mately quadratic. In practice, however, line searches are costly in

Figure 21. Cross section of the energy surface as defined by the intersection of the line search path in Figure 20 with the energy surfaceThe independent variable α is a one-dimensional parameter that is adjusted so as to minimize the value of the function E (x’, y’), where x’ and y’ are parame-terized in terms of α in Eq. 62. Point a corresponds to the initial point (when α is 0), and point c is the local one-dimensional minimum. Points b and d, along with a, bound the minimum and form the basis for an iterative search for the mini-mum.

-2 0 2 4 66

7

8

9

αE

( x′, y

′)

b

c

da(x0,y0)


4. Minimization

terms of the number of function evaluations that must be per-formed. The energy must be evaluated at 3–10 points to precisely locate the one-dimensional minimum, and thus extensive line searches are inefficient.


Only minimization algorithms used in MSI’s simulation engines (Table 14) are considered here:

Steepest descents

Conjugate gradient

Newton–Raphson methods

“Iteration” defined To be consistent in discussions of efficiency, a minimization iteration must be explicitly defined. That is, an iteration is complete when the direction vector is updated. For minimizers using a line search, each completed line search is therefore an iteration. Iterations should not be confused with function evaluations.

Table 14. Minimization algorithms used by MSI simulation engines

algorithm variantsimulation engine

CHARMm Discover OFF MMFF93

steepest descents √ √ √ √conjugate gradient Polak–Ribiere √ √

Fletcher–Reeves √ √ √ √Powell √ √

Newton–Raphson full, iterative √ √ √ √BFGS (quasi) √ √DFP (quasi) √ √truncated √ √ √ABNR √ √ √



Choosing the minimiza-tion algorithm(s)

Access to controls used to specify what minimizer(s) to use is pre-sented under General methodology for minimization.

Steepest descents

In the steepest-descents method, the line search direction is defined along the direction of the local downhill gradient –∇E(xi, yi). Figure 22 shows the minimization path followed by a

steepest-descents approach for the simple quadratic function. As expected, each line search produces a new direction that is perpen-dicular to the previous gradient; however, the directions oscillate along the way to the minimum. This inefficient behavior is charac-teristic of steepest descents, especially on energy surfaces having narrow valleys.

Figure 22. Minimization path following a steepest-descents pathWhen complete line searches starting from point a are used, the minimum is reached in about 12 iterations. Here, where a rigorous line search is carried out, approximately 8 function evaluations are needed for each line search using a quadratic interpolation scheme. Note how steepest descents consistently over-shoots the best path to the minimum, resulting in an inefficient, oscillating trajec-tory.

0.0 5.0-5.0-2.0

0.0

2.0

x

y

a


4. Minimization

Increased efficiency with truncated line searches

What would happen if the line search were eliminated and the position would simply be updated any time that the trial point along the gradient had a lower energy? The advantage would be that the number of function evaluations performed per iteration would be dramatically decreased. Furthermore, by constantly changing the direction to match the current gradient, oscillations along the minimization path might be damped.

The result of such a minimization is shown in Figure 23. The min-imization begins from the same point as in Figure 22, but each line search uses, at most, two function evaluations (if the trial point has a higher energy, the step size is adjusted downward and a new trial point is generated) (Levitt and Lifson, 1969). Note that the steps are more erratic here, but the minimum is reached in roughly the same number of iterations. The critical aspect, however, is that by avoiding comprehensive line searches, the total number of func-tion evaluations is only 10–20% of that used by the rigorous line search method.



A slow but robust method The exclusive reliance of steepest descents on gradients is both its weakness and its strength. Convergence is slow near the minimum because the gradient approaches zero, but the method is extremely robust, even for systems that are far from harmonic. It is the method most likely to generate a lower-energy structure regard-less of what the function is or where the process begins. Therefore, the steepest-descents method is often used when the gradients are large and the configurations are far from the minimum. This is commonly the case for initial relaxation of poorly refined crystal-lographic data or for graphically built models. In fact, as explained in the following sections, more advanced algorithms are often designed to begin with steepest descents as the first step.

Tip

Figure 23. Minimization path following a steepest-descents path without line searches

The searching starts from point a and converges on the minimum in about 12 iterations. Although the number of iterations is slightly larger than in Figure 22, the total minimization is five times faster since, on average, each iteration uses only 1.3 function evaluations. Note that, in most applications in molecular mechan-ics, the function evaluation is the most time-consuming portion of the calcula-tion.

0.0 5.0-5.0-2.0

0.0

2.0

x

ya


4. Minimization

Conjugate gradient

The reason that the steepest-descents method converges slowly near the minimum is that each segment of the path tends to reverse progress made in an earlier iteration. For example, in Figure 22, each line search deviates somewhat from the ideal direction to the minimum. Successive line searches correct for this deviation, but they cannot efficiently correct because each direction must be orthogonal to the previous direction. Thus, the path oscillates and continually overcorrects for poor choices of directions in earlier steps.

Increasing the efficiency of line searches by con-trolling the choice of new direction

It would be preferable to prevent the next direction vector from undoing earlier progress. This means using an algorithm that pro-duces a complete basis set of mutually conjugate directions such that each successive step continually refines the direction toward the minimum. If these conjugate directions truly span the space of the energy surface, then minimization along each direction in turn must by definition end in arriving at a minimum. The conjugate gradient algorithm constructs and follows such a set of directions.

In conjugate gradients, hi+1, the new direction vector leading from point i+1, is computed by adding the gradient at point i+1, gi+1, to the previous direction hi scaled by a constant γi:

Eq. 63

Polak–Ribiere method where γi is a scalar that can be defined in two ways. In the Polak–Ribiere method, γi is defined as:

Eq. 64

The conjugate gradient method (below) and the iterative and quasi Newton–Raphson methods assume that the conformation is close enough to a local minimum that the potential energy surface is very nearly quadratic. Hence, steepest descents should generally be used for the first 10–100 steps of minimization (depending on the size of the model and how distorted its starting conformation is). Please see also When to use different algorithms.

hi 1+ gi 1+ γihi+=

γi

gi 1+ gi–( )gi 1+

gi gi⋅---------------------------------------=



Fletcher–Reeves method And in the Fletcher–Reeves method (1964), γi is defined as:

Eq. 65

(Fletcher 1980). Although the two conjugate gradient methods have similar characteristics, one or the other might behave better in certain cases.

This direction is then used in place of the gradient in Eq. 62, and a new line search is conducted. This construction has the remarkable property that the next gradient, gi+1, is orthogonal to all previous gradients, g0, g1, g2, …, gi, and that the next direction, hi+1, is con-jugate to all previous directions, h0, h1, h2, …, hi. Thus, the term conjugate gradient is somewhat of a misnomer. The algorithm pro-duces a set of mutually orthogonal gradients and a set of mutually conjugate directions. This method converges in approximately N steps, where N is the numbr of degrees of freedom.

Powell method The Powell method (used in CHARMm and Cerius2•MMFF; see Powell 1977 and Gunsteren & Karplus 1980) is essentially a strat-egy for handling convergence problems.

Conjugate gradient best for large models

Conjugate gradients is the method of choice for large models because, in contrast to Newton–Raphson methods, where storage of a second-derivative matrix (N (N + 1) ⁄ 2) is required, only the previous 3N gradients and directions have to be stored. However, to ensure that the directions are mutually conjugate, more com-plete line search minimizations must be performed along each direction. Since these line searches consume several function eval-uations per search, the time per iteration may be longer for conju-gate gradients than for steepest descents. This is more than compensated for by the more efficient convergence to the mini-mum achieved by conjugate gradients.

Tip

γi

gi 1+ gi 1+⋅gi gi⋅

----------------------------=

The conjugate gradient method can be unstable if the conformation is so far away from a local minimum that the potential energy surface is not nearly quadratic. Steepest descents (Steepest descents) should generally be used for the first 10–100 steps of minimization. Please see also When to use different algorithms.


4. Minimization

Newton–Raphson methods

Iterative Newton-Raphson method

Using the second deriva-tives to accelerate con-vergence

As a rule, N 2 independent data points are required to numerically solve a harmonic function with N variables. Since a gradient is a vector N long, the best you can hope for in a gradient-based mini-mizer is to converge in N steps. However, if you can exploit sec-ond-derivative information, a minimization could ideally converge in one step, because each second derivative is an N × N matrix. This is the principle behind the variable metric minimiza-tion algorithms, of which Newton–Raphson is perhaps the most commonly used.

Another way of looking at Newton–Raphson is that, in addition to using the gradient to identify a search direction, the curvature of the function (the second derivative) is also used to predict where the function passes through a minimum along that direction. Since the complete second-derivative matrix defines the curvature in each gradient direction, the inverse of the second-derivative matrix can be multiplied by the gradient to obtain a vector that translates directly to the nearest minimum. This is expressed mathematically as:

Eq. 66

where rmin is the predicted minimum, r0 is an arbitrary starting point, A(r0) is the matrix of second partial derivatives of the energy with respect to the coordinates at r0 (also known as the Hessian matrix), and ∇ E(r0) is the gradient of the potential energy at r0.

Iteration required because energy surface is not harmonic

The energy surface is generally not harmonic, so that the mini-mum-energy structure cannot be determined with one Newton–Raphson step. Instead, the algorithm must be applied iteratively:

Eq. 67

Thus, the ith point is determined by taking a Newton–Raphson step from the previous point (i - 1). Similar to conjugate gradients, the efficiency of Newton–Raphson minimization increases as con-vergence is approached (Ermer 1976).

rmin r0 A 1– r0( )– E r0( )∇⋅=

ri ri 1– A 1– ri 1–( )– E ri 1–( )∇⋅=



Drawbacks of pure New-ton–Raphson method

As elegant as this algorithm appears, its application to molecular modeling has several drawbacks. First, the terms in the Hessian matrix are difficult to derive and are computationally costly for molecular forcefields. Furthermore, when a structure is far from the minimum (where the energy surface is anharmonic), the mini-mization can become unstable.

For example, when the forces are large but the curvature is small, such as on the steep repulsive wall of a van der Waals potential, the algorithm computes a large step (a large gradient divided by the small curvature) that may overshoot the minimum and lead to a point even further from the minimum than the starting point. Thus, the method can diverge rapidly if the initial forces are too high (or the surface too flat).

Finally, calculating, inverting, and storing an N × N matrix for a large system can become unwieldy. Even taking into account that the Hessian is symmetric and that each of the tensor components is also symmetric, the storage requirements scale as approximately 3N 2 for N atoms. Thus, for a 200-atom system, 180,000 words are required. The Hessian alone for a 1,000-atom system already approaches the limits of a Cray-XMP supercomputer, and a 10,000-atom system is currently intractable.

Pure Newton–Raphson is reserved primarily for calculations where rapid convergence to an extremely precise minimum is required, for example, from initial derivatives of 0.1 kcal mol-1 Å-1 to 10-8 kcal mol-1 Å-1. Such extreme convergence is necessary when performing vibrational normal mode analysis, where even small residual derivatives can lead to errors in the calculated vibrational frequencies.

Tip

Variants of iterative Newton-Raphson method

In addition to the iterative Newton–Raphson method, variants of the Newton method are available (Table 14): the quasi-Newton

The iterative Newton–Raphson method can be unstable if the conformation is so far away from a local minimum that the potential energy surface is not nearly quadratic. Steepest descents (Steepest descents) should generally be used for the first 10–100 steps of minimization. Please see also When to use different algorithms.


4. Minimization

(which includes the BFGS and DFP methods) and the truncated Newton methods. These variants, as well as others, are character-ized by the use of the general algorithm shown in Figure 24.

The differences among the various Newton methods revolve around:

♦ How A is calculated and how its positive-definite character is preserved during the minimization.

♦ How the search direction is determined and to what accuracy (i.e., φk).

♦ How the line search is carried out and how exact it is.

Quasi-Newton–Raphson The quasi-Newton–Raphson method follows the basic idea of the conjugate-gradients method by using the gradients of previous iterations to direct the minimization along a more efficient pathway. However, the use of the gradients

Figure 24. General algorithm for variants of Newton–Raphson method

1. Supply an initial guess r0.

2. Test for convergence.

3. Compute an approximation Hessian A that is positive-definite.

4. Solve for the search direction pk so that:

where φk is some prescribed quantity that controls the accuracy of the computed pk.

5. Compute an appropriate step length λk so that the energy decreases by a sufficient amount.

6. Increment the coordinates:

7. Go to Step 2.

Akpk gk+ φk gk<

xk 1+ xk λkpk+=



is within the Newton framework. In particular, a matrix B approx-imating the inverse of the Hessian (A-1) is constructed from the gradients using a variety of updating schemes. This matrix has the property that, in the limit of convergence, it is equivalent to A-1, so that, in this limit, the method is equivalent to the Newton–Raph-son method. Another property of B is that it is always positive-def-inite and symmetric by construction, so that successive steps in the minimization always decrease the energy.

Of the several different updating schemes for determining B, the two most common ones are the Broyden, Fletcher, Goldfarb, and Shanno (BFGS, also known as VA09A) and the Davidon, Fletcher, and Powell (DFP) algorithms. The BFGS method uses a Fletcher–Powell algorithm with approximate second derivatives.

DFP updating scheme Defining δ and γ as the changes in the coordinates and gradients for successive iterations, the approximate Hessians (B-1) are given by the following in the DFP method:

Eq. 68

BFGS updating scheme and in the BFGS method:

Eq. 69

In practice, the BFGS method is preferred over the DFP method, because BFGS has been shown to converge globally with inexact line searches, while DFP has not.

Advantages and disad-vantages

The quasi-Newton–Raphson method has an advantage over the conjugate-gradient method in that it has been shown to be qua-dratically convergent for inexact line searches. Like the conjugate-gradient method, the method also avoids calculating the Hessian. However, it still requires storage proportional to N 2 (N = number of degrees of freedom), and the updated Hessian approximation may become singular or indefinite even when the updating scheme guarantees hereditary positive-definiteness. Finally, the behavior may become inefficient in regions where the second derivatives change rapidly. Thus, this minimizer is used in practice

Bk 1+ BkδδT

δTγ---------

BkγγTBk

γTBkγ---------------------–+=

Bk 1+ Bk 1γT

Bkγ

δTγ-----------------+

δδT

δTγ---------

δγTBk BkγδT

+

δTγ-------------------------------------

–+=


4. Minimization

as a bridge between the iterative Newton–Raphson and the conju-gate-gradient methods.

Tip

Truncated Newton–Raphson The truncated Newton–Raphson method (Figure 25) differs from the quasi-Newton–Raphson method in two respects:

♦ First, the solution of the line search direction is done iteratively using the conjugate-gradient method.

♦ Second, the elements in the Hessian are not constructed from previous gradients.

The quasi Newton–Raphson method can be unstable if the conformation is so far away from a local minimum that the potential energy surface is not nearly quadratic. Steepest descents (Steepest descents) should generally be used for the first 10–100 steps of minimization. Please see also When to use different algorithms.



Figure 25. Flowchart of truncated Newton–Raphson method

1. Initialize variables for the outer Newton loop.2. Calculate gradient gk, Hessian Hk, and preconditioner Mk.

Test for Newton convergence (gkmax ≤ ε) and exit if true.

3. Determine line search direction pk by using the conjugate-gradient method to solve:

a. Initialize conjugate gradient variables for inner conjugate-gradients loop.b. Calculate conjugate-gradient gradient (r0 = -gk):

and construct the Newton line search direction using the conjugate-gradient direction:

c. Test for convergence of the conjugate-gradient inner loop:if:

then:

and go to Step 4.d. Begin next iteration of conjugate gradients.

Solve for Mzk = rk for zk and construct new conjugate direction:

4. Next iteration in the outer Newton loop:Determine length of step along line search direction and increment coordinates:

5. Go to Step 2.

Mk1– Hkpk Mk

1– gk–=

rj 1+ rj γjHdj–=

γj

rjTzj

djTHdj

---------------=

pj 1+ pj γjdj+=

rj 1+ φ gk≤

pk pj 1+=

dj 1+ zj 1+ βjdj+=

βj rj 1+T

zj 1+

rjTzj

-----------=

xk 1+ xk λkpk+=


4. Minimization

Increased stability and speed

By using the second derivatives to generate the Hessian, the mini-mization is more stable far away from the minimum or in regions where the derivatives change rapidly. Since solving Eq. 67 by inversion is not tractable for large models, the search direction is solved iteratively using the conjugate-gradient method. Further-more, to increase the speed in solving the search direction, the tol-erance for conjugate-gradient convergence is dependent on the proximity to the minimum. The tolerance is relatively large when the minimization is far from the solution and decreases during convergence. This is appropriate, because the dependency of con-vergence on the line search direction becomes greater at the end of the minimization. At the beginning, it is more efficient to take more less-well-defined Newton steps than to take fewer well-defined steps.

To further increase the convergence rate of the conjugate-gradient minimization, the Newton equation that is solved for each line search direction is preconditioned with a matrix M. The matrix M may range in complexity from the identity matrix (M = I) to M = H. In the first limit, the Newton equation remains unchanged. However, in the second, there is no saving in computational effort, because M-1 H = I must be solved.

Please see also When to use different algorithms.

ABNR similar to truncated Newton–Raphson

The adopted-basis Newton–Raphson method (ABNR) is similar to the truncated Newton–Raphson method. The ABNR method per-forms energy minimization using a Newton–Raphson algorithm applied to a subspace of the coordinate vector spanned by the dis-placement coordinates of the last positions. The second derivative matrix is constructed numerically from the change in the gradient vectors, and is inverted by an eigenvector analysis that allows the routine to recognize and avoid saddle points in the energy surface. At each step, the residual gradient vector is calculated and used to add a steepest-descents step, incorporating new direction into the basis set.

Because ABNR avoids the large storage requirements of the full Newton–Raphson second derivative method, larger systems can be minimized more efficiently. ABNR is the method of choice if storage is a problem.




Many issues are involved in designing an appropriate simulation strategy for a given model, some of which have to do only with the minimization algorithms themselves:

Minimizations with MSI simulation engines

When to use different algorithms

Convergence criteria

Significance of minimum-energy structure

Prerequisites One of the most important steps in any simulation is properly pre-paring the model to be simulated. Calculations on the fastest com-puter running the most efficient minimization algorithm may be worthless if the hydrogen is put on the wrong nitrogen or an important water molecule is omitted.

Unfortunately, it is impossible to provide a single recipe for a suc-cessful model—too much depends on the objectives and expecta-tions of each calculation. Are energies to be compared quantitatively? What is the hypothesis being tested? The effects of tethering, fixing, energy cutoffs, etc. on the results can be answered only by controlled preliminary experiments.

Considerations For simulation strategies that involve minimization, several con-siderations must be addressed, including:

♦ When to use constraints and restraints (When to use constraints/restraints).

♦ Which minimization algorithm(s) to use (When to use different algorithms).

♦ What criteria to use for judging convergence of the minimiza-tion (Convergence criteria).

♦ The significance of the minimum-energy structure and its cal-culated energy (Significance of minimum-energy structure).


4. Minimization

Minimizations with MSI simulation engines

Prerequisites To set up a minimization run, first:

1. Choose the desired forcefield if you don’t want to use the default forcefield (Forcefields).

2. Set up the forcefield and prepare your model (Preparing the Energy Expression and the Model).

Accessing minimization controls

Next:

3. Specify items such as the minimization algorithm(s) (Minimization algorithms and When to use different algorithms) and run-termination criteria (Convergence criteria) (unless you want your calculation to run under the default conditions).

To find the relevant controls in the different molecular model-ing programs:

♦ For Cerius2•OFF, go to the OFF METHODS deck of cards and choose the MINIMIZER card. Select the Run menu item to access the Energy Minimization control panel. You can access additional tools, such as for setting the minimization method, by clicking the Preferences… pushbuttons in this control panel.

♦ In the Cerius2•MMFF module, select the Run menu item to access the MMFF Energy Minimization control panel. You can access additional tools, such as for setting the minimization method, by clicking the Controls… pushbutton in this control panel or selecting Controls on the MMFF card.

♦ For Cerius2•Discover, go to the DISCOVER deck of cards. Select the Run menu item on the DISCOVER card to access the Run Discover control panel. Set the Task popup to Minimiza-tion. You can access additional tools, such as for setting the minimization method, by clicking the More… pushbutton to the right of this popup to open the Discover Minimize control panel.

♦ In the Insight•Discover_3 module, select the Calculate/Mini-mize command. Toggle More on to access additional controls. Set the controls as desired and select Execute.



Alternatively, for a simple minimization run, select the Strat-egy/Simple_Minimize command. Set the controls as desired.

♦ In the Insight•Discover module, select the Parameters/Mini-mize command. Set the controls as desired and select Execute.

♦ In QUANTA, select the CHARMm/Minimization Options menu item. Select the desired minimization method and other options in the CHARMm Minimization Setup dialog box, then click OK.

Discover and CHARMm offer additional functionality when run in standalone mode. (How to run Discover and CHARMm in standalone mode is documented separately—see Additional information.)

Specifying output 4. Specify the desired output:

♦ In Cerius2•OFF, select the Output menu item on the MINI-MIZER card.

♦ In the Cerius2•MMFF module, click the Output… button in the MMFF Energy Minimization control panel or select Output on the MMFF card.

♦ In Cerius2•Discover, set the Task popup in the Run Discover control panel to Minimization. Then click the Output… push-button to access the Discover Minimize Output control panel.

♦ In the Insight•Discover_3 module, use the Analyze/Output command (it may not be accessible until after you have exe-cuted the Calculate/Minimize command). Set the controls as desired and select Execute.

♦ In the Insight•Discover module, use the Run/Report and/or Run/Files commands. Set the controls as desired and select Execute.

♦ In QUANTA, limited control over output is available via the CHARMm/Initialization Options menu item.


Starting a minimization run Finally:


4. Minimization

5. Start the minimization run:

♦ In Cerius2•OFF, click the Minimize the Energy action button in the Energy Minimization control panel.

♦ In the Cerius2•MMFF module, click the MINIMIZE button in the MMFF Energy Minimization control panel.

♦ In Cerius2•Discover, set the Task popup in the Run Discover control panel to Minimization. Then click the RUN pushbut-ton.

♦ In the Insight•Discover_3 module, execute the D_Run/Run command.

Alternatively, for a simple minimization run, select Execute in the Strategy/Simple_Minimize command.

♦ In the Insight•Discover module, select the Run/Run command. Set the controls as desired, being sure that Run_Minimization is on and Run_Dynamics is off, and select Execute.

♦ In QUANTA, go to the Modeling window and click CHARMm Minimization.

Specific information For specific information on setting up and running minimizations with the various MSI simulation engines and on examining the results, please see the relevant documentation (see Available docu-mentation).

When to use different algorithms

The default minimizers in Discover and OFF use a cascade of appropriate minimization algorithms in sequence. However, you may want to exercise more control over your simulation.

Model size and distance from the minimum

The choice of which algorithm to use depends on two factors—the size of the model and its current state of optimization. The conju-gate gradient and steepest descents methods can be used with models of any size. Most Newton–Raphson methods cannot be used with very large models, because they need sufficient disk space to store a second-derivative matrix. (The ABNR method does not store a large second-derivative matrix.)

Until the derivatives are well below 100 kcal mol-1Å-1, it is likely that the point is sufficiently distant from a minimum that the



energy surface is far from quadratic. Algorithms that assume the energy surface to be quadratic (Newton–Raphson, quasi-Newton–Raphson, conjugate gradient) can be unstable when the model is far from the quadratic limit. The Newton–Raphson method is par-ticularly sensitive because it must invert the Hessian matrix.

Therefore, as a general rule, steepest descents is often the best min-imizer to use for the first 10–100 steps, after which the conjugate gradient and/or a Newton–Raphson minimizer can be used to complete the minimization to convergence. The truncated New-ton–Raphson minimizers are often the best Newton–Raphson methods for most applications.

Starting structures and choice of forcefield

For highly distorted structures, the presence of cross terms and Morse bond potentials in the forcefield can cause convergence problems. These functional forms can produce either small restor-ing forces or, more seriously, minima at nonphysical points on the potential energy surface.

Thus, in addition to using steepest descents for such distorted structures, you also ought to use a forcefield with a simple, qua-dratic functional form. How to choose and set up forcefields is cov-ered in Preparing the Energy Expression and the Model.

Conjugate gradient vs. Newton–Raphson and disk space

Several practical aspects of the conjugate gradient method are worth mentioning. First, the conjugate gradient algorithm requires convergence along each line search before continuing in the next direction. The gradient at step i+1 must be perpendicular to hi or the derivation guaranteeing a conjugate set of directions breaks down. Second, to start conjugate gradients, an initial direction h0 must be chosen that is equal to the initial gradient. Finally, addi-tional storage is required for an extra vector of N elements to hold the N components of the old gradient. For energy minimization in Cartesian space this would be the 3N derivatives of the energy with respect to the x, y, and z coordinates of each atom. This makes conjugate gradient the method of choice for systems that are too large for storing and manipulating a second-derivative matrix, as is required by the Newton–Raphson minimizers.

The general memory requirements of all the minimizers are listed in Table 15.


4. Minimization

aN = number of atoms (number of degrees of freedom).

Failure to converge Also, note that the derivation invokes a quadratic approximation. For nonharmonic systems, the conjugate gradient method can exhaustively minimize along the conjugate directions without converging. This condition indicates that the minimizer may have gotten stuck at a saddle point. If this occurs, you can restart the algorithm. Several minimizations may be required. For a detailed discussion of this algorithm, see the excellent text by Press et al. (1986) or the somewhat more formal treatment by Fletcher (1980).

Convergence criteria

Mathematical definition In the literature a wide variety of criteria have been used to judge minimization convergence in molecular modeling. Mathemati-cally, a minimum is defined as the point at which the derivatives of the function are zero and the second-derivative matrix is posi-tive definite. Nongradient minimizers can use only the increment in the energy and/or coordinates as criteria. In gradient minimiz-ers, derivatives are available analytically and should be used directly to assess convergence.

Application to chemical models

In a molecular minimization, the atomic derivatives may be sum-marized as an average, a root-mean-square (rms) value, or the largest value. The average, of course, must be an average of the absolute values of the derivatives, because the distribution of derivatives is symmetric about zero. A rms derivative is a better

Table 15. General storage requirements of minimization algorithms

algorithm variant memory needed for scales asa

steepest descents first derivatives 3N conjugate gradient Polak–Ribiere first derivatives, gradient from previous iteration 3N

Fletcher–Reeves first derivatives, gradient from previous iteration 3N Powell first derivatives, gradient from previous iteration 3N

Newton–Raphson full, iterative Hessian, eigenvectors (3N) 2

BFGS first derivatives, Hessian update, scratch vectors (3N) 2

DFP first derivatives, Hessian update, scratch vectors (3N) 2

truncated Hessian (3N) 2

ABNR first derivatives 3N



measure than the average, because it weights larger derivatives more, and it is therefore less likely that a few large derivatives would escape detection, which can occur with simple averages.

Regardless of whether you choose to report convergence in terms of the average or rms values of the derivatives, you should always check that the maximum derivative is not unreasonable. There can be no ambiguity about the quality of the minimum if all deriva-tives are less than a given value.

How close to absolute convergence is good enough?

The more difficult question is, What value of the average or rms derivative constitutes convergence? The specific value depends on the objective of the minimization. If you simply want to relax over-lapping atoms before beginning a dynamics run, minimizing to a maximum derivative of 1.0 kcal mol-1 Å-1 is usually sufficient. However, to perform a normal mode analysis, the maximum derivative must be less than 10-5, or the frequencies may be shifted by several wavenumbers.

Local or global minimum? There is no guarantee that the minimum you find is necessarily a global minimum.

Small models can be minimized to a global minimum. However, multiple minimizations from different starting conformations should be run to confirm that a global minimum has indeed been found.

Larger models can often be minimized to several different confor-mations that a molecule might assume at 0 K. A global minimum may never be found for large models, because of the complexity of the potential energy surface. However, these many minima can be useful in understanding the molecule’s conformational “space”.

Please see also Failure to converge.

Other termination criteria You generally also set a maximum number of iterations, so that runs that do not converge will nevertheless end within a reasonabe amount of time. That is, the run ends when either the convergence criteria or the maximum number of iterations is reached, which-ever occurs first.

The BTCL language can be used to access the Discover Minimize database during a minimization run, so as to implement sophisti-cated customized stopping strategies.


4. Minimization

Significance of minimum-energy structure

Calculated energy is rela-tive

In dealing with macromolecular optimization calculations, it is important to keep in mind the theoretical significance of the mini-mum-energy structure and its calculated energy. For all forcefields used in calculations of this type, the energy zero is arbitrary, and therefore, the total potential energy of different models cannot be compared directly. However, it is meaningful to make compari-sons of energies calculated for different configurations of chemi-cally identical models. In principle, the calculated energy of a fully minimized structure is the classical enthalpy at absolute zero, ignoring quantum effects (in particular the zero-point vibrational motion). For a model that is sufficiently small that its normal modes can be calculated, quantum corrections for zero-point energy and the free energy at higher temperatures can be taken into account (Hagler et al. 1979c).

Precautions in ligand-binding calculations

The minimized energies calculated for enzyme–substrate com-plexes can be used to estimate relative binding enthalpies, but there are two caveats:

♦ First, for a meaningful comparison of the relative binding of two different substrates, a complete thermodynamic cycle must be considered (Kirkwood 1935, Quirke and Jacucci 1982, Tembe and McCammon 1984, Mezei and Beveridge 1986).

In practical terms, this means that an enthalpy calculation must be made for the various substrates in water. Where the relative binding of two different enzymes to the same substrate is calcu-lated, the energy of each enzyme with solvent in the binding site must be calculated.

Entropy is usually neglected

♦ A second consideration for using minimization results to esti-mate relative binding strengths is that the entropy is neglected in such calculations. Direct calculation of entropy differences is a computationally intensive process, and only recently has it been taken into account correctly by calculations of relative free energies (Hagler et al. 1979c, Hwang and Warshel 1987, Warshel et al. 1986, Singh et al. 1987, Straatsma et al. 1986).

The extent of the errors introduced by neglecting entropic con-tributions in the simpler minimization calculations is difficult to estimate, although, as with the zero-point energy, the



entropy can be estimated for a model small enough that its nor-mal mode frequencies can be calculated (Hagler et al. 1979c).

What is the purpose of your calculation?

The relative importance of these fundamental considerations depends on the objective of the calculation. When studying the rel-ative binding in an enzyme active site of two substrates, one of which is flexible and the other rigid, entropic effects may be crucial for obtaining even qualitative agreement with experimental bind-ing constants. On the other hand, if a putative compound overlaps sterically with many active-site atoms and causes hundreds of kilocalories of strain energy even in a minimized structure, the compound can be rejected confidently.

The bottom line is that physical-chemistry common sense cannot be abandoned when you are setting up a calculation and interpret-ing the results.


An energy calculation is essentially just a zero-iteration minimiza-tion. It is used to calculate the energy of the current model struc-ture without changing any atom positions.

It is often useful to obtain this information before performing other operations such as running an energy minimization, performing molecular dynamics, or doing a conformational search. Then results obtained using different methods can be compared with the initial values.

A gradient calculation is used to calculate the forces on the current model structure’s atoms without changing any atom positions.

Specifying single-point energy and gradient cal-culations

To specify single-point energy and gradient calculations rather than a minimization (Minimizations with MSI simulation engines):

♦ In Cerius2•OFF, click the Calculate the Current Energy action button in the Energy Minimization control panel (the gradient is also calculated).

♦ In the Cerius2•MMFF module, click the Calculate Current Energy button in the MMFF Energy Minimization control panel.


4. Minimization

♦ In Cerius2•Discover, set the Task popup in the Run Discover control panel to Single Point Energy or to Gradient.

♦ In the Insight•Discover_3 module, select the Calculate/Mini-mize command. Set Iterations to 0 and select Execute.

♦ In the Insight•Discover module, select the Parameters/Mini-mize command. Set Max Steps to 0 and select Execute.

♦ In QUANTA, go to the Modeling window and click CHARMm Energy.



This section includes:

Application of minimization to vibrational theory

Vibrational frequencies

General methodology for vibrational calculations

Harmonic vibrational fre-quencies obtained from an equilibrium geometry

The vibrational frequencies and modes of a molecule are strictly dynamic properties. However, it is possible to calculate the har-monic vibrational frequencies of a model from just information at its equilibrium geometry by expanding the potential energy sur-face as a Taylor series, truncating after the second term, and con-sidering infinitesimal displacements. This harmonic approximation usually gives a good description of the true fre-quencies and normal modes and can be valuable for tasks ranging from evaluating the quality of a forcefield to understanding vibra-tional shifts induced by conformational changes or other interac-tions. The harmonic vibrational frequencies also can be used for zero-point vibrational corrections and for deriving vibrational free energy contributions. These effects can be important in comparing conformational energies and rotational barrier heights.

Uses of vibrational calcu-lations

Beyond these considerations, the vibrational frequencies can be used for two classes of problems:



♦ The first is for determining the shape of the potential energy surface; that is, the characterization of stable points as minima, transition states, or other points. For this purpose, the question of forcefield accuracy is less important. The qualitative, rather than quantitative, shape of the surface is all that is important.

♦ The second use is for comparison with experimental results. The vibrational frequencies and thermodynamic corrections depend strongly on the forcefield as well as on the fundamental harmonic approximation invoked. By nature, low-frequency modes are less harmonic. The torsion and nonbond interac-tions, which dominate low-frequency modes, are fundamen-tally anharmonic; hence the interpretation of the calculated low-frequency modes should take this into account. Unfortu-nately, these low-frequency modes make the largest contribu-tions to the vibrational entropy.

Application of minimization to vibrational theory

Kinetic energy of nuclei Following Wilson et al. (1980), the kinetic energy of the nuclei is:

Eq. 70

where the coordinates represent displacements from an equilib-rium structure. If the 3N Cartesian coordinates are replaced with 3N mass-weighted coordinates as follows:

Eq. 71

Simplified kinetic energy of nuclei

where mα is the mass of the atom associated with the α coordinate, and ∆xα runs over the y and z coordinates, as well as x, then the kinetic energy has the following simple form:

Eq. 72

T12--- mi td

d ∆x

2

tdd ∆y

2

tdd ∆z

2

+ +

i 1=

N

∑=

qi mα ∆xα=

T12---

td

dqi

2

i 1=

3N

∑=


4. Minimization

The second term in a Tay-lor series is needed to obtain vibrational infor-mation

When the potential energy of the system is expanded as a Taylor series in the same coordinates, it yields:

Eq. 73

V0 is simply a constant—the energy scale can be chosen so that V0 = 0. The definition of an equilibrium structure is that the force on each atom is zero. The second term in Eq. 73 also is zero, leaving the following second-order approximation of the potential energy:

Eq. 74

Combining energy and motion and solving for the mass-weighted coordi-nates

Using this approximation of the potential energy in Newton’s equations of motion (Eq. 4) yields the following simultaneous sec-ond-order differential equations:

Eq. 75

The solution to these equations can be of the form:

Eq. 76

Numerically solvable form of the function

where the Ai are related to the relative amplitudes of the vibra-tional motion, λ1⁄ 2 is proportional to the vibrational frequency, and δ is a phase. Substituting Eq. 76 in Eq. 75 yields a set of alge-braic equations:

Eq. 77

V V0Vδqiδ-------

0

qi

i 1=

3N

∑ 12--- δ2V

qi qjδδ---------------

0

qiqj …

i j, 1=

3N

∑+ +=

V12--- δ2V

qi qjδδ---------------

0

qiqj

i j, 1=

3N

∑=

t2

2

d

d qi δ2Vqi qjδδ---------------

0

qj

d 1=

3N

∑+ 0= i 1 2 … 3N,,,=

qi Ai λ1 2/ t δ+( )cos=

δ2Vqi qjδδ---------------

0

δi jλ– Ai

i 1=

3N

∑ 0= j 1 2 … 3N,,,=



where δij is a Kronecker delta, which equals one if i = j and zero otherwise. This is an eigenvalue problem that is readily solved numerically by standard techniques. The second derivatives of the potential energy, often called the force constants, can be analytically evaluated for most energy expressions used in molecular mechan-ics, in terms of the Cartesian coordinates of the atoms. A simple transformation to the mass-weighted coordinates then gives the values needed in Eq. 77.

Vibrational frequencies

Eigenvalues converted to vibrational frequencies

The simulation engine determines vibrational frequencies by cal-culating the second derivative matrix, mass weighting it, and then diagonalizing it to obtain the eigenvalues. These eigenvalues are then converted to vibrational frequencies in wavenumbers as fol-lows:

Eq. 78

Evaluating the quality of the calculation

where the conversion factor F converts the units from kcal mol-1 to wavenumbers. Of the 3N coordinates used to calculate the energy and vibrational frequencies, six correspond to net translations and rotations of the model (five for linear systems). These six modes have no restoring force and therefore have vibrational frequencies of zero for a minimized structure. Due to numerical inaccuracies, the simulation engine reports these frequencies with small values, which are typically less than 0.1 cm-1. If the structure is not per-fectly minimized, the first-order terms in the Taylor expansion of the potential surface in Eq. 73 do not vanish. In turn, they intro-duce terms that couple the net rotations of the model with the internal motions and both perturb the internal vibrational fre-quencies and give apparent frequencies for the three rotations. Thus, the magnitude of the six “zero” frequencies is a good indica-tion of the quality of the calculation.

Transition states

The model must be opti-mized before doing a vibration calculation

For calculating vibrational frequencies, the model must be mini-mized and the gradients must be zero. This does not mean that the configuration of the model must be at a minimum, but rather that it must be at a stable point on the surface. If the structure is not at

νi F λ i=


4. Minimization

a minimum, but rather is at a saddle point or transition state in one or more directions, this is reflected in the eigenvalues and the reported vibrational frequencies. For a saddle point, at least one eigenvalue is negative, which means that the curvature of the sur-face along at least one normal mode is negative. By convention, the imaginary frequencies for such modes are reported as negative.

Therefore, the reported vibrational frequencies describe the char-acter of the stable point on the surface. If all frequencies are real, it is a minimum; if one frequency is imaginary, the structure is in a simple transition state; if two or more frequencies are imaginary, a double or more complicated transition state is indicated. The nor-mal modes corresponding to the frequencies can be analyzed to understand the reaction path going through such a transition state.

Thermodynamics

Quantum mechanical solution…

The quantum mechanical solution for the vibrational energy of a set of uncoupled harmonic oscillators, which corresponds to the classical treatment outlined above, is:

Eq. 79

…used to correct the vibrational energy deter-mined with a forcefield

where the summation is over all the vibrational frequencies, ni is the vibrational quantum number for each vibration, h is Planck’s constant, and νi is the vibrational frequency. This leads to the fol-lowing correction to the classical forcefield energy:

Eq. 80

Free energy correction where k is the Boltzmann constant and T is the temperature. The first term, hνi ⁄ 2, is the zero-point correction; the second term cor-rects for the average thermal population of vibrational levels at the temperature T. This leads to a vibrational free energy correction of:

Evib ni12---+

hνi

i

∑=

Evib12---

1ehνi /kT 1–-------------------------+ hνi

i

∑=



Eq. 81

Vibrational entropy and a vibrational entropy of:

Eq. 82

General methodology for vibrational calculations

Specifying a vibration cal-culation

To specify a vibration calculation for a model that is already well minimized (Minimizations with MSI simulation engines):

♦ In the Cerius2, use the IR/RAMAN module.

♦ In the Insight•Discover_3 module, select the Calculate/Vibra-tional command. Set the controls desired and select Execute.

Vibrational calculations in FDiscover and CHARMm are avail-able only in standalone mode.

Model must be small and very well minimized

Computation of the harmonic vibrational frequencies requires the storage and diagonalization of the second-derivative matrix, which has dimensions of 3N × 3N, where N is the number of atoms. The work involved in the diagonalization scales as N 3 and quickly becomes prohibitively expensive for more than a few hundred atoms. The frequencies are valid only for well minimized models having maximum derivatives no greater than approximately 0.001 kcal mol-1 Å-1.

Use a forcefield that was designed for vibrational calculations

The forcefield is a major consideration. Most forcefield develop-ment has emphasized structures and energetics rather than vibra-tional frequencies. As a result, the frequencies calculated with forcefields (see Forcefields) such as AMBER, MM2, CHARMm, and, to a lesser extent, CVFF, may often be in error by several hundred wavenumbers. The inclusion of cross terms such as bond–bond and bond–angle terms is crucial for the accurate reproduction of experimental frequencies. The CVFF forcefield includes such cross terms and was, in part, parameterized to reproduce experimental frequencies, which explains its moderately good performance for vibrational calculations. The second-generation forcefields MM3 and CFF were explicitly designed to evenly weight vibrational fre-quencies as well as structural and energetic properties. Therefore,

Avib

hνi

2-------- kT ln 1 e

hνi /kT––

+

i

∑=

Svib Evib Avib–( ) T⁄=


4. Minimization

they provide the most reasonable and consistent results, usually within 50–100 cm-1. This error of up to approximately 100 cm-1 appears to be the current limit of general-purpose, transferable forcefields.


5 Molecular Dynamics

While minimization computes the forces on the atoms and changes their positions to minimize the interaction energies, dynamics computes forces and moves atoms in response to the forces.

Molecular dynamics solves the classical equations of motion for a system of N atoms interacting according to a potential energy forcefield as described in Forcefields. Dynamics simulations are useful in studies of the time evolution of a variety of systems at nonzero temperatures, for example, biological molecules, poly-mers, or catalytic materials, in a variety of states, for example, crys-tals, aqueous solutions, or in the gas phase.

This chapter explains To perform the most reasonable and realistic dynamics simula-tions, you should read this entire chapter, which includes informa-tion on:

Integration algorithms

The choice of timestep

Integration errors

Statistical ensembles

Temperature

Pressure and stress

Types of dynamics simulations

Constraints during dynamics simulations

Dynamics trajectories

General methodology for dynamics calculations

Related information Forcefields and Preparing the Energy Expression and the Model focus on the representation of the potential energy surface and the useful ways that it can be biased through the addition of restraints and


5. Molecular Dynamics

constraints, as well as other information on preparing the model system for calculations.

Use of simulation engines for calculating normal modes is found under Vibrational calculation.

Specific information For specific information on setting up and running dynamics cal-culations with the various MSI simulation engines, please see the relevant documentation (see Available documentation).

Table 16. Finding information in Chapter 5


Repeating a dynamics simulation. Defined initial coordinates and random initial veloc-ities.

Integrators used by MSI simulation engines. Table 17.Thermodynamic ensembles handled by MSI

simulation engines.Table 19.

Obtaining thermodynamic properties of a model.

Equilibrium thermodynamic properties.

Temperature-control methods used by MSI simulation engines.

Table 20.

Producing true canonical ensembles. Nosé é and Nosé–Hoover dynamics.Pressure- and stressecontrol methods used by

MSI simulation engines.Table 21.

Types of dynamics simulations readily set up with MSI simulation engines.

Table 22.

Dynamics runs with periodic minimization. Quenched dynamics.Controlled temperature change during

dynamics.Simulated annealing.

Finding a common configuration of related models.

Consensus dynamics.

Impulse dynamics. Impulse dynamics.Simulated dynamics in a viscous fluid. Langevin dynamics.Dynamics of localized parts of a model. Stochastic boundary dynamics.SHAKE and RATTLE algorithms. Constraints during dynamics simulations.Equilibration and data-collection stages of a

dynamics simulation.Stages and duration of dynamics simulations.

How long is long enough for a dynamics sim-ulation.

Has equilibrium been achieved?; How long should the simulation be?.


Some uses of dynamics calculations

The major applications of molecular dynamics are:

♦ Performing conformational searches.

During dynamics simulations, a system undergoes conforma-tional and momentum changes so that different parts of the phase space accessible to the model can be explored. The con-formational search capability of dynamics is one of its most important uses.

♦ Generating statistical ensembles.

By providing several mechanisms for controlling the tempera-ture and pressure of simulated systems, molecular dynamics allows you to generate statistical ensembles from which vari-ous energetic, thermodynamic, structural, and dynamic prop-erties can be calculated. For such studies, it is important that the calculation visit various conformational states with the correct statistical frequency.

♦ Studying the motions of molecules.

Although modern crystallography has provided a window into the static structure of molecules both small and large, the thought of intermolecular collisions and conformational varia-tion is always present. After all, binding of substrates by pro-teins, folding of proteins and peptides into unique shapes, the dynamic behavior of polymers, and chemical reactions them-selves would be inconceivable without the concept of molecu-lar motion.

Studies of model motions can be used to derive properties such as diffusion coefficients.

Non-dynamics approaches are also used

Other approaches to simulating molecular motion and generating conformational searches exist.

Setting up a dynamics simulation. Dynamics with MSI simulation engines.Continuing or restarting a dynamics run. Restarting a dynamics simulation.Using a previous dynamics run to start a new

simulation.Restarting a dynamics simulation.

Table 16. Finding information in Chapter 5




For example, a dynamics trajectory can be constructed from a set of normal modes to represent the vibrations of a model. While this is a fast method, it is restricted to harmonic motion about a single energy minimum.

An approach to doing conformational searches is the Monte Carlo method. While this method can sample conformational space so as to produce meaningful statistical ensembles, it does not provide dynamic information about the model, since particles of the model system are simply moved randomly according to some statistical rules.



Introduction

Criteria of good integrators in molecular dynamics

Integrators in MSI simulation engines

Introduction

Newton’s equation of motion applied to atoms

At its simplest, molecular dynamics solves Newton’s familiar equation of motion:

Eq. 83

where Fi is the force, mi is the mass, and ai is the acceleration of atom i.

The force on atom i can be computed directly from the derivative of the potential energy V with respect to the coordinates ri:

Eq. 84

What is a trajectory? Notice that classical equations of motion are deterministic. That is, once the initial coordinates and velocities are known, the coordi-

Fi t( ) miai t( )=

ri∂∂V

– mi ti2

2

∂

∂ ri=



nates and velocities at a later time can be determined. The coordi-nates and velocities for a complete dynamics run are called the trajectory. (However, trajectories are sensitive to initial conditions, so the same simulation run with a different simulation engine or on a different computer does not produce an identical trajectory.)

The finite-difference method

A standard method of solving an ordinary differential equation such as Eq. 84 numerically is the finite-difference method. The general idea is as follows. Given the initial coordinates and veloc-ities and other dynamic information at time t, the positions and velocities at time t + ∆t are calculated. The timestep ∆t depends on the integration method as well as the system itself.

Defined initial coordinates and random initial veloci-ties

Although the initial coordinates are determined in the input file or from a previous operation such as minimization, the initial veloci-ties are randomly generated at the beginning of a dynamics run, according to the desired temperature. Therefore, dynamics runs cannot be repeated exactly, except for forcefield engines (CHARM standalone, Discover) that allow you to set the random number seed to the value that was used in a previous run.

More details on the initial velocities are provided under Tempera-ture.

Criteria of good integrators in molecular dynamics

Molecular dynamics is usually applied to a large model. Energy evaluation is time consuming and the memory requirement is large. To generate the correct statistical ensembles, energy conser-vation is also important.

Thus, the basic criteria for a good integrator for molecular simula-tions are as follows:

♦ It should be fast, ideally requiring only one energy evaluation per timestep.

♦ It should require little computer memory.

♦ It should permit the use of a relatively long timestep.

♦ It must show good conservation of energy.



Integrators in MSI simulation engines

Integrators provided in MSI simulation engines were chosen according to the above criteria. Only dynamics algorithms used in MSI’s simulation engines (Table 17) are considered here:

Verlet leapfrog integrator

Verlet velocity integrator

ABM4 integrator

Runge–Kutta-4 integrator

aFDiscover only, not in CDiscover.bCDiscover only, not in FDiscover.

Choosing the dynamics algorithm(s)

To specify the dynamics integrator:

♦ The Cerius2•Dynamics Simulation module always uses the Verlet leapfrog integrator.

♦ In the Cerius2•Discover and Insight•Discover_3 modules, the default integrator is the Verlet velocity method. If you really want to change it, you can write out the command input file, edit the BTCL dynamics command statement with a text editor, and then use that file for your run.

Alternatively (in Insight•Discover_3), select the Language_Control/Command_Comment command. Set the Comment Type to Command. Enter integration_method = ABM4 or integration_method = Runge_Kutta in the Command/Com-

Table 17. Dynamics integrators used by MSI simulation engines

integratorsimulation engine

CHARMm Discover OFF

Verlet leapfrog √ √a √Verlet velocity √ √b

ABM4 √2

Runge–Kutta–4 √2



ment entry box and select Execute. Be sure that you insert this stage at the correct point in your command input file.

♦ The Insight•Discover module always uses the Verlet leapfrog integrator.

♦ CHARMm allows you to choose between the Verlet leapfrog and velocity methods only when run in standalone mode.

Verlet leapfrog integrator

Advantages of Verlet methods

Variants of the Verlet (1967) algorithm of integrating the equations of motion (Eq. 84) are perhaps the most widely used method in molecular dynamics. The advantages of Verlet integrators is that these methods require only one energy evaluation per step, require only modest memory, and also allow a relatively large timestep to be used.

The leapfrog algorithm The Verlet leapfrog algorithm is as follows:

Given r(t), v(t –∆t/2), and a(t), which are (respectively) the position, velocity, and acceleration at times t, t –∆t/2, and t, compute:

where f (t + ∆t) is evaluated from -dV/dr at r (t + ∆t).

Disadvantage of Verlet leapfrog method

The Verlet leapfrog method has one major disadvantage: the posi-tions and velocities calculated are half a timestep out of synchrony.

Verlet velocity integrator

The Verlet velocity algorithm overcomes the out-of-synchrony shortcoming of the Verlet leapfrog method. The Verlet velocity algorithm is as follows:

The velocity algorithm Given r(t), v(t), and a(t), which are (respectively) the position, velocity, and acceleration at time t, compute:

v t12---∆t+

v t12---∆t–

∆ta t( )+=

r t ∆t+( ) r t( ) ∆tv t12---∆t+

+=

a t ∆t+( ) f t ∆t+( )m

--------------------=



ABM4 integrator

ABM4, which stands for Adams–Bashforth–Moulton fourth order, is a predictor and corrector method. It is a fourth-order method, meaning that the truncation error is to the fifth order of the timestep used.

This method requires two energy evaluations per step and has to make use of the results of the previous three steps. It is thus not self starting—the first three steps are generated by the Runge–Kutta method. More memory has to be used, because previous informa-tion has to be stored.

ABM4’s algorithm The algorithm is as follows:

Let:

where subscripts (not shown above) 0, 1, -1, -2, and -3 indicate y at times t, t + ∆t, t - ∆t, t - 2∆t, and t - 3∆t.

Predictor step The predictor is:

Now evaluate y1′, using y1predicted, which involves one energy

evaluation.

Corrector step The corrector is:

Now evaluate y1′, using y1corrected, which involves another

energy evaluation.

r t ∆t+( ) r t( ) ∆tv t( ) ∆t2a t( )2

------------------+ +=

a t ∆t+( ) f t ∆t+( )m

--------------------=

v t ∆t+( ) v t( )12---∆t a t( ) a t ∆t+( )+[ ]+=

y r t( ) v t( ),= and y ′ v t( ) a t( ),=

y1predicted

y0∆t24------ 55y0 ′ 59y 1– ′– 37y 2– ′ 9y 3– ′–+( ) O ∆t5( )+ +=

y1corrected

y0∆t24------ 9y1 ′ 19y0 ′ 5y 1– ′– y 2– ′+ +( ) O ∆t5( )+ +=



Runge–Kutta-4 integrator

Robust, but disadvan-tages

Runge–Kutta-4 stands for the fourth-order Runge–Kutta method, which is one of the oldest numerical methods for solving ordinary differential equations. The method is self starting but requires four energy evaluations per step.

From testing done at MSI, the timestep has to be very small. This is thus not a very suitable integrator for molecular simulation.

However, the method is very robust, meaning that it can deal with almost all kinds of equations, including stiff ones. This integrator is used to generate the trajectory for the first three steps for ABM4.

Since we do not recommend using this integrator, the algorithm is not presented here. Details can be found in Press et al. 1986.


A key parameter in the integration algorithms is the integration timestep ∆t. To make the best use of the computer time, a large timestep should be used. However, too large a timestep causes instability and inaccuracy in the integration process.

Relation of timestep to molecular vibration

The timestep used depends on the model as well as the integrators. The main limitation imposed by the model is the highest-fre-quency motion that must be considered. A vibrational period must be split into at least 8–10 segments for models to satisfy the Verlet assumption that the velocities and accelerations are constant over the timestep used.

In most organic models, the highest vibrational frequency is that of C–H bond stretching, whose period is on the order of 10-14 s (10 fs). The integration timestep should therefore be about 0.5–1 fs. If you use the SHAKE or RATTLE constraint algorithm (Constraints during dynamics simulations), a longer timestep is possible.

If you are studying simple model liquids or solids and are not interested in internal modes, much longer timesteps may be used, e.g., up to 20 fs. A timestep of about 5 fs should be adequate for ionic material models.



Appropriate for the inte-grator

The timestep must also be appropriate to the integrator. For the ABM4 method, the timestep should be about half that needed for the Verlet algorithm. The Runge–Kutta-4 method seems to require a much smaller timestep than the other methods.

Setting the timestep To specify the length of the timestep:

♦ In Cerius2•OFF, go to the DYNAMICS SIMULATION card in the OFF METHODS deck of cards. Click the Run menu item to access the Dynamics Simulation control panel. Enter the desired time step (in ps) in the Dynamics Time Step entry box.

♦ In Cerius2•Discover, click the Run menu item in the DIS-COVER card to access the Run Discover control panel. Set the Task popup to Dynamics and click the More… pushbutton to the right of the Task popup to open the Discover Dynamics control panel. Enter values for two of the Total time, Steps, and Time step entry boxes (the third value is computed from the other two) in the Equilibration and Production sections of the latter control panel.

♦ In the Insight•Discover_3 module, select the Calculate/Dynamics command. Toggle More on to access additional con-trols, and change the value in the Time Step fs entry box.

♦ In the Insight•Discover module, select the Parameters/Dynam-ics command. Change the value (in fs) in the Time Step entry box.

♦ In QUANTA, Choose the CHARMm/Dynamics Options menu item. You can change the Time Step in any of the setup dialogs that are accessed by clicking a radio button and clicking OK.

Integration errors

If the chosen timestep is too small, no harm is done, except for the waste of computer time. However, if the timestep is too large for the calculation conditions, the simulation can “blow up”.

Two examples illustrate how the timestep, temperature, and inte-gration algorithm affect the results. The first example is a simula-tion of the collision of two hydrogen atoms travelling towards each other. The second is an examination of energy conservation in

Integration errors


a simple harmonic oscillator when different integrators and timesteps are used.

Example 1—Two colliding hydrogen atoms

The stability of the numerical integration with respect to the time step can be tested directly by integrating (over distance) the forces used by dynamics and comparing the integral with the analytical energy. The error in this integral as a function of time step is an indication of the intrinsic limitations of molecular dynamics.

Timestep slightly too large To illustrate this, consider the collision of two atoms. Figure 26 plots the true potential energy of the van der Waals potential between two hydrogens along with the energy integrated numer-ically from the forces. The time step used is 1 fs at a temperature of 300 K. The temperature is set by assigning an initial velocity of 1500 m s-1 (0.015Å fs-1) to one of the hydrogens along the vector connecting them. This velocity is the most probable velocity of a hydrogen atom at 300 K.

As the two atoms approach each other, the integration agrees well with the analytic curve. However, after the atoms collide, the inte-grated energy is significantly higher than the true energy. This behavior is due to the atom moving too quickly through a rapidly changing energy function. When the atoms are far apart, the energy change is smallest and therefore, the forces are smallest and most linear. As the atoms approach, they speed up and take larger and larger steps, until they reach their highest velocity at the energy minimum. Unfortunately, this is precisely where the forces start changing the fastest. Thus, when the atoms should be taking large steps (far apart), they are taking the smallest steps, and when they should take small steps (near the minimum), they take the largest steps. The consequence is that the particles “step through” the energy barrier momentarily. Of course, once a new force is cal-culated at the extrapolated coordinate, the trajectory is rapidly cor-rected. However, it is too late for the energy integral—some energy has been gained.

In this example, the total energy rose by about 0.02 kcal mol-1. Whether this is a reasonable error depends on how closely the exact motions need to be reproduced.



Shorter timesteps mean more computational cost

Figure 27 shows the same curve as Figure 26, but with timesteps of 0.33 and 0.10 fs. Both give better results than 1.0 fs, but it is not clear whether the extra time required to calculate the smaller timesteps is worthwhile.

Figure 26. Numerical integration of energy from molecular dynamics hydrogen-collision trajectory, 1 fs timestep

The integrated energy calculated numerically from a dynamics trajectory of two colliding hydrogen atoms (circles) is compared with the analytical energy curve (thick line). Simulation done with FDiscover.

300K1 fs timestep

2 3 4 5-0.02

-0.01

0.00

0.01

0.02

H–H distance (Å)

ene

rgy

(kc

al m

ol-1

)

Integration errors


A very large timestep leads to artifactual behavior

The consequences of too large a time step are much more dramatic. Doubling the time step from 1 to 2 fs results in an unusual artifact for the hydrogen system (Figure 28). In this case, the integration error affects not only the potential energy, but the kinetic energy as

Figure 27. Numerical integration of energy from molecular dynamics hydrogen-collision trajectory, 0.33 and 0.1 fs timesteps

Energy integration errors decrease with smaller time steps. Compared to 0.016 kcal error with 1 fs time steps, the 0.33 fs time step has a 0.006 error, and the 0.1 fs time step a 0.001 error. The cost for increased accuracy is the computational burden to compute more steps. For most simulations, a 1-fs time step is a good compromise between numerical accuracy and computational efficiency. Sim-ulation done with FDiscover.

300K0.33 fs timestep

-0.01

0.00

0.01

0.02

300K0.1 fs timestep

2 3 4 5-0.02

-0.01

0.00

0.01

H–H distance (Å)

ene

rgy

(kc

al m

ol-1

)



well. Momentum is removed, so that the hydrogens no longer have the velocity needed to escape after the collision. The atoms are trapped forever (barring an inverse error that could impart momentum). The atoms now vibrate back and forth (only 2 cycles are plotted in Figure 28), and each cycle incurs an additional inte-gration error.

An even higher timestep can cause the system to explode

Increasing the time step another factor of 2 to 4 fs (Figure 29) finally causes the system to “blow up”. The timestep is so long that the atoms deeply interpenetrate each other ’s steeply repulsive wall between steps. The resulting force is now so large that the atoms fly off at a speed of about 1 Å fs-1 (or 105 m s-1). An equiva-lent temperature would be hundreds of thousands of degrees.


A 2-fs time step causes the rebounding force to be underestimated, robbing the colliding Hs of sufficient escape velocity and resulting in the two Hs being “bound” by their van der Waals forces. Simulation done with FDiscover.

ene

rgy

(kc

al m

ol-1

)

300K2 fs timestep

2 3 4 5-0.02

0.00

0.02

0.04

0.06

H–H distance (Å)

0.08

0.10

Integration errors


Effect of temperature To complete the analysis of integration errors, it is instructive to compare the effects of increasing kinetic energy on the stability of the numerical integration. Figure 30 shows that there is essentially no difference in the error between 300 and 1200 K. Going as high as 30,000 K merely almost doubles the error, indicating that dynamics simulations are not as sensitive to the temperature as to the timestep.


With a time step of 4 fs, the Hs travel too far in a single step, interpenetrating each other’s van der Waals radii before the forces are recalculated. By this time, the forces are so large that the Hs are flung apart at a temperature equivalent to hundreds of thousands of degrees. Simulation done with FDiscover.

ene

rgy

(kc

al m

ol-1

)

300K4 fs timestep

2 3 4 5-0.05

0.00

0.05

0.10

0.15

H–H distance (Å)

0.20



Example 2 — Energy conservation of a harmonicoscillator

Verlet velocity vs. ABM4 To test the conservation of energy in a simulation, 10 ps of molec-ular dynamics was performed on a harmonic oscillator having an equilibrium length of 0.75 Å and period of 7.5 fs. As can be seen in Table 18, the Verlet velocity method can use a larger timestep than the ABM4 method. Although the Verlet algorithm starts to show instability at 1 fs, ABM4 starts to fail at 0.25 fs.

For the Verlet velocity integrator, the standard deviation in the total energy is proportional to ∆t2, as predicted by the theory. This is a simple verification that the integrator has been implemented correctly.

Figure 30. Integration errors for hydrogen-collision trajectories at several temperatures

Integration errors observed for an H–H collision for an initial velocity equal to the mean velocity appropriate for 300 K (squares), 1200 K (circles) and 30,000 K (tri-angles). Simulation done with FDiscover.

2 3 4 5-0.02

-0.01

0.00

0.01

0.02

H–H distance (Å)

ene

rgy

(kc

al m

ol-1

)

0.03

0.04




You can control the tem-perature and pressure

Integrating Newton’s equations of motion allows you to explore the constant-energy surface of a system. However, most natural phenomena occur under conditions where a system is exposed to external pressure and/or exchanges heat with the environment. Under these conditions, the total energy of the system is no longer conserved, and extended forms of molecular dynamics are required.

Purpose of the calculation Several methods are available for controlling temperature and pressure. Depending on which state variables (the energy E, enthalpy H (i.e., E + PV), number of particles N, pressure P, stress S, temperature T, and volume V) are kept fixed, different statistical ensembles can be generated. A variety of structural, energetic, and dynamic properties can then be calculated from the averages or the fluctuations of these quantities over the ensemble generated.

Available thermodynamic ensembles

Both isothermal (exchange heat with a temperature bath to main-tain a constant thermodynamic [not kinetic] temperature) and adi-abatic (do not exchange heat) ensembles are available:

Table 18. Energy conservation for different timesteps with the Verlet velocity and ABM4 integratorsRun conditions: 10,000 fs, constant-energy (NVE), harmonic oscillator, initial energy 0.296 kcal mol-1, CDiscover 94.0.

integratortimestep

fsfinal energy kcal mol-1

average energy kcal mol-1

standard deviation kcal mol-1

Verlet velocity 1.0 0.327 0.325 0.021Verlet velocity 0.5 0.296 0.302 0.005Verlet velocity 0.25 0.298 0.298 0.001ABM4 0.25 0.693 0.467 0.114ABM4 0.10 0.299 0.297 0.001



aIn all ensembles, the number of particles is conserved.bOnly for periodic systems, because volume is undefined in nonperiodic systems. For all space-group symmetries unless otherwise noted.cOnly for cubic, orthorhombic, and triclinic unit cells.dCDiscover only, not in FDiscover.

Choosing the thermody-namic ensemble

To access the controls used for specifying the thermodynamic ensemble:

♦ In Cerius2•OFF, go to the DYNAMICS SIMULATION card on the OFF METHODS deck of cards. Click the Run card menu item to access the Dynamics Simulation control panel. Select the radio button next to the desired ensemble. You can access additional controls relevant to each ensemble by clicking the Preferences… button to the right of the ensemble.

♦ In Cerius2•Discover, click the Run menu item on the DIS-COVER card to access the Run Discover control panel. Set the Task popup to Dynamics and click the More… pushbutton to the right of the Task popup. The controls available depend on what Ensemble and Thermostat you choose and on whether the current model is periodic or not.

♦ In the Insight•Discover_3 module, select the Calculate/Dynamics command. Only the ensembles appropriate to your model system (periodic or nonperiodic) are displayed in the parameter block. Toggle More on to access additional controls relevant to the chosen ensemble.

♦ In the Insight•Discover module, select the Parameters/Dynam-ics command. Choose between NVT and NPT ensembles by

Table 19. Thermodynamic ensembles handled by MSI simulation engines

ensembleasimulation engine

CHARMm Discover OFF

Constant temperature, constant volume (NVT) √ √ √Constant temperature, constant pressure (NPT)b √c √ √Constant temperature, constant stress (NST)2 √d √Constant energy, constant volume (NVE) √ √ √Constant pressure, constant enthalpy (NPH)2 √4 √Constant stress, constant enthalpy (NSH)2 √4 √



toggling Constant_Pressure off or on, respectively. The NVE ensemble is accessible only through DSL commands.

♦ In QUANTA, set up and apply periodic boundary conditions by choosing the CHARMm/Periodic Boundaries menu item.

NVE ensemble

Some energy drift The constant-energy, constant-volume ensemble (NVE), also known as the microcanonical ensemble, is obtained by solving the standard Newton equation without any temperature and pressure control. Energy is conserved when this (adiabatic) ensemble is generated. However, because of rounding and truncation errors during the integration process, there is always a slight fluctuation or drift in energy.

When the Verlet leapfrog integrator is used, only r (t) and v (t – 1 ⁄ 2 ∆t) are known at each timestep. Thus, the potential and kinetic energies at each timestep are also half a step out of syn-chrony. Although the difference between the kinetic energies half a timestep apart is small, this can also contribute to the fluctuation in the total energy.

Although the temperature is not controlled during true NVE dynamics, you might want to use NVE conditions during the equilibration phase (Stages and duration of dynamics simulations) of your simulation. For this purpose, Cerius2•Discover and Cerius2•Dynamics Simulation allow you to hold the temperature within specified tolerances by periodic scaling of the velocities.

When to use it True constant-energy conditions (i.e., without temperature contol) are not recommended for equilibration because, without the energy flow facilitated by temperature control, the desired temper-ature cannot be achieved.

However, during the data collection phase, if you are interested in exploring the constant-energy surface of the conformational space, or for other reasons do not want the perturbation introduced by temperature- and pressure-bath coupling, this is a useful ensem-ble.

Results The results can be used (Equilibrium thermodynamic properties) to calculate the thermodynamic response function (Ray 1988).



NVT ensemble

The constant-temperature, constant-volume ensemble (NVT), also referred to as the canonical ensemble, is obtained by controlling the thermodynamic temperature. Direct temperature scaling should be used only during the initialization stage (Stages and duration of dynamics simulations), since it does not produce a true canonical ensemble (it is not truly isothermal). Any of the other temperature-control methods available (How temperature is controlled) is used during the data collection phase.

When to use it This is the appropriate choice when conformational searches of models are carried out in vacuum without periodic boundary con-ditions. (Without periodic boundary conditions, volume, pressure, and density are not defined and constant-pressure dynamics can-not be carried out.)

Even when periodic boundary conditions are used, if pressure is not a significant factor, the constant-temperature, constant-volume ensemble provides the advantage of less perturbation of the trajec-tory, due to the absence of coupling to a pressure bath.

NPT and NST ensembles

Periodic systems The constant-temperature, constant-pressure ensemble (NPT) allows control over both the temperature and pressure. The unit cell vectors are allowed to change, and the pressure is adjusted by adjusting the volume (i.e., the size and also, in some programs, the shape of the unit cell). This method applies only to periodic sys-tems.

The constant-temperature, constant-stress ensemble (NST) is an extension of the constant-pressure ensemble. In addition to the hydrostatic pressure which is applied isotropically, the constant-stress ensemble allows you to control the xx, yy, zz, xy, yz, and zx components of the stress tensor.

Control of run conditions Pressure can be controlled by the Berendsen, Andersen, or Par-rinello–Rahman method (How pressure and stress are controlled). However, only the size, and not the shape, of the unit cell can be changed with the Berendsen and Anderson methods (Berendsen method of pressure control and Andersen method of pressure control).



Stress can be controlled by the Parrinello–Rahman method (Par-rinello–Rahman method of pressure and stress control), since it allows both the cell volume and its shape to change.

Temperature can be controlled by any method available (How tem-perature is controlled) (except, of course, the temperature scaling method, since it is not truly isothermal).

When to use it NPT is the ensemble of choice when the correct pressure, volume, and densities are important in the simulation. This ensemble can also be used during equilibration to achieve the desired tempera-ture and pressure before changing to the constant-volume or con-stant-energy ensemble when data collection starts.

Results The NST ensemble is particularly useful if you want to run a sim-ulation at incremented tensile loads to study the stress–strain rela-tionship in polymeric or metallic materials.

If the forcefield being used yields a high pressure at the experi-mental volume, it may be more realistic to simulate at the experi-mental pressure rather than the experimental volume. High simulated pressure is a sign that the system is unduly compressed, which restricts atomic motions, artefactually slowing down the dynamic relaxations.

NPH and NSH ensembles

Periodic systems The constant-pressure, constant-enthalpy ensemble (NPH, Ander-sen 1980, see also Andersen method of pressure control) is the ana-logue of constant-volume, constant-energy ensemble, where the size of the unit cell is allowed to vary. In the constant-pressure (or -stress), constant-enthalpy ensemble (NPH or NSH, Parrinello and Rahman 1981, see also Parrinello–Rahman method of pressure and stress control), both the size and shape of the unit cell are allowed to vary (meaning that external stress can be applied). These meth-ods apply only to 3D periodic systems.

Enthalpy H, which is the sum of E and PV, is constant when the pressure is kept fixed without any temperature control. Although the temperature is not controlled during true (adiabatic) NPH or NSH dynamics, you might want to use these conditions during the equilibration phase (Stages and duration of dynamics simulations) of your simulation. For this purpose, Cerius2•Discover and



Cerius2•Dynamics Simulation allow you to hold the temperature within specified tolerances by periodic scaling of the velocities.

Results The natural response functions (specific heat at constant pressure, thermal expansion, adiabatic compressibility, and adiabatic com-pliance tensor) are obtained (Equilibrium thermodynamic properties) from the proper statistical fluctuation expressions of kinetic energy, volume, and strain (Ray 1988).

Equilibrium thermodynamic properties

Precautions Since the ensembles are artificial constructs, they produce aver-ages that are consistent with one another when they represent the same state of the model. Nevertheless, the fluctuations vary in dif-ferent ensembles. Some of the fluctuations are related to thermo-dynamic derivatives, such as the specific heat or the isothermal compressibility.

Caution

The transformation and relation between different ensembles has been discussed in greater detail by Allen and Tildesley (1987).

Obtaining equilibrium thermodynamic proper-ties

One of the objectives of molecular dynamics is to obtain the equi-librium thermodynamic properties of a model. If a microscopic dynamic variable A takes on values A(t) along a trajectory, then the following time average:

Eq. 85

yields the thermodynamic value for the selected variable. This dynamic variable can be any function of the coordinates and momenta of the particles of the model.

Time averaging for first-order properties

Through time averaging, you can calculate the first-order proper-ties of a system (such as the internal energy, kinetic energy, pres-sure, and virial). Similarly, using microscopic expressions in the

In practice, obtaining accurate fluctuations to calculate physical quantities is difficult, and this approach should be used with caution.

A OLPT ∞→

T--- A t( ) td

T

∫=

Temperature


form of fluctuations of these first-order properties, you can also calculate thermodynamic properties of a system. These include the specific heat, thermal expansion, and bulk modulus.

In the thermodynamic limit, the first-order properties obtained in one ensemble are equivalent to those obtained in other ensembles (differences are on the order of 1/N).

Ensemble-dependent second-order properties

However, second-order properties such as specific heats, com-pressibilities, and elastic constants differ between ensembles. For example, the specific heat at constant pressure differs from the spe-cific heat at constant volume.

Therefore, it is important to use the appropriate ensemble when performing simulations to obtain these properties.

Temperature


How temperature is calculated

How temperature is controlled

Relation of temperature and velocity

Temperature is a state variable that specifies the thermodynamic state of the system and is also an important concept in dynamics simulations. This macroscopic quantity is related to the micro-scopic description of simulations through the kinetic energy, which is calculated from the atomic velocities.

The temperature and the distribution of atomic velocities in a sys-tem are related through the Maxwell–Boltzmann equation:

Eq. 86

This well known formula expresses the probability f (v) that a mol-ecule of mass m has a velocity of v when it is at temperature T. Figure 31 shows this distribution at various temperatures.

f v( )dvm

2πkt-----------

3 2/e

mv2

2kT----------–

4πv2dv=



The x, y, z components of the velocities, on the other hand, have Gaussian distributions:

Eq. 87

How initial velocities are generated

The initial velocities are generated from the Gaussian distribution of vx, vy, and vz. The Gaussian distribution is generated from a ran-dom number generator and a random number seed.

How temperature is calculated

Temperature is a thermodynamic quantity, which is meaningful only at equilibrium. It is related to the average kinetic energy of the system through the equipartition principle. This principle states that every degree of freedom (either in momenta or in coordi-nates), which appears as a squared term in the Hamiltonian, has an

Figure 31. Maxwell–Boltzmann distribution of velocity of water at various temperatures

Distribution of model velocities at equilibrium as predicted by the Maxwell–Bolt-zmann equation. The simulation program assigns random initial velocities to a system of atoms such that the overall distribution of velocities matches a Max-well–Boltzmann distribution for the desired temperature.

pro

ba

bili

ty (

¥ 10

00)

300K

0 1000 20000

1

2

speed (m s-1)

3100K

600K

1000K

g vx( )dvxm

2πkT-------------

1 2/e

mvx2–

2kT-------------

dvx=

Temperature


average energy of kT/2 associated with it. This is true for momenta pi which appear as pi

2/2m in the Hamiltonian.

Relation of kinetic energy, degrees of freedom, and temperature

Hence we have:

Eq. 88

The left side of Eq. 88 is also called the average kinetic energy of the system, Nf is the number of degrees of freedom, and T is the thermodynamic temperature. In an unrestricted system with N atoms, Nf is 3N because each atom has three velocity components (i.e., vx, vy, and vz).

Instantaneous kinetic temperature

It is convenient to define an instantaneous kinetic temperature function:

Eq. 89

The thermodynamic tem-perature

The average of the instantaneous temperature Tinstan is the ther-modynamic temperature T.

Temperature is calculated from the total kinetic energy and the total number of degrees of freedom. For a nonperiodic system:

Temperature in nonperi-odic systems…

Eq. 90

Six degrees of freedom are subtracted because both the translation and rotation of the center of mass are ignored.

…and in periodic systems And for a periodic system:

Eq. 91

pi2

2m-------

i

N

∑ K⟨ ⟩Nf kBT

2---------------= =

Tinstan2K

Nf kB-----------=

3N 6–( )kBT

2-------------------------------

mivi2

2------------

i 1=

N

∑=

3N 3–( )kBT

2-------------------------------

mivi2

2------------

i 1=

N

∑=



Only the three degrees of freedom corresponding to translational motion can be ignored, since rotation of a central cell imposes a torque on its neighboring cells.

How temperature is controlled

Although the initial velocities are generated so as to produce a Maxwell–Boltzmann distribution at the desired temperature, the distribution does not remain constant as the simulation continues. This is especially true when the system does not start at a mini-mum-energy configuration of the model. This occurs often, since the model is commonly minimized only enough to eliminate any hot spots.

During dynamics, kinetic energy is changed to potential energy as the minimized structure changes to the thermal equilibrium struc-ture, and the temperature also changes.

Need to control tempera-ture

To maintain the correct temperature, the computed velocities have to be adjusted appropriately. In addition to maintaining the desired temperature, the temperature-control mechanism must produce the correct statistical ensemble. This means that the prob-ability of occurrence of a certain configuration obeys the laws of statistical mechanics.

For example, in order for constant-temperature, constant-volume dynamics to generate the canonical ensemble, P(E) (i.e., the proba-bility that a configuration with energy E will occur) must be pro-portional to exp(-E/kBT), also called the Boltzmann factor.

Methods of controlling temperature

Only temperature-control methods used in MSI’s simulation engines (Table 20) are considered here:

Direct velocity scaling

Berendsen method of temperature-bath coupling

Nosé and Nosé–Hoover dynamics

Andersen method

Temperature


aTemperature scaling is not generally used to control temperature during a simulation, but to quickly change the simulated temperature to the desired value. It does not produce the correct statistical ensemble.bReferred to as T_DAMPING in Cerius2•Dynamics Simulation.cReferred to as Nosé in the Cerius2•Discover and Insight•Discover_3 mod-ules and Hoover in Cerius2•Dynamics Simulation.dCDiscover only, not in FDiscover.

Choosing the tempera-ture-control method(s)

To access the controls used for specifying the temperature-control method:

♦ In Cerius2, go to the DYNAMICS SIMULATION card on the OFF METHODS deck of cards. Click the Run card menu item to access the Dynamics Simulation control panel. You can set the target temperature for velocity scaling with the Required Temperature entry box. The Preferences… buttons for each sta-tistical ensemble and type of dynamics simulation give access to additional temperature-control methods and parameters.

♦ In Cerius2•Discover, click the Run menu item on the DIS-COVER card to access the Run Discover control panel. Set the Task popup to Dynamics and click the More… pushbutton to the right of the Task popup. Set the Ensemble popup in the Dis-cover Dynamics control panel to NVT or NPT and select the Thermostat. Other controls depend on which thermostat you choose.

♦ In the Insight•Discover_3 module, select the Calculate/Dynamics command. Controls for setting the temperature con-trol method are displayed in the parameter block when you toggle More on and when an ensemble requiring temperature control is chosen.

Table 20 Temperature-control methods used by MSI simulation engines

methodsimulation engine

CHARMm Discover OFF

Velocity scalinga √ √ √Berendsen temperature bathb √ √ √Nosé √Nosé–Hooverc √d √Andersen √4



♦ In the Insight•Discover module, velocity scaling is automati-cally used during the equilibration stage (Stages and duration of dynamics simulations) and the Berendsen method during the data-collection stage. You can change the relaxation time used with the Berendsen method by selecting the Parameters/Vari-ables command, toggling Timtmp on, and entering a value in the TIM_TMP entry box.

♦ In QUANTA, select the CHARMm/Dynamics Option menu item. Choose Setup Detailed Dynamics and click OK. You can also specify how atomic velocities are assigned or scaled.

Direct velocity scaling

Direct velocity scaling is a drastic way to change the velocities of the atoms so that the target temperature can be exactly matched whenever the system temperature is higher or lower than the tar-get by some user-defined amount.

Important

Implementation In Discover, the velocities of all atoms are scaled uniformly as fol-lows:

Eq. 92

In the Cerius2•Dynamics Simulation module, the rescale factor is:

Eq. 93

where Tinst = the instantaneous temperature, Tr = the required tem-perature, Tavg = the average kinetic temperature, and the term under the root is positive. Otherwise, rescaling is not done.

CHARMm allows you to assign velocities based on values in a comparison coordinate file or to assign either a uniform or a Gaus-sian distribution of velocities to the atoms. The Gaussian distribu-tion is recommended.

Direct velocity scaling can not be used to generate realistic thermodynamic ensembles, since it suppresses the natural fluctuations of a system.

vnew

vold-----------

2 Ttarget

Tsystem----------------=

2Tinst---------- Tr Tavg–( ) 1+

Temperature


Achieving equilibrium Direct temperature scaling adds (or subtracts) energy from the sys-tem efficiently, but it is important to recognize that the fundamen-tal limitation to achieving equilibrium is how rapidly energy can be transferred to, from, and among the various internal degrees of freedom of the model. The speed of this process depends on the energy expression, the parameters, and the nature of the coupling between the vibrational, rotational, and translational modes. It also depends directly on the size of the system, larger systems tak-ing longer to equilibrate.

Berendsen method of temperature-bath coupling

After equilibration, a more gentle exchange of thermal energy between the system and a heat bath can be introduced through the Berendsen et al. (1984) method (referred to as temperature damp-ing in Cerius2•Dynamics Simulation), in which each velocity is multiplied by a factor λ given by:

Eq. 94

where ∆t is the timestep size, τ is a characteristic relaxation time, T0 is the target temperature, and T the instantaneous temperature.

To a good approximation, this treatment gives a constant-temper-ature ensemble that can be controlled, both by adjusting the target temperature T0 and by changing the relaxation time τ (generally between 0.1 and 0.4 ps).

This is a simple approach that does not use Hamiltonians.

Nosé and Nosé–Hoover dynamics

Produces true canonical ensembles…

Nosé dynamics (Nosé 1984a, 1984b, 1991) is a method for perform-ing constant-temperature dynamics that produces true canonical ensembles both in coordinate space and in momentum space. The Nosé–Hoover formalism (referred to as Nosé in Discover and as Hoover in Cerius2•Dynamics Simulation) is based on a simplified reformulation by Hoover (1985), which eliminates time scaling and therefore yields trajectories (Dynamics trajectories) in real time and with evenly spaced time points. The method is also called the Nosé–Hoover thermostat.

λ 1 ∆tτ-----–

T T0–

T---------------

1 2/

=



…through use of a ficti-tious mass

The main idea behind Nosé–Hoover dynamics is that an addi-tional (fictitious) degree of freedom is added to the model, to rep-resent the interaction of the model with the heat bath. This fictitious degree of freedom is given a mass Q. The equations of motion for the extended (i.e., model plus fictitious) system are solved. If the potential chosen for that degree of freedom is correct, the constant-energy dynamics (or the microcanonical dynamics, NVE) of the extended system produces the canonical ensemble (NVT) of the real model.

Hamiltonian and equa-tions of motion for this extended system

The Hamiltonian H* of the extended system is:

Eq. 95

Equations of motion for the real-atom coordinates q and moments p, as well as for the fictitious coordinates S and momentum ζ (where φ is the interaction potential) are:

Eq. 96

Eq. 97

Eq. 98

where Q = the user-defined q_ratio × a constant × g × T;g = number of degrees of freedom;T = temperature.

Magnitude of the fictitious mass and computational efficiency

The choice of the fictitious mass Q of that additional degree of free-dom is arbitrary but is critical to the success of a run. If Q is too small, the frequency of the harmonic motion of the extended degree of frequency is too high. This forces a smaller timestep to be used in integration. However, if Q is too large, the thermaliza-

H* pi

2

2mi--------- φ q( )

Q2----ζ2 gkTlnS+ + +

i

∑=

td

dqi pi

mi-----=

td

dpi

qiddφ

– ζpi–=

tddζ

pi2

mi------∑ gkBT–

Q------------------------------------=

Temperature


tion process is not efficient—as Q approaches infinity, there is no energy exchange between the heat bath and the model.

The choice of Q should therefore be based on a balance between the stability of the solution and the highest-frequency motions of the model.

Suggested value for ficti-tious mass…

Q should be different for different models—Nosé (1991) suggests that Q should be proportional to gkBT, where g is the number of degrees of freedom in the model, kB is the Boltzmann constant, and T is the temperature.

To determine the proportionality constant, studies were done with a model consisting of a box of liquid argon containing 343 atoms at 87 K. Setting Q to 2.5 10-5 kcal mol-1 fs-2 for this model yielded good results. This proportionality constant, together with the gkBT term, was then used to generate Q for a box of water and an amor-phous cell of polypropylene, which also yielded satisfactory results.

Discover ordinarily chooses Q automatically. However, you may choose to multiply this calculated Q by a user-defined Q ratio.

…or for the relaxation time In the Cerius2•Dynamics Simulation module, the factor that you can control directly is τ, a relaxation time for the model (τ2 is directly proportional to Q).

For simple fluid models, τ can be chosen as the second moment of the velocity autocorrelation function. For covalently bonded mod-els, however, it is better to choose τ based on the frequencies involved in the model.

A rule of thumb, therefore, is to choose τ on the same order as the smallest time scale (highest frequency) of your model.

Timestep to use Tests on polypropylene indicate that Nosé–Hoover dynamics needs a timestep of 0.5 fs with the velocity Verlet method in order to approach within 3% of the target temperature of 298 K. In con-trast, 1.0 fs is sufficient for the direct velocity scaling method of temperature control.

For the Nosé–Hoover method, the smaller the timestep, the closer it approaches the target temperature. Apparently, the need for a smaller timestep to achieve accuracy is unrelated to Q, as long as Q is in the appropriate range (because the error in reaching the tar-



get temperature remains the same when Q is increased or decreased by a factor of 2).

Accurate integrator For energy to be conserved, the Nosé–Hoover method also requires an accurate integrator. Thus, the ABM4 integrator, if avail-able, with 0.5 fs as the timestep should be used.

Andersen method

One version of the Andersen method of temperature control involves randomizing the velocities of all atoms at a predefined “collision frequency”. (The other version involves choosing one atom at each timestep and changing its velocity according to the Boltzmann distribution.)

The CDiscover program implements the first version. The pre-defined frequency is proportional to N 2/3, where N is the number of atoms in the system. Although this frequency is calculated by the program, you can change it.

Pressure and stress


Units and sign conventions for pressure and stress

How pressure and stress are calculated

How pressure and stress are controlled

What are pressure and stress?

Pressure is another basic thermodynamic variable that defines the state of the system. It is a familiar concept, defined as the force per unit area. Standard atmospheric pressure is 1.013 bar, where 1 bar = 105 Pa. A single number for the pressure implies that pressure is a scalar quantity, but in fact, pressure is a tensor of the more gen-eral form (McQuarrie 1976):

Pressure and stress


Eq. 99

Each element of the tensor is the force that acts on the surface of an infinitesimal cubic volume that has edges parallel to the x, y, and z axes. The first subscript refers to the direction of the normal to the plane on which the force acts, and the second subscript refers to the direction of the force.

In an isotropic situation, where forces are the same in all directions and there is no viscous force, the pressure tensor is diagonal. With the same diagonal elements, the tensor can be written as:

Eq. 100

where the scalar quantity p is the equivalent hydrostatic pressure.

Sometimes, (especially in materials science studies), the the stress tensor, or stress, is used in preference to the pressure tensor (its negative). The diagonal elements are known as the tensile stress, and the nondiagonal elements are the shear stress.

The changes in unit-cell lattice parameters and volume resulting from the stress can be obtained from an analysis of dynamics tra-jectory output file data (Dynamics trajectories). Multiple dynamics runs can be performed at varying stress or pressure values, and the strains obtained can be used to plot a stress–strain curve.

Units and sign conventions for pressure and stress

Pressure and stress can be expressed in many different units. The most common ones are bar and GPa. In SI units, 1 bar = 105 N m-2 and 1 GPa = 109 N m-2. Hence, 1 GPa = 104 bar. Pressure is usually expressed in bars, but in materials science, stress is often expressed in terms of GPa.

P

Pxx Pxy Pxz

Pyx Pyy Pyz

Pzx Pzy Pzz

=

P p

1 0 0

0 1 0

0 0 1

=



In the CDiscover program, constant-stress dynamics has GPa as the default unit. Since the FDiscover program is geared towards controlling pressure, the unit used is bar. In the Cerius2•Dynamics Simulation module, GPa are the only units used for pressure and stress.

Although pressure and stress are defined with the same physical quantities, they have opposite sign conventions. Positive pressure implies a compressive force pushing the system inward, but posi-tive stress means a force acting outward to expand the system.

How pressure and stress are calculated

Pressure is calculated through the use of the virial theorem (Allen and Tildesley 1987). Like temperature, pressure is a thermody-namic quantity and is, strictly speaking, meaningful only at equi-librium.

Relation of pressure, tem-perature, volume and internal virial

Thermodynamic pressure, thermodynamic temperature, volume, and internal virial can be related in the following way:

Eq. 101

And W is defined as:

Eq. 102

Volume (and pressure) is known only for periodic models

Note that pressure is defined only when the system is placed in a container having a definite volume. In a computer simulation, the unit cell under periodic boundary conditions is viewed as the con-tainer. Volume, pressure, and density can be calculated only when the model is recognized as periodic.

Instantaneous pressure Analogous to the temperature, an instantaneous pressure function P can be defined so that thermodynamic pressure is the average of the instantaneous values:

PV NkbT23--- W⟨ ⟩+=

W12--- ri fi⋅

i 1=

N

∑=

Pressure and stress


Eq. 103

where P is the instantaneous pressure and T is the instantaneous kinetic temperature, which is related to the instantaneous kinetic energy K of the system as:

Eq. 104

The instantaneous pressure function can be written as:

Eq. 105

Instantaneous pressure tensor

As mentioned above, pressure is a tensor and its components can also be expressed in tensorial form. Eq. 101 can be recast in the form of:

Eq. 106

In detail, the two terms on the right-hand side of the equation are:

Eq. 107

PNkbT

V--------------

23---

WV------+=

T2

3Nkb-------------K=

P2

3V------- K W+( )=

P1V--- mivivi

T

N

∑ rifiT

N

∑+=

miviviT

i 1=

N

∑

mivixvix

i

∑ mivixviy

i

∑ mivixviz

i

∑

miviyvix

i

∑ miviyviy

i

∑ miviyviz

i

∑

mivizvix∑ mivizviy∑ mivizviz∑

=



Eq. 108

Instantaneous hydrostatic pressure

where rix, vix, and f ix indicate the x components of the position, velocity, and force vectors of the ith atom, respectively.

From the definition of the instantaneous pressure tensor, the instantaneous hydrostatic pressure is calculated as 1/3 the trace of the pressure tensor (Pxx + Pyy + Pzz).

Forces on the image atoms

When periodic boundary conditions are used, atoms in the unit cell interact not only with the other atoms in the unit cell but also with their translated images. Forces on the images in the virial W must be included correctly. If the interaction is pairwise, using Newton’s third law, W can be written as:

Eq. 109

instead of:

Eq. 110

Berendsen et al. (1984) use the rij ⋅ fij formalism by evaluating the virial and the kinetic energy tensor based on the centers of mass, which is valid because the internal contribution to the virial is can-celed (on the average) by the internal kinetic energy. Because of the way forces are evaluated, rescaling of coordinates is also done on the basis of the centers of mass of the models.

How coordinates are scaled in response to pres-sure changes

The Discover program, however, calculates pressure on an atomic basis and performs atomic scaling. The major advantage of using atomic scaling of the coordinates is that atom overlapping can be avoided. Such overlap can occur if centers of mass are moved

rif iT

i 1=

N

∑

rixfix

i

∑ rixfiy

i

∑ rixfiz

i

∑

riyfix

i

∑ riyfiy

i

∑ riyfiz

i

∑

rizfix∑ rizfiy∑ rizfiz∑

=

W12--- ri jfi j

>

∑=

W12--- rifi∑=

Pressure and stress


instead of individual atoms. In addition, for large models having internal flexibility, atomic scaling yields a smoother response to pressure changes.

Explicit or implicit images In the Discover program, when explicit images (also called ghost atoms) are used under periodic boundary conditions, the rij ⋅ fij for-malism is used, with the explicit images included.

In the FDiscover program when minimum images (implicit images) are used under periodic boundary conditions, the rij ⋅ fij formalism is used so that forces between real and image atoms can be properly accounted for.

How pressure and stress are controlled

As with temperature (How temperature is controlled), the pressure (and stress) -control mechanism must produce the correct statisti-cal ensemble. This means that the probability of occurrence of a certain configuration obeys the laws of statistical mechanics.

Methods of controlling pressure

Only pressure-control methods used in MSI’s simulation engines (Table 21) are considered here:

Berendsen method of pressure control

Andersen method of pressure control

Parrinello–Rahman method of pressure and stress control

aCDiscover only, not in FDiscover.

Table 21. Pressure- and stress-control methods used by MSI simulation engines


CHARMm Discover OFF

Berendsen pressure “bath” √ √Andersen √a √Parrinello–Rahman √1 √



Important

Choosing the pressure- and/or stress-control method(s)

To access the controls used for specifying the pressure- and stress-control methods:

♦ The Cerius2•Dynamics Simulation module automatically uses the Parrinello–Rahman method to control stress and the Ander-sen method to control pressure.

♦ In the Cerius2•Discover and Insight•Discover_3 modules, the Parrinello–Rahman method is the default for pressure (and stress) control. If you really want to change it, you can write out the command input file, edit the BTCL dynamics command statement with a text editor, and then use that file for your run.

Alternatively (for Insight•Discover_3), select the Language_Control/Command_Comment command. Set the Comment Type to Command. Enter pressure_control_method = Andersen_cp or pressure_control_method = Berendsen_pc in the Command/Comment entry box and select Execute. Be sure that you insert this stage at the correct point in your command input file.

♦ The Insight•Discover and QUANTA•CHARMm modules always uses the Berendsen pressure-control method.

Berendsen method of pressure control

Pressure changes can be accomplished by changing the coordi-nates of the particles and the size of the unit cell in periodic bound-ary conditions.

How it works The Berendsen method (Berendsen et al. 1984) couples the system to a pressure “bath” to maintain the pressure at a certain target. The strength of coupling is determined both by the compressibility of the system (using a user-defined variable γ) and by a relaxation time constant (a user-defined variable τ ). At each step, the x, y, and z coordinates of each atom are scaled by the factor:

With the Berendsen and Andersen methods the volume can change, but there is no change in the shape of the cell; thus only the pressure is controlled. With the Parrinello–Rahman method, the cell’s shape can change, and therefore both pressure and stress can be controlled.

Pressure and stress


Eq. 111

where ∆t is the time step, P is the instantaneous pressure, and P0 is the target pressure. The Cartesian components of the unit cell vec-tors are scaled by the same factor µ.

Change in cell size but not shape

Note that this method (as implemented) changes the cell uni-formly, so that the size of the cell is changed, but not its shape. Therefore, for simulations such as crystal phase transitions, where both the cell size and shape are expected to change, this method is not appropriate.

Disadvantages of the method

The pressure fluctuation has been observed to be large during test runs with the Discover program and in studies in the literature (Brown and Clark 1984).

Negative pressures sometimes occur because the virial can be neg-ative, even though this defies the usual sense that pressure is a positive number. The calculated pressure depends on the cutoff distances used in the simulation.

Precautions To compensate for the missing long-range part of the potential contributions to the energy, pressure at r > rc can be estimated by assuming the radial distribution g (r) ~ 1 (uniform distribution) in that region. With this assumption and the known form of nonbond potentials, the correction can be estimated analytically (Allen and Tildesley 1987).

However, this correction is not built into the calculation. If the cut-off distance is too short, the calculated pressure may be wrong. Therefore in practice, you should test the effect of cutoff on pres-sure by gradually increasing the cutoff and choosing the appropri-ate cutoff accordingly.

Andersen method of pressure control

Change in cell size but not shape

With the Andersen method (1980) of pressure control, the volume of the cell can change, but its shape is preserved by allowing the cell to change isotropically.

Uses The Anderson method is useful for liquid simulations since the box could become quite elongated in the absence of restoring

µ 1∆tτ--------γ P P0–[ ]+

1 3/

=



forces if the shape of the cell were allowed to change. A constant shape also makes the dynamics analysis easier.

However, this method is not very useful for studying materials under nonisotropic stress or phase transitions, which involve changes in both cell lengths and cell angles (for these conditions, the Parrinello–Rahman method should be used).

How it works The basic idea of the method is to treat the volume V of the cell as a dynamic variable in the system. The Lagrangian of the system is modified so that it contains a term in the kinetic energy with a user-defined mass M and a potential term which is the pV poten-tial derived from an external pressure Pext acting on volume V of the system.

Parrinello–Rahman method of pressure and stress control

Both size and shape of cell can change

The Parrinello-Rahman method of pressure and stress control can allow simulation of a model under externally applied stress. This is useful for studying the stress–strain relationship of materials. Both the shape and the volume of the cell can change, so that the internal stress of the system can match the externally applied stress.

Summary of derivation The method is presented in detail by Parrinello and Rahman (1981) and is only summarized here.

The Lagrangians of the system are modified such that a term rep-resenting the kinetic energy of the cell depends on a user-defined mass-like parameter W. An elastic energy term pΩ is related to the pressure and volume or the stress and strain of the system. The equations of motion for the atoms and cell vectors can be derived from this Lagrangian. The motion of the cell vectors, which deter-mines the cell shape and size, is driven by the difference between the target and internal stress.

With hydrostatic pressure only:

Eq. 112L12--- mis

·i ′Gs·

i 1=

N

∑ φ rij( )

j i>

N

∑i 1=

N

∑–12---WTrh· ′h· pΩ–+=



where h = a ⋅ b ⋅ c is the cell vector matrix, G = h′h, ri = hsi, φ is the interaction potential, and Ω is the volume of the cell. The dots above some symbols indicate time derivatives and the primes indicate matrix transposition. Tr is the trace of a matrix.

With stress, the elastic term pΩ is replaced by:

Eq. 113

where S is the applied stress, Ω0 is the initial volume, and ε is the strain; ε is also a tensor, defined as (h0′-1 Gh0

-1 - 1)/2.

Choice of the mass vari-able

Note the user-defined variable W, which determines the rate of change of the volume/shape matrix.

A large W means a heavy, slow cell. In the limiting case, infinite W reverts to constant-volume dynamics. A small W means fast motion of the cell vectors. Although that may mean that the target stress can be reached faster, there may not be enough time for equilibration.

In tests, too small a W also induced artificial periodic motions of the cell. A value of 20 seems to give satisfactory results for a cell of 76mers of polyethylene.


In addition to relatively simple dynamics simulations, it is also possible to bias or control the dynamics run in several ways and/or to combine dynamics and minimization in one simulation. These types of dynamics runs can be used in conjunction with one or more of the various thermodynamic statistical ensembles (as appropriate, see Statistical ensembles).

Several such types of dynamics simulations are discussed in this section:

Quenched dynamics

Simulated annealing

Consensus dynamics

Impulse dynamics

p Ω Ω0–( ) Ω0Tr S p–( )ε+



Langevin dynamics

Stochastic boundary dynamics

Multibody order-N dynamics

aAvailable through Insight•Discover_3 module for CDiscover, via DSL only for FDiscover; not yet available in Cerius2•Discover.bTruncated (only one T-to-0 K half-cycle), and referred to as “quenched dynamics” in the CHARMm documentation.cAvailable in standalone mode only.dAvailable in standalone mode of CDiscover only.

Choosing the type of dynamics

To access the controls used for specifying types of dynamics simu-lations:

♦ In Cerius2, go to the DYNAMICS SIMULATION card on the OFF METHODS deck of cards. Click the Run card menu item to access the Dynamics Simulation control panel. Check the check box next to the desired type of simulation. You can access additional controls relevant to each type by clicking the Prefer-ences… button to its right.

♦ To set up complicated dynamics runs with the Insight• Discover_3 module, you need to set up several stages of mini-mization (if desired) and dynamics, using the Calculate/Mini-mize and Calculate/Dynamics commands. You probably also need to write and read appropriate files at various stages of the run with the Language_Control/File_Control command. In

Table 22. Types of dynamics simulations easily set up with MSI simulation engines


CHARMm Discover OFF

Quenched dynamics √a √Simulated annealing √b √1 √Consensus dynamics √c

Impulse dynamics √d √Langevin dynamics √Stochastic boundary dynamics √



addition, you may need to set up control loops (such as If and Foreach statements) with the Language_Control/Looping_Control command.

Alternatively, to produce a phi/psi map (showing energy as a function of rotation about two torsions of a model), you can use the Strategy/Phi_Psi_Map command.

♦ In QUANTA, select the CHARMm/Dynamics Option menu item. Use the Setup Heating, Setup Equilibration, and Setup Simulation radio buttons to access dialogs in which you set temperatures as desired for simulated annealing for these stages.

Langevin dynamics is set up by clicking the Setup Detailed Dynamics button. You can combind Langevin dynamics with simulated annealing by setting the Bath Temperature to 0.

Stochastic boundary conditions are set by selecting the CHARMm/Constraints Options/Stochastic Bdry Settings menu item to specify these constraints and then selecting the CHARMm/Constraints Options/Stochastic Bdry On menu item to activate them.

Exploring conformational space using dynamics

A common limitation of classical minimization algorithms is that they usually locate a local minimum close to the starting configu-ration, which is not necessarily the global minimum (Local or global minimum?). This is because the minimizers discussed under Mini-mization algorithms are specifically designed to ignore configura-tions if the energy increases.

By using the available thermal energy to climb and cross confor-mational energy barriers, dynamics provides insight into the accessible conformational states of the molecule that would be inaccessible to classical minimization.

The results can be assessed by plotting selected data in 3D format (e.g., a phi/psi map for rotation about the peptide bond) or by minimizing selected structures that are generated during the dynamics run (see Quenched dynamics).



Quenched dynamics

How it works In quenched dynamics, periods of dynamics are followed by a quench period in which the structure is minimized. You can spec-ify the simulation time between quenches and the number of min-imization steps. The quenched structure can then be written to a trajectory file (Dynamics trajectories), and dynamics continues with the prequenched structure.

Uses Quenched dynamics is a way to search conformational space for low-energy structures.

Simulated annealing

How it works In simulated annealing, the temperature is altered in time incre-ments from an initial temperature to a final temperature and back again. This cycle can be repeated. The temperature is changed by adjusting the kinetic energy of the structure (by rescaling the velocities of the atoms).

Simulated annealing can be combined with quenched dynamics. That is, at the end of each temperature cycle, the lowest-energy structure of that cycle can be minimized and saved in a trajectory file (Dynamics trajectories). Annealing continues using the last structure and velocities from the previous cycle.

Tip

Simulated annealing cannot be combined with impulse dynamics (Impulse dynamics).

Uses Annealing allows the energy of the structure to be changed grad-ually, without allowing the structure to become trapped in a con-formation that has a lower energy than nearby conformations but a higher energy than more distant conformations (that is, in a local energy minimum).

Use the quartic form of the nonbond term for simulated annealing studies in which the initial coordinates are unknown. These simulations can involve atoms moving through other atoms and for this it is essential that the nonbond term not go to infinity.



Consensus dynamics

How it works Consensus dynamics can be thought of as an extension of the teth-ering restraint technique (Template forcing, tethering, quartic droplet restraints, and consensus conformations). In tethering, a model sys-tem serves as a fixed template. The “moving” system, driven by molecular mechanics or dynamics, is then forced to conform to the template by applying restraints.

In contrast, the consensus technique allows the “template” to respond to changes in the “moving” system, by treating all the models as “moving templates”. The net result is that both models change so that their structures become similar. Consensus dynam-ics can be applied to more than two models simultaneously.

Uses Consensus restraints have several uses. One example is the deter-mination of structural similarities among a set of homologous compounds—perhaps several similar compounds that bind to a particular receptor. If you hypothesize that an apparently homolo-gous region of the compounds is responsible for the binding, then you would want to find a configuration for this region that is com-patible with relatively low-energy conformations for all the com-pounds. By applying consensus restraints to the homologous region of each compound, you can use dynamics followed by min-imization to find a likely binding configuration.

Impulse dynamics

How it works Impulse dynamics allows you to assign initial directional veloci-ties to selected atoms before carrying out dynamics.

Impulse dynamics can be used only with constant-energy, con-stant-volume dynamics (NVE ensemble) and cannot be combined with simulated annealing (Simulated annealing).

Uses You can use impulse dynamics to push interacting molecules over energy barriers before allowing the structure to relax.



Langevin dynamics

How it works The Langevin dynamics method (McCammon et al. 1976, Levy et al. 1979) approximates a full molecular dynamics simulation of a system by eliminating unimportant or uninteresting degrees of freedom. The effects of the eliminated degrees of freedom are sim-ulated by mean and stochastic forces.

Friction coefficients must be assigned to selected atoms before starting the dynamics run. Langevin dynamics includes a con-stant-temperature bath.

Uses For example, instead of simulating hundreds of solvent molecules surrounding the solute molecules, the solvent can be ideally repre-sented as a viscous fluid described in terms of dissipative and fluc-tuative equations.

Stochastic boundary dynamics

How it works The stochastic boundary molecular dynamics method (Brooks et al. 1985a) uses a combination of both Langevin dynamics and Newtonian dynamics. With this method, the model is partitioned into a reaction region where Newtonian dynamics simulation is run, a buffer region where Langevin dynamics is run, and a reser-voir region.

In this way, atoms distant from the specific interactive sites in a large macromolecular system can be effectively eliminated from extensive analysis.

Uses This allows detailed studies of spatially localized portions of inter-acting models. Enzyme–substrate interactions at the active site can be effectively studied using this technique.

Multibody order-N dynamics

The Insight•MBOND module and the standalone MBOND/CHARMm program (documented separately) is used for multi-body order-N dynamics, in which models are substructured into a set of interconnected rigid and flexible bodies, as well as atomistic regions. The rigid bodies move as units, and the deformations of



flexible bodies are represented by sets of low-frequency compo-nent modes. The atomistic regions are simulated by conventional dynamics. This method is another way of eliminating unimportant or uninteresting degrees of freedom and thus allowing a longer timestep to be used.


Constraints and restraints (Applying constraints and restraints) can be selectively applied during a dynamics run to save computing time and/or focus the simulation on more interesting parts of your model. For example, atoms can be specified as fixed or movable, and restraints can be applied in order to pull certain atoms towards one another.

Improving computational efficiency

In addition, certain types of constraints are applied only during dynamics, to increase computational efficiency. This section includes information only on constraints that are used only in dynamics simulations:

The SHAKE algorithm

The RATTLE algorithm

Setting constraints during dynamics

To access the controls used for setting up SHAKE or RATTLE dynamics:

♦ In the Cerius2•Discover module, select the Run menu item in the DISCOVER card to open the Run Discover control panel. In this control panel, set the Task popup to Dynamics and click the More… pushbutton to the right of the Task popup to open the Discover Dynamics control panel. In this panel, check the Rattle check box and click the More… pushbutton to its right to open the Rattle control panel. Use the latter to specify whether bonds and/or angles should be constrained.

♦ In the Insight•Discover_3 module, select the Calculate/Dynamics command, toggle More on, and then toggle Rattle on.

Note



♦ In QUANTA, select the CHARMm/SHAKE Options menu item. Set the controls as desired and click OK.

Effect on timestep and energy conservation

Bond vibrations constitute the highest frequencies in the system and thus determine the largest timestep that can be used during dynamics. If the bonds are constrained, longer timesteps can be used during dynamics, because of the absence of these high-fre-quency bond vibrations.

In a test run on crambin, a timestep as long as 3 fs could be used with the RATTLE algorithm, but dynamics without RATTLE or SHAKE can have a timestep of no more than 1 fs. Furthermore, energy conservation for the 3 fs timestep with RATTLE was as good as that at 1 fs without RATTLE or SHAKE.

Computational costs Although constraining bonds allows the timestep to be increased without incurring significant overhead for carrying out RATTLE iteratively, constraining angles does not really help to reduce the computational cost, because it takes a long time for the iterative procedure to converge. In some circumstances, it actually takes much longer than without RATTLE or SHAKE.

Tip

The SHAKE algorithm

Availability The SHAKE method (Ryckaert et al. 1977), which is a popular algo-rithm for introducing distance constraints during molecular dynamics simulations, is used in CHARMm to constrain the har-monic stretching of bonds.

For full access to the RATTLE functionality, you need to use BTCL statements. You can write out a command input file from Cerius2 or Insight, edit the BTCL rattle command statement(s) with a text editor, and then use that file for your run. Alternatively (in Insight), select the Language_Control/Command_Comment command. Set the Comment Type to Command. Enter the desired rattle command statement(s) in the Command/Comment entry box and select Execute.

We do not recommend that angle constraints be used even though this functionality is available. However, if angle constraints have to be used, you should use a larger tolerance than with bonds.



Allowed constraints during dynamics

SHAKE may be applied to all bonds or only to bonds containing hydrogens, and to no angles, all angles, or only angles containing hydrogens.

Tip

Effect on timestep A twofold increase in the time step (to 0.001 ps) is enabled.

The RATTLE algorithm

What is the RATTLE algo-rithm

Constraints can be applied during dynamics runs via the RATTLE algorithm, which is the velocity version of SHAKE. The RATTLE procedure is to go through the constraints one by one, adjusting the coordinates so as to satisfy each in turn. The procedure is iter-ated until all the constraints are satisfied to within a given toler-ance. Unlike SHAKE, RATTLE (Andersen 1983) makes sure that velocities are adjusted also, to satisfy the constraints, and is suit-able for use with the velocity version of Verlet integrators.

Allowed constraints during dynamics

RATTLE can be used to constrain bonds, angles spanned by two constrained bonds, as well as the distance between any pair of atoms in periodic and nonperiodic systems. It can be used with the constant-volume ensemble.

Note

Dynamics in fixed-geome-try water

Another major use of RATTLE is to enable the use of a fixed-geom-etry water model. With RATTLE, you can use the SPC, TIP3P, or current water models. The initial configurations of the water mol-ecules are set to the equilibrium geometry of the SPC or TIP3P model or to the current geometry of water in your model.

To maintain a high degree of accuracy during the dynamics simulation, as well as to provide a larger integration time step, it is best to apply SHAKE only to bonds containing hydrogens.

For the time being, RATTLE cannot be used with constant-pressure dynamics, since that involves calculating the pressure on a molecule basis, which is not currently available.



Dynamics trajectories

What are trajectories? The results of dynamics simulations can be saved in trajectory files, which are essentially a series of snapshots of the simulation taken at regular intervals. Trajectories can include data such as model structure, minimized-model structure (in quenched dynamics, see Quenched dynamics), temperature, energies, volume, pressure, cell parameters, and stress.

Uses Trajectory files can be used for analysis of the results and also for continuing an interrupted dynamics simulation.

You can use trajectory files for computing the average structure and for analyzing fluctuations in geometric parameters, thermo-dynamic properties, and time-dependent processes. The time series can be evaluated for global properties such as the radius of gyration and the number density of the model. Examples of exper-imental quantities that can be calculated using correlation func-tions are frictional coefficients, IR line widths, fluorescence depolarization rates, spectral densities (the Fourier transform of the correlation function), and NMR relaxation times.

Animations You can also animate trajectories, to view how the model behaved during a dynamics run.

Typically, the displayed model’s conformation is updated during a run so you can monitor progress visually. The frequency with which the display is updated can affect the time needed to perform a dynamics simulation, especially with large models.

With the Cerius2•Dynamics Simulation module, some informa-tion also can be displayed in the form of graphs that are updated as the run proceeds.

Genera l methodo logy fo r dynamicscalculations

To perform dynamics simulations, you need to know something about the general strategy of all types of dynamics simulations,



and you need to know the general procedure for setting up a dynamics run.

This section includes information on:

Stages and duration of dynamics simulations

Dynamics with MSI simulation engines

Restarting a dynamics simulation

Prerequisites One of the most important steps in any simulation is properly pre-paring the system to be simulated. Calculations on the fastest com-puter running the most efficient dynamics algorithm may be worthless if the hydrogen is put on the wrong nitrogen or an important water molecule is omitted.

Unfortunately, it is impossible to provide a single recipe for a suc-cessful model—too much depends on the objectives and expecta-tions of each calculation. Are conformational changes that are far removed in conformational space from the area being focused on expected or interesting? What is the hypothesis being tested? The effects of specific dynamics variables, as well as tethering, fixing, energy cutoffs, etc., on the results can be answered only by con-trolled preliminary experiments.

Stages and duration of dynamics simulations

Dynamics simulations are usually carried out in two stages, equil-ibration and data collection (or “production”). The duration of each stage depends on the system as well as on the purpose of the run.

Equilibration stage For the equilibration stage, you typically assign random velocities to atoms in the model according to the Maxwell–Boltzmann distri-bution around the desired target temperature. Temperature con-trol during the equilibration stage is usually by direct velocity scaling (Direct velocity scaling). Depending on your model and the purposes of your simulation, you may bring the simulated system up to the target temperature relatively quickly or in gradual steps.



The purpose of equilibration is to prepare the system so that it comes to the most probable configuration consistent with the tar-get temperature and pressure.

When a system is large, it may take a long time to equilibrate because of the vast conformational space it has to search. The con-tour of the energy surface is another factor to consider. If energy barriers between various local minima and the global minimum are high, barrier crossing is difficult and may take longer.

Has equilibrium been achieved?

One way to judge whether a model has equilibrated is to plot the various thermodynamic quantities, such as energy, temperature and pressure, versus time. When equilibration has been achieved, these quantities fluctuate around their averages, which remain constant over time. This is a necessary but not sufficient test, because it is not unusual for a sudden conformational change to occur after a long period of time.

Another way to check equilibration is to start the calculation with different initial conformations and different initial velocities. Con-vergence to similar conformations and properties from different initial values is a good indicator that equilibration has occurred.

Production (data-collec-tion) stage

After equilibrating the system at the target temperature and pres-sure, you can begin the production stage, during which data and statistics are collected. Temperature control during the production stage is generally by a more gentle, realistic method than direct velocity scaling (How temperature is controlled). You may use differ-ent thermodynamic ensembles (Statistical ensembles) during the equilibration and production stages.

Depending on the purpose of your study (Types of dynamics simu-lations), you may run several data-collection simulations under different conditions after a single equilibration run or stage.

How long should the simu-lation be?

One commonly asked question is how long to collect the statistics. The answer depends on both the properties being calculated and the model being simulated.

Some quantities, such as internal energy, temperature, and pres-sure, converge readily, since they are calculated from averages and the dominant contributions are from the most probable states. Other quantities, like specific heat or isothermal compressibilities, are more difficult to obtain, since they depend on the fluctuations.



One empirical way to find out how long is long enough for a dynamics simulation is to monitor the change in the desired quan-tities over time.

The length of a trajectory needed to calculate a property depends on the time variation of the property under consideration. If the property is a slowly varying function, the dynamics integration should be extended to cover several periods.

Characteristic durations for common events in real molecules are listed in Table 23.

If a property varies randomly about a mean value with a decay time of t and the simulation is run for length T, the variance in the estimate is proportional to (t/T)0.5. When multiple independent fragments are present (for example, each water molecule in a sol-vent simulation), averaging can be done over the fragments to improve sampling.

Dynamics with MSI simulation engines

Prerequisites To set up a dynamics run, first:

1. Choose the desired forcefield if you don’t want to use the default forcefield (Forcefields).

Table 23. Durations of some real molecular events

Event Approximate duration

Bond stretching. 1–20 fsElastic domain modes. 100 fs to several psWater reorientation. 4 psInter-domain bending. 10 ps–100 nsGlobular protein tumbling. 1–10 nsAromatic ring flipping. 100 µs to several secondsAllosteric shifts. 2 µs to several secondsLocal denaturation. 1 ms to several seconds



2. Set up the forcefield and prepare your model (Preparing the Energy Expression and the Model).

3. Run a preliminary minimization

The model usually needs to be minimized (Minimization) to remove strains that might cause abnormally large forces on some atoms and therefore result in unrealistic dynamics simu-lations.

Purpose of the run Next:

4. Specify items such as the dynamics algorithm(s) (Integration algorithms), time step (The choice of timestep), and temperature- and pressure-control methods (How temperature is controlled and How pressure and stress are controlled) (unless you want your calculation to run under the default conditions).

You generally run two dynamics simulations in sequence (or a single two-stage dynamics run). The first run or stage is for equilibrating the model under the desired conditions and the second, for collecting data and statistics (see Stages and duration of dynamics simulations. In Discover and CHARMm, these two runs are generally both set up at the same time, by repeating Steps 4 and 5 for each stage before starting the run.

Additional runs or stages may be required, depending on the purpose of the simulation (see Types of dynamics simulations). Depending on the type of run and the simulation engine, you may be able to set up all stages at the same time, by repeating Steps 4 and 5 for each stage before starting the run. If not, the procedure for restarting runs that have ended is outlined under Restarting a dynamics simulation.

Accessing dynamics con-trols

To find the relevant controls in the molecular modeling pro-grams:

♦ For the Cerius2•Dynamics Simulation module, go to the OFF METHODS deck of cards and choose the DYNAMICS SIMU-LATION card. Select the Run card item to access the Dynamics Simulation control panel. You can access additional tools, such as for setting variables for the dynamics methods and types, by clicking any of several Preferences… buttons in this control panel.



In addition, you may select Dynamics Controls on the DYNAMICS SIMULATION card. These controls, however, are typically left at their default values.

♦ In the Cerius2•Discover module, select the Run menu item in the DISCOVER card to open the Run Discover control panel. In this control panel, set the Task popup to Dynamics and click the More… pushbutton to the right of the Task popup to open the Discover Dynamics control panel. Use the controls in the Equilibration and Production sections of this control panel to set up the equilibration and data-collection stages of the simu-lation.

Additional control over the equilibration stage is available by clicking the More… pushbutton in the Equilibration section to open the Equilibration Options control panel.

If you want a minimization stage to precede the equilibration stage, check the Pre-Minimize check box. The associated More… pushbutton opens the Discover Minimize control panel (Minimization).

♦ In the Insight•Discover_3 module, select the Calculate/Dynamics command. Toggle More on to access additional con-trols. Set the controls as desired and select Execute for each dynamics stage you want to include in your run. You may set up quite complicated simulations by also using other com-mands in the Calculate and Language_Control pulldowns.

Alternatively, if you want to run only a simple minimization followed by dynamics, select the Strategy/Simple_Min_Dyn command and change any desired parameters.

♦ In the Insight•Discover module, select the Parameters/Dynam-ics command. Set the controls as desired and select Execute. You may also want to set some other parameters with the Parameters/Variables command.

♦ In QUANTA, select the CHARMm/Dynamics Option menu item. You can use the Setup Heating, Setup Equilibration, and Setup Simulation radio buttons to access dialogs in which you set controls as desired for warming and equilibrating the sys-tem and then running the data-collection stage of the simula-tion. Alternatively, you can use the Setup Detailed Dynamics button to access a dialog with additional controls.



Discover and CHARMm offer additional functionality when run in standalone mode. (How to run Discover and CHARMm in standalone mode is documented separately—see Available documentation.)

Specifying output 5. Specify the desired output:

♦ In Cerius2•Dynamics Simulation, click the Output… button in the Dynamics Simulation control panel or select Output on the DYNAMICS SIMULATION card to specify what information to display during the simulation.

Click the Trajectory… button in the Dynamics Simulation con-trol panel or select the Trajectory/Output menu item on the DYNAMICS SIMULATION card to write out various types of trajectory files.

♦ In the Cerius2•Discover module, select the Run menu item in the DISCOVER card to open the Run Discover control panel. In this control panel, set the Task popup to Dynamics and click the Output… pushbutton to open the Discover Dynamics Out-put control panel. You can specify output in the form of stan-dard output, table, and/or trajectory files and specify the frequency of output and the type of averaging.

♦ In the Insight•Discover_3 module, use the Analyze/Output command (it may not be accessible until after you have exe-cuted the Calculate/Dynamics command). Set the controls as desired and select Execute. Do this for every stage for which you want output in the form of standard output, table, archive, and/or history files. You can periodically write frames to an archive file with the Language_Control/File_Control com-mand.

♦ In the Insight•Discover module, use the Run/Report and/or Run/Files commands. Set the controls as desired and select Execute.

♦ In QUANTA, you can specify output files with any of the dia-logs accessed by selecting the CHARMm/Dynamics Option menu item.




Verifying the run instruc-tions

6. Review what you have requested for the run:

Since dynamics simulations are often quite complex and time consuming, you should review your run specifications before starting the simulation.

♦ In the Cerius2•Discover module, select the Run menu item in the DISCOVER card to open the Run Discover control panel. In this control panel, click the Input… pushbutton to open the Discover Input File control panel. Click the Save Strategy to .inp File action button.

♦ In the Insight•Discover_3 module, select the Setup/List com-mand, set List Options to Input_File, and select Execute. The command input file is listed in the textport. Then you can read the input file by using any text editor, issuing the UNIX more command, or clicking the Edit .inp File action button.

♦ In the Insight•Discover module, select the Parameters/Dynam-ics or the Run/Run command. Toggle the List option on and select Execute to view the parameters you have set so far.

To view the complete command input file, select the Run/Run command. Set the controls as desired, being sure that Run_Dynamics is on and Computation Mode is set to Command File, and select Execute. (Run_Minimization can be on or off, depending on whether you want to specify that minimization be performed before the dynamics equilibration stage.) Read the input file with any text editor or the UNIX more command.

Starting a dynamics run Finally:

7. Start the dynamics run:

♦ In the Cerius2•Dynamics Simulation module, click the RUN DYNAMICS button in the Dynamics Simulation control panel. (If you are starting a new simulation. Restarting a dynamics sim-ulation, be sure to click the Reset button in this control panel if you need to reinitialize the atom velocities rather than continu-ing from some previous run.)

♦ In the Cerius2•Discover module, click the RUN pushbutton in the Run Discover control panel.

♦ In the Insight •Discover_3 module, execute the D_Run/Run command.



Alternatively, if you want to run only a simple minimization followed by dynamics, select Execute in the Strategy/Simple_Min_Dyn parameter block.

♦ In the Insight•Discover module, select the Run/Run command. Set the controls as desired, being sure that Run_Dynamics is on and Computation Mode is set to Interactive or Batch, and select Execute. (Run_Minimization can be on or off, depend-ing on whether you want minimization to be performed before the dynamics equilibration stage.)

♦ In QUANTA, click CHARMm Dynamics in the Modeling menu window.

Specific information For specific information on setting up and running dynamics with the various MSI simulation engines, and on analyzing the results, please see the relevant documentation (see Available documenta-tion).


You can continue a dynamics run without a break from the previ-ous stage of the simulation, restart an interrupted run from where it ended, or start a new dynamics run from a particular point in a previous run.

When needed With the Insight•Discover_3 module, continuing a run from one stage to the next (e.g., switching from equilibration to data collec-tion or setting up complicated simulations) is straightforward: all stages can be specified when you set up the simulation, before starting the run. However, you may need to continue or restart a run, for any number of reasons.

Cerius2•Dynamics Simulation and CHARMm enable stages only for limited types of simulations, and Cerius2•Discover and Insight•Discover allow you to specify only a simple two-stage equilibrium and data-collection run (with or without preliminary minimization): to set up some other type of simulation, you must wait until a run has ended before setting up subsequent runs.

Read this section if you need to continue or restart a simulation that has already ended.



Files and precautions The trajectory and other input file(s) that are required depend on what simulation engine is being used, and the exact procedure depends on whether you a continuing a run without a break from the last conformation of an immediately preceding run, restarting an interrupted run from the last conformation, or using an inter-mediate point of an earlier run as the basis for a new simulation.

If you changed the model’s conformation (for example, by minimi-zation) since your previous dynamics simulation ended, you need to reassign the initial (random) velocities, since the old velocities do not apply to the new coordinates. The Insight•Discover_3 and Cerius2•Dynamics Simulation modules can detect this situation and reinitialize velocities automatically.

You may need to analyze the data from a dynamics run before doing a restart from other than its final conformation. You may, for example, want to find and use the lowest-energy conformer as the starting point for a new simulation or may want to restart a run with a particular set of velocities.

Please see the relevant specific documentation (Available documen-tation).


Briefly, the general procedure with each simulation engine is:

♦ In the Cerius2•Dynamics Simulation module, specify the tra-jectory file with the Dynamics Trajectory Input control panel, which is accessed by selecting the Trajectory/Input item from the DYNAMICS SIMULATION card. Use this control panel to restart an interrupted run at its final conformation, to specify the conformation number at which to begin a new run, and to specify the type of data to use from the previous run.

To restart the simulation at the last step of an immediately pre-ceding run, click the RUN DYNAMICS button in the Dynam-ics Simulation control panel.

To start a new simulation from the end or some point before the end of a previous run, specify a new dynamics method, algo-rithm, output, etc., if desired (see Purpose of the run). Then click the RUN DYNAMICS button.

To re-randomize the atomic velocities before starting a new run, click the Reset button in the Dynamics Simulation control panel before clicking the RUN DYNAMICS button.



♦ In the Insight•Discover_3 module, select the Language_Con-trol/File_Control command. Set Select File Type to Dyn_Restart and File Operation to Retrieve. Select Execute to set up a stage to read the dynamics restart file.

If you want to start the new simulation from some point before the end of a previous run, next set Select File Type to Archive_File (or History_File) and set the Frame No to a constant or a variable. (A variable can be can be initialized with the Language_Control/Command_Comment command and used, for example, in a Loop statement set up with the Language_Control/Looping_Control command.) Select Execute to set up a stage to control reading frame(s) from the desired file.

Specify any other dynamics parameters if desired, as outlined above in Steps 4 and 5. Velocity (in the Dynamics Calculate parameter block) should be set to Current if you want to con-tinue or restart the previous run. Set it to Create if you want to re-randomize the atomic velocities before starting a new run.

Review the command input file and start the run as outlined in Steps 6 and 7.

♦ With the Cerius2•Discover or Insight•Discover module, you need to use a text editor to prepare an appropriate command input file for continuing, starting, or restarting a run. Also make sure that all required files are in the current run directory.

In Cerius2•Discover, use the prepared command input file by selecting the Run menu item in the DISCOVER card and click-ing the Input… pushbutton in the Run Discover control panel. Then select the desired file using the browser controls in the Discover Input File control panel and click the Run .inp File action button.

In Insight•Discover, use the prepared command input file by selecting the Run/Run command, toggling Strategy on, and entering that file’s name in the Input_File_Name entry box. Select Execute.

♦ In QUANTA, select the CHARMm/Dynamics Option menu item. All the dialog boxes accessed via the radio buttons allow you to start the simulation from the beginning or restart it from the restart file. You can also use the CHARMm/Settings/Restore menu item to reload a CHARMm setup file.



Discover and CHARMm offer additional functionality when run in standalone mode. (How to run Discover and CHARMm in standalone mode is documented separately—see Available documentation.)


6 Free Energy


Forcefield-based calculation of relative or absolute free energy should be considered an “expert” application. We recommend using the FDiscover program for relative and absolute free energy calculations. Although it is possible to run relative free energy cal-culations with CHARMm, that program is slower and less flexible.

This chapter explains Relative free energy—theory and implementation

Absolute free energy

Re la t ive f ree energy — theor y andimplementation

This section includes Finite difference thermodynamic integration (FDTI)

Relative free energy—methodology

To compare chemically distinct systems, the FDiscover program can be used to calculate the relative free energy difference between chemically unique species. This approach uses the finite difference thermodynamic integration (FDTI) algorithm of Mezei et al. (1987). Since FDTI does not require analytical derivatives of the Hamiltonian with respect to the coupling parameter, it is more suited for the complex coupling formalisms required in chemical perturbations. It also has been shown to possess better conver-gence properties than other methods.

Finite difference thermodynamic integration (FDTI)

FDTI combines aspects of both the perturbation method (PM) and thermodynamic integration (TI) to improve the convergence and


6. Free Energy

accuracy of free energy calculations. A review of the properties of the perturbation method helps to appreciate the advantages of the FDTI approach.

Perturbation method (PM) The free energy A is related to the partition function Q by the equa-tion:

Eq. 114

If Q1 and Q2 are the partition functions for States 1 and 2, the dif-ference in free energy between these states is:

Eq. 115

Defining Ei(p,r) as the energy function corresponding to state i, the ratio of partition functions is related to the expectation value of exp[–(E2 – E1)/kBT] by:

Eq. 116

where P1 is the Boltzmann probability function for State 1:

Eq. 117

This can be expressed more compactly in bracket notation as:

Eq. 118

where the subscript 1 indicates that State 1 is considered the refer-ence state (that is, the energy difference ∆E is computed relative to an ensemble of structures for State 1). Thus, the ratio of the parti-tion functions can be computed from an ensemble average of the energy difference between a reference state and a perturbed state.

The free energy change is then given directly as:

A kBT ln Q–=

∆A A2 A1– kBT lnQ2

Q1------–= =

Q2

Q1------ exp E2 E1–( )/kBT–[ ] P1 r pdd∫∫=

P1

exp E1/kBT–[ ]Q1

-----------------------------------=

Q2

Q1------ exp ∆E/kBT–[ ]⟨ ⟩

1=

Relative free energy—theory and implementation


Eq. 119

This approach is accurate only when the energy of the initial and final states differs by a small amount, on the order of 2 kBT. Larger energy differences than this lead to such small values for the expo-nential term that the statistical uncertainty overwhelms the observable. For small changes, this method is very attractive because it calculates the complete free energy difference in one cal-culation.

To calculate free energy differences larger than 2 kBT, several sequential runs are performed, each computing the free energy change over a subinterval. Errors introduced by failure to con-verge at each step are propagated—each successive calculation adds its contribution to the previously accumulated total.

FDTI Thermodynamic integration of the relative free energy assumes that the free energy change can be expressed as an integral:

Assuming that a suitable coupling parameter λ can be found, which adequately describes a continuous conversion between the two states, the above equation can be integrated numerically. FDTI employs the perturbation formalism to numerically compute the derivatives of the free energy function with respect to the coupling parameter. Using Eq. 119, it is possible to compute the change in free energy (∆Ai) for a perturbation δλ away from the ith λ point (λi ± δλ):

Eq. 120

Computing ∆Ai for many different values of λ spanning the inter-val from 0 to 1, dividing each ∆Ai by δλ, and then numerically inte-grating over the interval, the total free energy change ∆A can be estimated. Mathematically, this is summarized as:

∆A A2 A1– kBT ln exp ∆E/kBT–[ ]⟨ ⟩1

–= =

∆AA λ( )δ

λδ------------- λd

0

1

∫=

∆Ai A λi( ) A λ i ± λδ( )– kBT ln exp E λ i( ) E λ i ± λδ( )–( )/kBT–[ ]⟨ ⟩ i–= =


6. Free Energy

Eq. 121

where k is the number quadrature points in the numerical integra-tion. Notice that ∆Ai ⁄ δλ can be computed both for a forward (+δλ) and a backward (–δλ) perturbation. Computing both at the same time takes no more computer time and is a measure of con-vergence (both should be equal for suitable values of δλ). The Dis-cover program averages the two values for the instantaneous value of δA ⁄ δλ at λi.

Method used in FDiscover The value of ∆λi in Eq. 121 for a given i depends on the numerical integration scheme used. For example, a simple trapezoidal rule approach would make each ∆λi equal. More sophisticated integra-tion methods may allow each ∆λi to be different. The FDiscover program uses a Gaussian–Legendre quadrature method (Press 1986) that chooses the values needed, given the total number of intervals specified. Hence, you never explicitly specify the λ inter-vals.

Advantages The advantages of FDTI are:

♦ Unlike the perturbation method (PM), large changes in free energy can be calculated in fewer steps.

♦ Unlike thermodynamic integration (TI), analytical derivatives of the Hamiltonian with respect to the coupling parameter are not needed.

♦ As has been suggested by Mezei, FDTI may converge faster than either TI or PM.

Incorporation of the cou-pling parameter λ

The first step in any free energy calculation is parameterizing the energy function to provide a continuous change between the ther-modynamic states that are being compared. The relative free energy function as implemented in the FDiscover program is parameterized at the level of the forcefield parameters themselves. For example, the energy E of a harmonic bond as a function of the bond length b is:

Eq. 122

∆A kBTln exp E λi( ) E λ i ± λδ( )–( )/kBT–[ ]⟨ ⟩

i

λδ---------------------------------------------------------------------------------------------∆λ i

i 1=

k

∑–=

E b( ) K b b0–( )2=



where K is the force constant and b0 is the reference bond length. In a free energy calculation E(b) may be different for the initial A and final B states (i.e., each state has different force constants and reference bond lengths). The function is reformulated as a function of the coupling parameter:

Eq. 123

As λ changes from 0 to 1, the bond description gradually changes from a State A bond to a State B bond.

Relative free energy—methodology

This describes the steps involved in performing an in vacuo calcu-lation to solve the free energy difference between methanol and ethane.

Background of example problem

A seminal free energy calculation reported by Jorgensen and Ravi-mohan (1985) showed that free energy perturbation techniques seem to accurately predict the difference in solvation energy between methanol and ethane. This calculation has since been repeated in several laboratories, using independent programs and forcefields, and has become a de facto benchmark for free energy calculations (Singh et al. 1987; Fleischman and Brooks 1987).

Construction of the ther-modynamic cycle

Consider the following thermodynamic cycle for the solvation of ethane and methanol:

E b λ;( ) λKA

1 λ–( )KB

+ b λb0A

1 λ–( )b0B

+[ ]–

2

=

CH3CH3 g( ) CH3CH3 aq( )→

CH3OH g( ) CH3OH aq( )→

∆A1

∆A2

∆A3 ↓ ↓ ∆ A4


6. Free Energy

This cycle requires that:

Eq. 124

The right side of this equation is the difference in solvation free energy that can be measured experimentally. However, the left side is more convenient to compute (making water appear around a solute is a much larger perturbation than changing a methyl group into a hydroxyl group). Either side can be chosen, because both sides are equal.

If there is no difference in intramolecular energy between ethane and methanol, then ∆A3 is zero and the gas phase calculation can be ignored. Although this is intuitively incorrect, a suitable force-field could treat the intramolecular energy of ethane and methanol identically. For example, if the nonpolar hydrogens were not rep-resented explicitly, no atom pairs would have more than two inter-vening bonds. Most forcefields calculate intramolecular nonbond interactions only for atom pairs separated by more than two inter-vening bonds. Furthermore, if the solutes are treated as rigid mod-els, the energy would not be affected by deformations in bond lengths or angles.

In this exercise all hydrogens are explicitly included. In addition, variations in degrees of freedom are allowed for both the solvent and solute molecules.

Including all degrees of freedom requires both a vacuum and a sol-vent calculation. The computational penalty for an additional gas phase calculation is minor; the solvent calculation is two orders of magnitude more intensive. Moreover, because setting up the free energy calculation is virtually identical for both calculations, the gas phase calculation provides a tractable example that yields experience and confidence in performing free energy calculations without wasting an inordinate amount of computer time.

Defining the λ = 0 and λ = 1 states

To set up the calculation, you must first design the chemical per-turbation for methanol going to ethane. You must first choose which of these models corresponds to the λ = 0 state. The λ = 0 state must always have enough atoms to accommodate both molecular extremes, even if invisible “dummy” atoms must be added as placeholders for atoms that appear as the perturbation proceeds.

∆A4 ∆A3– ∆A2 ∆A1–=



Ethane is the logical choice for the λ = 0 state, because methanol can be completely subsumed within ethane. However, to illustrate that the choice is arbitrary, the calculation in the opposite direc-tion, for methanol going to ethane, is described.

The λ = 0 state is provided to FDiscover as would any model for simulation, in ordinary coordinate (.car) and molecular data (.mdf) files. In the FDiscover command input file (.inp), you can specify which atoms are being perturbed and their limiting states with the Warp command. The limiting states are defined completely by the potential atom type, the partial atomic charge, and the mass of each atom.

Analyzing the output of a methanol-to-ethane free energy calculation

The free energy summary table printed out at the end of the com-pleted free energy loop (after the last resume) contains the com-plete history. Throughout the run, a partial summary table is output after each resume and contains all the available data.

Assessing convergence Graphing the Boltzmann factor [exp –(∆E ⁄ kT)] vs. time aids in assessing convergence during a given calculation. Systematic drift in exp –(∆E ⁄ kT) over time is symptomatic of nonconvergence.

To simplify analysis, various quantities are output during free energy calculations to a file with the extension .tot. Figure 32 plots

Figure 32. The ensemble-averaged quantity from Eq. 120 plotted vs. time for three values of λ

Each point plotted is actually an average of 10 points sampled over 100 fs.

10 12 14 16 18 20

exp

(– ∆

E/k

T)

Time (ps)

0.8

1.1

1.0

0.9 λ = 0.056λ = 0.44λ = 0.94


6. Free Energy

Column 5 of the .tot file [exp –(∆E ⁄ kT)] for the first, third, and last λ points for the methanol-to-ethane gas phase calculation. Note that there is a slight upward drift in the curve for the first λ point, although the others seem to have converged.


This section includes Theory and implementation

Example: Fentanyl

Analysis of results

Theory and implementation

This section describes a technique for convenient, precise, and numerically efficient determination of absolute free energies of stable or unstable constrained model conformations. The tech-nique of absolute free energy is general and can be applied in a trans-parent manner to systems in a vacuum or in solution, under any conditions of volume and/or temperature.

This approach is a special case of the thermodynamic integration approach to free energy calculations, which is itself a general method for computing the change in free energy upon going from one thermodynamic state to another. Absolute free energy simply constrains one of these states to be a model system for which the absolute free energy is known analytically. By integrating from a known, albeit model, state to the final real state, the absolute free energy becomes the sum of the numerically computed thermody-namic integration step and the analytical absolute free energy of the model state.

Uses of absolute free energy calculations

Evaluating absolute free energies for particular conformations is an important goal for several reasons:

♦ Absolute free energy values of different thermodynamic states can be compared directly without having to devise a transition pathway between them, as is necessary in relative free energy calculations.



♦ The poor convergence properties associated with reversible structural changes in relative free energy calculations can be avoided.

This section includes a description of the essential characteristics of the absolute free energy method, including a derivation of the ideal solid model and the thermodynamic integration method.

Derivation for ideal sys-tems

The absolute free energy algorithm depends on defining a model for which the partition function in Eq. 114 can be derived analyti-cally. The model implemented in the FDiscover program is an ideal solid. That is, the atoms in the system are constrained har-monically to a lattice (analogous to a solid) and do not interact with each other (analogous to the familiar ideal gas). The Hamilto-nian for such a system is:

Eq. 125

where the first summation is simply the kinetic energy (including a reciprocal of Planck’s constant for each degree of freedom, a quantum effect), and the second is a harmonic function constrain-ing each atom to a corresponding lattice point (xi

0, yi0, zi

0) with a force constant of Ki. Note that there are no terms for interactions between particles. This simplification is what makes an analytical solution possible. Substituting this Hamiltonian into the partition function gives:

Eq. 126

Returning to Eq. 114, the free energy of the model ideal solid can be written as:

HIdeal Solid1

h3N---------

12mi--------- pxi

2 pyi

2 pzi

2+ +( )

i

N

∑=

Ki xi xi0

–( )2d yi yi

0–( )

2zi zi

0–( )

2+

i

N

∑+

Q2mi

Ki---------

πkBT

h-------------

2 3 2⁄

i 1=

N

∏=


6. Free Energy

Eq. 127

A remarkable result of this formula is that it does not depend on coordinates—neither the model coordinates nor the lattice site coordinates. It depends only on the mass of the atoms and the force constant used for the harmonic constraint. This property has important practical implications for free energy calculations, mak-ing it possible to choose whatever set of lattice site coordinates gives the best convergence properties for the subsequent thermo-dynamic integration step.

Thermodynamic integration—derivation for real systems

As shown above, integrating the partition function analytically may be possible for simple Hamiltonians. However, for more real-istic systems that include many nonbond and bond interactions between atoms, an analytic solution is impossible. As in the rela-tive free energy calculation (see FDTI), we use thermodynamic integration to determine the change in free energy:

Eq. 128

Substituting the equation for the free energy (Eq. 114) into Eq. 128, we can write the difference in free energy as a function of the par-tition function Q(λ):

Eq. 129

Without defining explicitly how the Hamiltonian depends on λ, we can write the difference in the energies as:

Aideal solid32---kBT ln

2mi

Ki---------

πkBT

h---------------

2

i 1=

N

∑–=

A1 A0–A λ( )∂

λ∂------------- λd

0

1

∫=

A1 A0–kBT ln Q λ( )–[ ]∂

λ∂---------------------------------------- λd

0

1

∫ kBT1

Q λ( )-----------

Q λ( )∂λ∂-------------- λd

0

1

∫–= =



Eq. 130

The problem of computing a free energy difference is thus simpli-fied to that of computing the expectation value of a derivative of the Hamiltonian. Since an expectation value is, according to Gibbs’ postulate (an axiom of statistical mechanics), the ensemble average of the quantity, it is easily computed as the average of that quantity over a suitably equilibrated set of snapshots of the system.

These snapshots are typically generated from either a molecular dynamics or a Monte Carlo simulation. Thus, appropriately aver-aging the results of a molecular dynamics trajectory enables Eq. 130 to be evaluated. This is far easier than calculating the gen-eral partition function of Eq. 129.

Eq. 130 is perfectly general for any classical system and is the fun-damental equation of all thermodynamic integration methods. However, we have not yet described what the Hamiltonian is or how it can be parameterized in terms of λ. The Hamiltonian could be parameterized in an infinite number of ways. Because a func-tion of state (the free energy) is being integrated, the path between the end points is arbitrary. For absolute free energy calculations, it turns out that the simplest form is perfectly adequate.

What is the Hamiltonian? The parameterization scheme adopted by the FDiscover program is straightforward. The Hamiltonian is defined as a linear combi-nation of the two potential energy functions that describe the extreme states:

Eq. 131

Here, K(p) is the kinetic energy (because the two states are com-pletely defined by their respective potential energy functions, the kinetic energy does not have to be coupled to λ), V0 is the normal potential energy function (including bonds, angles, torsions, etc.), and VH is a harmonic oscillator site potential given by:

A1 A0–H p r λ, ,( )∂

λ∂------------------------- λd

0

1

∫=

H λ( ) K p( ) 1 λ–( )V0 λVH+ +=


6. Free Energy

Eq. 132

Parameterization of the Hamiltonian

where Ki is the ith atom’s spring constant, ri is the ith atom’s instan-taneous coordinate, and ri

0 the reference lattice of the noninteract-ing atoms (often referred to as an Einstein solid). This choice of parameterization is motivated by the template-forcing concept in the FDiscover program and by previous free energy evaluations (Hoover and Ree 1967). In Eq. 131, λ is greater than 0 and less than 1, and describes an energy–space path between a system described by an unadulterated molecular forcefield (λ = 0) and one repre-sented by independent harmonic oscillators (λ = 1).

Eq. 132 is thus the reference system for which an absolute free energy can be directly calculated. The potential VH restricts the exploration of phase space to a region defined by atomic mean-squared displacements relative to the Einstein solid. Conse-quently, the choice of both the reference state and spring constants, in general, affects the calculated free energies. For most cases, the energy-minimized coordinates of a mechanically stable structure are used for the reference Einstein solid. Using Eq. 127 for the free energy of the Einstein solid A1 and Eqs. 130 through 132, the abso-lute free energy of the real state is:

Absolute free energy of the real state

Eq. 133

Computational consider-ations and precautions

The simple form of Eq. 133 makes evaluating its ensemble average straightforward. A comprehensive dynamics trajectory for a given value of λ represents an ensemble of structures for that λ. By com-puting and averaging the values of (V0 – VH) for each structure in this ensemble, the integrand can be numerically estimated for any given value of λ. By performing several such calculations for many

VH Ki ri ri0

–( )2

i

∑=

A0 A1 λ∂∂

K p( ) 1 λ–( )V0 λHH+ +[ ] λd

0

1

∫–=

A1 V0 VH– λd

0

1

∫+=



values of λ between 0 and 1, eventually the function can be numer-ically integrated.

Several practical simulation considerations are apparent from Eqs. 131 and 133.

For example, when λ is close to zero, the calculated ensemble of structures includes configurations far from the reference state (the ensemble is generated according to Eq. 131). Therefore, fluctua-tions of VH are large when VH is evaluated for this ensemble.

Conversely, when λ is close to one, the structures generated are minimally influenced by the real forcefield. These ensembles can include structures with distorted bonds, angles, and even overlap-ping atoms, leading to large fluctuations in V0 in Eq. 133. Large fluctuations of (V0 – VH) decrease the precision with which its inte-gral can be calculated. The overall effect is to destabilize the inte-gral and, in extreme cases, cause it to diverge.

These undesirable effects can be minimized by reasonable choices of reference structures, spring constants, and integration algo-rithm. For example, the FDiscover program uses a Gaussian–Leg-endre quadrature algorithm to integrate Eq. 133. This has the useful property that one need not evaluate the function at the boundaries (where the divergence is worst).

Choice of the reference state

The choice of the reference state is the most critical step in an abso-lute free energy calculation. The reference state determines where the sampling of configuration space is centered. Choosing refer-ence states that are only slightly different (for example, 0.1 Å rms in coordinates) can change the free energy significantly if the ref-erence state has any residual strain.

Best results are obtained when the reference state represents a min-imum-energy conformation on the normal energy surface. If the reference structure is a minimum for both V0 and VH, there is no impetus for the conformation to wander away, which, as described above, is the primary cause of divergence.

For non-minimum configurations such as the saddle point c in Figure 34, excess strain can be removed by minimizing with tor-sional restraints (see Example: Fentanyl).

Limitations The absolute free energy technique is primarily used to evaluate the free energy of different conformations of the same model. This is particularly difficult to do by perturbation methods alone, since


6. Free Energy

a path from one conformation to another may be difficult to con-struct and model (for example, a path converting an α-helix into a β-sheet). Although free energy methods are path independent (because what is being evaluated is the difference between ther-modynamic state functions), the integration through the path must be performed reversibly, which is problematic for large struc-tural rearrangements.

Example: Fentanyl

This example presents a calculation of the free energies of various conformational states of fentanyl (Figure 33). Previous studies of

fentanyl have examined the conformational behavior around the anilido nitrogen and identified two main classes of energy minima in the 2D energy map (Tolleaneare et al. 1986). Figure 34 shows such a map, generated by contouring the energies obtained from a flexible-geometry minimization study performed with the Dis-cover program. These minima are separated by approximately 10 kcal mol-1 energy barriers and differ from each other by about 1 kcal mol-1. The free energy of the states represented by the min-ima and the barriers between them will be calculated by the abso-lute free energy method.

Figure 33. Definitions of φ1 and φ2 for fentanyl

C1

C2 C4

C5N3

O

φ1 = C1–C2–N3–C4

φ2 = C2–N3–C4–C5



The reference state has two roles in the free energy calculation. A trivial contribution is the free energy it contributes as an ideal solid. This contribution is trivial because it does not depend on the conformation of the reference structure. It depends only on the masses of its atoms and the spring constants constraining each atom to the lattice (see Eq. 127).

The nontrivial role is that it determines which part of configura-tional space is sampled during dynamics. In this example, the free energies of states a, b, and c (Figure 34) of fentanyl are compared. The reference states for a and b correspond to completely mini-mized structures at these points. Point c in Figure 34 represents the

Figure 34. Two-Dimensional energy map of fentanyl as a function of the two dihedral angles defined in Figure 33

Locations a and b correspond to the two unique minima, and c is at a transition state.

-100 0 100

-100

0

100

φ1

φ2

ba

c

180 42

610

8 6 4 2


6. Free Energy

free energy at a barrier. Obviously, it would be incorrect to mini-mize this reference structure—it would simply roll down into one of the adjacent wells. The lowest-energy structure at the saddle point can be constructed by forcing the dihedral angles to adopt the values of the saddle point (using the force torsion command in the standalone version of the FDiscover program, or the Con-straint/TorsionForce command in the Insight version) while mini-mizing the rest of the model (this, of course, is how the energy map in Figure 34 was generated in the first place).

It is important to choose the lowest-energy structure consistent with a given conformational state, so that the free energy calcu-lated does not contain excess enthalpy. Constraining the sampling of configurational space to a rather small region localized around the reference structure limits the distribution of energies that is sampled. Moving the reference structure to a slightly higher-energy conformation would sample different energies and change the total free energy.

Consider Figure 35, where the history of φ1 and φ2 is superimposed on the φ1 ⁄ φ2 energy map for one of the λ ensembles. Note how only the local region around the reference structure is sampled. While this may be unsettling (what is the meaning of a free energy that does not represent all of configurational space), remember that you are interested in the difference in free energy between closely related states (a, b, and c differ mainly by rotation of two dihedral angles). To achieve this resolution, the sampling must be restricted accordingly.



Hints for setting up the absolute free energy cal-culation

In the Insight version of the FDiscover program, an absolute free energy calculation is set up using the Parameters/Absolute com-mand. The system is first minimized, then the atoms are tethered to the minimized locations. The Parameters/Minimize and Parameters/Dynamics commands can be used to control the initial minimization, as well as the dynamics used to sample the confor-mational space.

In the standalone version, the reference state is defined in the FDis-cover input file (.inp), with all atoms being tethered. Using the

Figure 35. Time history of the dihedral angles φ1 and φ2 superimposed on the 2D energy map of Figure 34

Two trajectories are plotted, one for a minimum corresponding to state a in Figure 34, the other for the transition state (c). Note that only a small fraction of the conformational space of the model needs to be sampled to estimate the free energy of a conformer. Each trajectory corresponds to the first λ value, i.e., when the potential function is most like the standard potential.

-100 0 100

-100

0

100

φ1

φ2

0 1842

6 8 6 4 2

10


6. Free Energy

tether command allows you to use different reference structures easily. Once the tethered atom list is generated, the model could undergo dynamics to randomize the starting point just prior to starting the free energy calculation.

There is an important exception to this rule. Once a free energy cal-culation is initiated, the reference structure is saved into a coordi-nate file with the extension .cli. Thus, a permanent record is maintained of the reference state for each free energy calculation. This file is used by the FDiscover program to restart free energy calculations that may have been interrupted. The file format is identical to the .car and .cor files so that they can be used inter-changeably.

One important property of the .cli file is that, if the file exists when the free energy calculation begins (for example, from an earlier or different run), the reference state is read from the file regardless of any prior tethered atom list generated. This does not mean that if a .cli file is provided, a tethered atom list command is not needed. Rather, if the .cli file is present, no matter what conformation is used for the tethered atom list command, the actual reference structure used is that in the .cli file. Be aware then, when doing new free energy calculations requiring different reference states that any existing .cli files that would override the internally gener-ated reference state must be removed.

The magnitude of the spring constants in Eq. 132 indirectly affects the free energy calculated by controlling the volume of phase space being sampled during dynamics. Large spring constants bias the sampling to the immediate neighborhood of the reference structure. Smaller values allow the model to sample more space.

The spring constants also affect the error associated with a calcula-tion. Ensembles that contain structures far from the reference state make greater contributions to the free energy, by virtue of the greater fluctuations in (V0 – VH) for this ensemble.

The FDiscover program provides two methods for assigning spring constants:

1. All atoms are assigned the same value. You may want to assign large Ki values when you are studying a mechanically unstable state, to constrain dynamics to phase space near the reference structure. In the FDiscover standalone version this choice is specified by including the keyword assigning in the free



energy command along with the specification of a spring con-stant. In the Insight environment, the force constant is con-trolled by changing the Spring Constant parameter in the Parameters/Absolute command.

2. Spring constants can be estimated from the definition of mean-squared displacements under a harmonic potential. In this case, the following expression is used:

Eq. 134

and the default is for each atom to receive a unique value of Ki.

Method 1 is useful for studying mechanically unstable states, for example, the extended state of decaglycine. Method 2 is more appropriate for studying mechanically stable states, for example, the energy-minimized α-helix or hairpin structures of decaglycine. Typical values of the spring constants range from 1 to 50 kcal mol-1 Å-1.

If possible, you should let Method 2 choose the spring constants automatically. However, if the conformational state is not in a local minimum, you must explicitly specify a spring constant. An esti-mated value corresponding to a nearby minimum is a good start-ing point. The error associated with an explicit choice should be monitored, as well as the configurational space sampled for this calculation. Error analysis is discussed under Analysis of results.

Another choice that has to be made is how many λ intervals to use in the integration. This of course depends ultimately on the level of precision needed and the behavior of the integral and cannot be anticipated for all cases.

For small flexible models such as fentanyl, using a medium spring constant (50 kcal mol-1 Å-1), the use of 10 intervals allows a reason-ably behaved integral to be estimated to with a statistical error of ± 0.5 kcal mol-1.

For careful work, the behavior of the integral (that is, how (V0 –VH) varies as a function of λ) must be examined. Rapid changes in

(V0 – VH) are indicative of systematic errors. Some regions of the integral may need to be computed for longer times and with more intervals, to achieve the desired accuracy.

Ki

kBT

∆x20

-----------------=


6. Free Energy

The number of λ intervals typically ranges from 4 to 12. If adequate precision cannot be achieved with 12 intervals, look for and correct systematic factors destabilizing the integral (i.e., Are the spring constants too weak? Is the reference state at too high an energy? Is the system completely equilibrated?).

Analysis of results

Available output Although a free energy calculation is done as a single FDiscover run, the calculation actually consists of several independent cycles of dynamics equilibration (initialize dynamics) and data collec-tion (resume dynamics)—as many cycles as λ intervals. The results of these dynamics calculations are stored independently for each λ value, in three files.

The standard output file (extension .out) is the most important. Among other things, it includes a history of average energy values (potential and kinetic) during each dynamics cycle. Data needed for the actual free energy calculation are collected only during the resume dynamics phase. The actual free energy results are in the SUMMARY OF FREE-ENERGY CALCULATION table included at the end of each resume dynamics output. The total free energy can be calculated only at the conclusion of all the λ cycles. How-ever, the incomplete table is output at the end of each λ interval as an intermediate result.

A second output file with the extension .tot contains only the instantaneous values of the kinetic energy plus the parameterized potential energy for the entire run (all values of λ). The frequency of output into this file is controlled by the sampling option in the absolute free energy command and is, by default, output every step. The .tot file is particularly convenient as input to plotting and statistical analysis programs (such as SAS or RS1).

The third source of analysis information is the dynamics history file (extension .his). This file contains all the coordinates and veloc-ities for each λ run. Because a free energy calculation consists of several λ cycles, each with its own initialize command, there can be several history files. To allow for several history files with the same root name, an option has been added to the initialize com-mand to change the extension from .his to a number (e.g., FENTA-NYL.1, FENTANYL.2, etc.). This is specified by including the



keyword save in the initialize command. If save is not included, the Discover program keeps only the history file corresponding to the last λ run.

Assessing and minimizing statistical and systematic errors

A major source of systematic error in these calculations is lack of convergence (that is, failure to equilibrate long enough to achieve thermodynamic equilibrium at each λ value) and insufficient sam-pling of configurational space. Other sources of systematic error include inaccuracies in the force field (both in the functional form and the parameters) and quantum mechanical effects.

Random errors are a natural consequence of free energy calcula-tions. The statistical distribution of states available to a molecule at a given temperature is precisely what defines its entropy. Measur-ing entropy is an inherently statistical process that can be quanti-fied with standard random error analysis procedures. Statistical errors are calculated for the average of (V0 – VH) at each λ value.

Systematic error/conver-gence

Careful analysis of the random errors reported by the FDiscover program helps point to likely systematic errors. For example, rapid changes in error as a function of λ could be caused by a fail-ure to completely equilibrate.

Although repeating the entire run with longer equilibration and sampling times is an obvious solution, it is not a particularly effi-cient one. The integral computed under both short and long con-ditions may be indistinguishable up to λ = 0.8. It is wasteful to compute all intervals at the worst-case level.

For such situations, the absolute free energy command in the standalone version of the FDiscover program allows you to per-form a free energy calculation over just a part of the interval. If you are using the Insight interface to the FDiscover program, the parameters Lower_lambda, Upper_lambda, and Quadrature Points in the Parameters/Absolute command control the points sampled, allowing you to break the calculations into ranges.

You could perform two independent free energy calculations. The first calculation would use short equilibration and sampling times to compute the free energy from λ = 0 to 0.8. The second calcula-tion would compute the free energy from λ = 0.8 to 1.0 with longer equilibration and sampling times.

The total free energy is then just the sum of the two. Note that, when summing the free energies from these two runs, the ideal


6. Free Energy

solid contribution must be included only once. Mathematically, the integral is simply being broken into pieces:

Eq. 135

Absolute free energy files and output

The .cli file contains the free energy comparison list, i.e., the tem-plate coordinates prevalent during tethering. This file can be used as input for further simulations with the same template coordi-nates, by specifying file filename with the calculate command in the standalone version of the FDiscover program.

The .tot file contains a history of instantaneous parameterized energies from the dynamics simulations.

Table of spring constants A table of spring constants is printed before commencing the first dynamics simulation for the integration over λ. The data tabulated are absolute atom #, harmonic force constant, and (optionally) mean-square displacement and mean harmonic energies as deduced from spring-constant-generation procedures utilizing molecular dynamics.

Dynamics running aver-ages

Dynamics running averages contain additional categories for (V0 – VH) and <V0>.

Summary of averages per λ

Each table of dynamics averages generated by resume is followed by a table summarizing contributions to the free energy integral, including statistical errors and variances on the value dF for each lambda value.

The final page of the output contains a SUMMARY OF FREE ENERGY CALCULATIONS. It also includes a total free energy summary, giving the absolute free energy of the ideal solid (the λ = 1 state of the template). Each λ value has an associated output table in the .out file.

λ V0 VH–d

0

1

∫ λ V0 VH–d

0

0.8

∫ λ V0 VH–d

0.8

1.0

∫+=


A References

Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids, Claren-don Press, Oxford Science Publications (1987).

Andersen, H. C. “Molecular dynamics simulations at constant pressure and/or temperature”, J. Chem. Phys., 72, 2384 (1980).

Andersen, H. C. “Rattle: A “velocity” version of the Shake algo-rithm for molecular dynamics calculations”, J. Comp. Physics, 52, 24–34 (1983).

Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. “Molecular dynamics with coupling to an exter-nal bath”, J. Chem. Phys., 81, 3684–3690 (1984).

Born, M.; Oppenheimer, J. R. Ann. Physik, 84, 457 (1927).

Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J., Swami-nathan, S., and Karplus, M., J Comp. Chem., 4, 187–217 (1983).

Brooks, C. L., III; Brunger, A. T.; Karplus, M. Biopolymers, 24, 843 (1985a).

Brooks, C. L., III; Montgomery; Pettitt, B.; Karplus, M. “Structural and energetic effects of truncating long range interactions in ionic and polar fluids”, J. Chem. Phys., 83, 5897–5908 (1985b).

Brown, D.; Clark, J. H. R. “A comparison of constant energy, con-stant temperature, and constant pressure ensembles in molec-ular dynamics simulation of atomic liquids”, Molecular Physics, 51, 1243–1252 (1984).

Burchart, E. de Vos Studies on Zeolites; Molecular Mechanics, Frame-work Stability, and Crystal Growth, Ph.D. Thesis, Technische Universiteit Delft (1992).

Casewit, C. J.; Colwell, K. S.; Rappé, A. K., J. Am. Chem. Soc., 114, 10035 (1992a).


A. References

Casewit, C. J.; Colwell, K. S.; Rappé, A. K., J. Am. Chem. Soc., 114, 10046 (1992b).

Catlow, C. R. A.; Norgett, M. J. “Lattice structure and stability of ionic materials”, personal communication (1976).

Dauber–Osguthorpe, P.; Roberts, V. A.; Osguthorpe, D. J.; Wolff, J.; Genest, M.; Hagler, A. T. “Structure and energetics of ligand binding to proteins: E. coli dihydrofolate reductase–trimetho-prim, a drug–receptor system”, Proteins: Structure, Function and Genetics, 4, 31–47 (1988).

Deem, M. W.; Newsam, J. M.; Sinha, S. K.. “The h = 0 term in Cou-lomb sums by the Ewald transformation”, J. Phys. Chem., 94, 8356–8359 (1990).

Demontis, P.; Yashonath, S.; Klein, M. L. “Localization and mobil-ity of benzene in sodium-Y zeolite by molecular dynamics cal-culations”, J. Phys. Chem., 93, 5016 (1989).

Ding, H. Q.; Karasawa, N.; Goddard, W. A., III “Atomic level sim-ulations on a million particles: The cell multipole method for Coulomb and London nonbond interactions”, J. Chem. Phys., 97, 4309 (1992).

Dinur, U.; Hagler, A. T. “Approaches to empirical force fields”, in Reviews of Computational Chemistry, Vol. 2; Lipkwowitz, K. B.; Boyd, D. B. Eds.; VCH: New York; Chapter 4 (1991).

Ermer, O. “Calculation of molecular properties using force fields. Applications in organic chemistry”, Structure and Bonding, 27, 161–211 (1976).

Ewald, P. P. Ann. d. Physik, 64, 253 (1921).

Fletcher, R. Practical Methods of Optimization, Vol. 1, Unconstrained Optimization, John Wiley & Sons: New York (1980).

Fletcher, R.; Reeves, C. M. Comput. J., 7, 149 (1964).

Garofalini, S. H. J. Amer. Ceram. Soc., 67, 133 (1984).

Garofalini, S. H.; Zirl, D. M., J. Amer. Ceram. Soc., 73, 2848 (1990).

Greengard, L.; Rokhlin, V. I. “A fast algorithm for particle simula-tions”, J. Comp. Phys., 73, 325 (1987).

Gunsteren, W. F.; Karplus, M. J. Comp. Chem., 1, 266 (1980).


Ha, S. N.; Giammona, A.; Field, M.; Brady, J. W. “A revised poten-tial-energy surface for molecular mechanics studies of carbo-hydrates”, Carbohydrate Research, 180, 207–221 (1988).

Hagler, A. T.; Dauber, P.; Lifson, S. “Consistent force field studies of intermolecular forces in hydrogen bonded crystals. III. The C=O…H–O hydrogen bond and the analysis of the energetics and packing of carboxylic acids”, J. Am. Chem. Soc., 101, 5131–5141 (1979a).

Hagler, A. T.; Ewig, C. S. “On the use of quantum energy surfaces in the derivation of molecular force fields”, Comp. Phys. Comm., 84, 131–155 (1994).

Hagler, A. T.; Lifson, S.; Dauber, P. “Consistent force field studies of intermolecular forces in hydrogen bonded crystals. II. A benchmark for the objective comparison of alternative force fields”, J. Am. Chem. Soc., 101, 5122–5130 (1979b).

Hagler, A. T.; Stern, P. S.; Sharon, R.; Becker, J. M.; Naider, F. “Com-puter simulation of the conformational properties of oligopep-tides. Comparison of theoretical methods and analysis of experimental results”, J. Am. Chem. Soc., 101, 6842–6852 (1979c).

Halgren, T. A., J. Amer. Chem. Soc., 114, 7827–7843 (1992).

Halgren, T. A.; Nachbar, R. B. “The Merck molecular force field: IV. Conformational energies and geometries for MMFF94,” J. Comp. Chem., 19, 587–615 (1996).

Harvey, S. C., “Treatment of electrostatic effects in macromolecular modeling”, Proteins: Structure, Function, and Genetics, 5, 78–92 (1989).

Hill, J. -R.; Sauer, J. “Molecular mechanics potential for silica and zeolite catalysts based on ab initio calculations. 1. Dense and microporous silica”, J. Phys. Chem., 98, 1238–1244 (1994).

Homans, S. W. “A molecular mechanical force field for the confor-mational analysis of oligosaccharides: Comparison of theoret-ical and crystal structures of Man α1–3 Man β1–4 GlcNAc”, Biochemistry, 29, 9110–9118 (1990).

Hoover, W. “Canonical dynamics: Equilibrium phase–space distri-butions”, Phys. Rev., A31, 1695–1697 (1985).


A. References

Hwang, J. K.; Warshel, A. “Semiquantitative calculations of cata-lytic free energies in genetically modified enzymes”, Biochem-istry, 26, 2669–2673 (1987).

Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. “Derivation of Class II force fields. 2. Derivation and characterization of a Class II force field, CFF93, for the alkyl functional group and alkane molecules”, J. Amer. Chem. Soc., 116, 2515–2525 (1994).

Kao, J.; Allinger, N. L., J. Amer. Chem. Soc., 99, 975 (1977).

Karasawa, N.; Goddard, W. A., III “Acceleration of convergence for lattice sums”, J. Phys. Chem., 93, 7320–7327 (1989).

Karasawa, N.; Goddard, W. A., III “Force fields, structures, and properties of polyvinylidene fluoride crystals,” Macromole-cules, 25, 7268 (1992).

Kirkwood, J. G. “Statistical mechanics of fluid mixtures”, J. Chem. Phys., 3, 300–313 (1935).

Kitson, D. H.; Hagler, A. T. “Theoretical studies of the structure and molecular dynamics of a peptide crystal“, Biochemistry, 27, 5246–5257 (1988).

Kohler, A. E.; Garofalini, S. H. Langmuir, 10, 4664 (1994).

Levitt, M.; Lifson, S. J. Molec. Biol., 46, 269 (1969).

Levy, R. M.; McCammon, J. A.; Karplus, M. Chem. Phys. Lett., 64, 4 (1979).

Lifson, S.; Hagler, A. T.; Dauber, P., J. Amer. Chem. Soc., 101, 5111 (1979).

Liljefors T.; Tai, J. C.; Li, S.; Allinger, N. L., J. Comp. Chem., 8, 1051 (1987).

Maple, J.; Dinur, U.; Hagler, A. T. “Derivation of force fields for molecular mechanics and dynamics from ab initio energy sur-faces”, Proc. Nat. Acad. Sci. USA, 85, 5350–5354 (1988).

Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Dinur, U.; Waldman, M.; Ewig, C. S; Hagler, A. T. “Derivation of Class II force fields. 1. Methodology and quantum force field for the alkyl func-tional group and alkane molecules”, J. Comput. Chem., 15, 162–182 (1994a).


Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. “Deriva-tion of Class II force fields. 3. Characterization of a quantum force field for the alkanes”, Israel J. Chem., 34, 195 –231 (1994b).

Maple, J. R., Thacher, T. S., Dinur, U.; Hagler, A. T. “Biosym force field research results in new techniques for the extraction of inter- and intramolecular forces”, Chemical Design Automation News, 5 (9), 5–10 (1990).

Mayo, S. L.; Olafson, B. D.; Goddard, W. A. III “DREIDING: A generic force field”, J. Phys. Chem., 94, 8897–8909 (1990).

McCammon, J. A.; Gelin, B. R.; Karplus, M.; Wolynes, P. G. Nature, 262, 325–326 (1976).

McQuarrie, Statistical Mechanics, Harper & Row, Chapter 3 (1976).

Mezei, M.; Beveridge, D. “Free energy simulations”, Ann. N.Y. Acad. Sci., 482, 1–23 (1986).

Miller, G. W.; Knaebel, K. S.; Ikels, K. G., AIChE J., 33, 194 (1987).

Momany, F. A.; Carruthers, L. M.; McGuire, R. F.; Scheraga, H. A., J. Phys. Chem., 78, 1595 (1974).

Momany, F. A.; Rone, R. J. Comp. Chem., 13, 888–900 (1992).

Némethy, G.; Pottle, M. S.; Scheraga, H. A., J. Chem. Phys., 87, 1883 (1983).

Nosé, S. “A molecular dynamics method for simulations in the canonical ensemble”, Molec. Phys., 52, 255–268 (1984a).

Nosé, S. “A unified formulation of the constant temperature molecular dynamics methods”, J. Chem. Phys., 81, 511–519 (1984b).

Nosé, S. “Constant temperature molecular dynamics methods”, Prog. Theoret. Phys. Supplement, 103, 1–46 (1991).

Parrinello, M.; Rahman, A. “Polymorphic transitions in single crystals: A new molecular dynamics method”, J. Appl. Phys., 52, 7182–7190 (1981).

Pickett, S. D.; Nowak, A. K.; Thomas, J. M.; Peterson, B. K.; Swift, J. F. P.; Cheetham, A. K.; Denouden, C. J. J.; Smit, B.; Post, M. F. M. “Mobility of adsorbed species in zeolites—A molecular dynamics simulation of xenon in silicalite”, J. Phys. Chem., 94, 1233 (1990).


A. References

Powell, M. J. D. “Restart Procedure for the Conjugate Gradient Method,” Mathematical Programming, 12, 241 (1977).

Press, W. H.: Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. In Numerical Recipes, The Art of Scientific Computing; Cambridge University Press, Cambridge (1986).

Quirke, N.; Jacucci, G. “Energy difference functions in Monte Carlo simulations: Application to (1) the calculation of free energy of liquid nitrogen. II. The calculation of fluctuation in Monte Carlo averages”, Mol. Phys., 45, 823–838 (1982).

Rappé, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A.; Skiff, W. M. “UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations”, J. Amer. Chem. Soc., 114, 10024 (1992).

Rappé, A. K.; Colwell, K. S.; Casewit, C. J., Inorg. Chem., 32, 3438 (1993).

Rappé, A. K.; Goddard, W. A., J. Phys. Chem., 95, 3358 (1991).

Ray, J. R. Comp. Phye. Rep., 8, 109 (1988) and references therein.

Reidl, D., International Tables for Crystallography: A Space-Group Symmetry, Vol. A; Dordrecht, Holland (1983).

Rigby, D.; Sun, H.; Eichinger, B. E. “Computer simulations of poly(ethylene oxides): Force field, PVT diagram and cycliza-tion behavior”, Polymer, 44, 311 (1998).

Root, D.M., C.R. Landis, and T. Cleveland, J. Am. Chem. Soc., 1993, 115, 4201-4209.

Root, D.M., Ph.D. thesis, University of Wisconsin, Madison, 1997.

Rosenthal, A. B.; Garofalini, S. H., J. Amer. Ceram. Soc., 70, 821 (1987).

Rosenthal, A.B.; Garofalini, S. H., J. Non-Crys. Solids, 107, 65 (1988).

Ryckaert, J.–P.; Ciccotti, G.; Berendsen, H. J. C. “Numerical Integra-tion of the Cartesian equations of motion of a system with con-straints: Molecular dynamics of n-alkanes”, J. Comp. Phys., 23, 327–341 (1977).

Schmidt, K. E.; Lee, M. A. “Implementing the fast multipole method in three dimensions”, J. Stat. Phys., 63, 1223 (1991).


Shi, S.; Yan, L.; Yang, Y.; Fisher, J. ESFF Forcefield Project Report II; MSI, San Diego.

Singh, U. C; Brown, F. K.; Bash, P. A.; Kollman, P. A. “An approach to the application of free energy perturbation methods using molecular dynamics: Applications to the transformations of methanol to ethane, oxonium to ammonium, glycine to ala-nine, and alanine to phenylalanine in aqueous solution and to H3O+(H2O)3 NH4

+(H2O)3 in the gas phase”, J. Am. Chem. Soc., 109, 1607–1614 (1987).

Soules, T. F., J.Chem. Phys., 71, 4570 (1979).

Soules, T. F.; Varshneya, A. K., J. Amer. Ceram. Soc., 64, 145 (1981).

Sprague, J. T.; Tai, J. C.; Yuh, Y.; Allinger, N. L., J. Comp. Chem., 8, 581 (1987).

Straatsma, T. P.; Berendsen, H. J. C.; Postma, J. P. M. “Free energy of hydrophobic hydration: A molecular dynamics study of noble gases in water”, J. Chem. Phys., 85, 6720–6727 (1986).

Sun, H. J. Comp. Chem., 15, 752 (1994).

Sun, H. “Ab initio calculations and force field development for computer simulation of polysilanes”, Macromolecules, 28, 701 (1995).

Sun, H. “The parameterization and validation of a condensed-phase optimized ab initio forcefield” (in preparation).

Sun, H.; Mumby, S. J.; Maple, J. R.; Hagler, A. T. “An ab initio CFF93 all-atom force field for polycarbonates”, J. Amer. Chem. Soc., 116, 2978–2987 (1994).

Sun, H.; Rigby, D. “Polydimethylsiloxane: Ab initio force field: Structural, conformational and thermophysical properties”, Spectrochimica Acta (in press).

Sun, H.; Ren, P.; Fried, J. R. "The COMPASS Force Field: Parame-terization and Validation for Polyphosphazenes" Computa-tional and Theoretical Polymer Science, 8 (1/2), 229 (1998).

Sun, H. "COMPASS: An ab Initio Force-Field Optimized for Con-densed-Phase Applications - Overview with Details on Alkane and Benzene Compounds" J. Phys. Chem., (1998 in press).


A. References

Tembe, B. L.; McCammon, J. A. “Ligand–receptor interactions”, Comput. Chem., 8, 281–283 (1984).

Tesar, A. A.; Varshneya, A. K. J. Chem. Phys., 87, 2986 (1987).

Tosi, M. P., Solid State Physics, 16, 107 (1964).

van Beest, B. W. H.; Kramer, G. J.; van Santen, R. A., Phys. Rev. Lett., 64, 1955 (1990).

Verlet, L. “Computer experiments on classical fluids. I. Thermody-namical properties of Lennard–Jones molecules”, Phys. Rev., 159, 98–103 (1967).

Waldman, M.; Hagler, A. T. “New combining rules for rare gas van der Waals parameters”, J. Comput. Chem., 14, 1077–1084 (1993).

Warshel, A.; Sussman, F.; King, G. “Free energy of charges in sol-vated proteins: Microscopic calculations using a reversible charging process”, Biochemistry, 25, 8368–8372 (1986).

Watanabe, K.; Austin, N.; Stapleton M. R. “Investigation of the air separation properties of zeolites types A, X, and Y by Monte Carlo simulations,” Molec. Sim., 15, 197–221 (1995).

Weiner, S. J.; Kollman, P. A.; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona, G.; Profeta, S. Jr.; Weiner, P. “A new force field for molecular mechanical simulation of nucleic acids and pro-teins”, J. Am. Chem. Soc., 106, 765–784 (1984).

Weiner, S. J.; Kollman, P. A.; Nguyen, D. T.; Case, D. A. “An all atom forcefield for simulations of proteins and nucleic acid”, J. Comp. Chem., 7, 230–252 (1986).

Wiberg, K. B.; Murcko, M. A., J. Am. Chem. Soc., 111, 4821 (1989).

Williams, D. E., J. Phys. Chem., 45, 3370 (1966).

Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations, Dover, New York (1980).

Woodcock, L. V., Advances in Molten Salt Chemistry; Plenum, New York (1975).

Yashonath, S.; Thomas, J. M.; Nowak, A. K.; Cheetham, A. K. “The siting, energetics and mobility of saturated hydrocarbons inside zeolitic cages—Methane in zeolite-Y”, Nature, 331, 601 (1988).


Zirl, D. M.; Garofalini, S. H. Phys. Chem. Glasses, 30, 155 (1989).


A. References


B Forcefield Terms and Atom Types

This appendix explains This appendix includes tables of atom types for these forcefields:

Table 25 . Atom types—AMBER

Table 26 . Atom types—Homans

Table 27 . Atom types—CFF91

Table 28 . Atom types—CHARMm

Table 29 . Atom types—CVFF

Table 30 . Atom types—CVFF_aug

Table 31 Atom types for the first three rows of the periodic table—ESFF 4)

Table 27 . Atom types—CFF91

This appendix also includes a table of definitions of individual terms found in various forcefields (Forcefield term definitions).

Examining forcefield files You can examine the forcefield files directly for atom types and other information (for information on how to edit these files, please see Editing a forcefield):

♦ For the Insight molecular modeling program, human-readable forcefield files are found in $BIOSYM_LIBRARY/*.frc

♦ For Cerius2•OFF, human-readable forcefield files are found in the C2_installation_directory/Cerius2-Resources/FORCE-FIELD directory (the Cerius2-Resources/FORCE-FIELD direc-tory also appears in the directory in which you run Cerius2).

♦ For CHARMm, the human-readable forcefield parameter files are located in $CHM_DATA/*PRM. The atom types are in the $CHM_DATA/MASSES.RTF file.


B. Forcefield Terms and Atom Types

Forcefield term definitions

Table 24. Common potential terms in major forcefields supported by MSI

name illustrated form of the termforce-fielda

quadratic bond-stretching

AMBER, CHARMm, UFF

quartic bond-stretching

CFF

Morse bond-stretching

CVFF, ESFF

quadratic angle-bending

AMBER, CHARMm, CVFF

quartic angle-bending

CFF

cosine angle-bending

variousESFF, UFF

single-cosine torsion or similar

AMBER, CHARMm,CVFF

three-term cosine tor-sion

CFF

cosine-Fou-rier torsion

UFF

sin–cos tor-sion

ESFF

k r r0–( )2

k2 r r0–( )2 k3 r r0–( )3 k4 r r0–( )4+ +

k 1 eα– rb rb

0–( )( )

–2

k θ θ0–( )2

k2 θ θ0–( )2 k3 θ θ0–( )3 k4 θ θ0–( )4+ +

k 1 nφ φ0–( )cos+( )

k1 1 φ φ01–( )cos–[ ] k2 1 2φ φ02–( )cos–[ ]k3 1 3φ φ03–( )cos–[ ]

++

k 1 nφcos±( )

kφθ1sin2 θ2sin2

θ10

sin2 θ20

sin2-------------------------------- sign

θ1sinn θ2sin2

θ10

sinn θ20

sinn-------------------------------- nφ[ ]cos+

Forcefield term definitions


improper cosine out-of-plane

or similarAMBER, CVFF, UFF

improper quadratic out-of-plane

CHARMm

improper square out-of-plane, imprope

CVFF

Wilson (or umbrella) out-of-plane

CFF, ESFF, UFFUFF

pyrimid-height out-of-plane

not used

none

6–9 van der Waals or

CFF, ESFF, UFF

6–12 van der Waals or

AMBER, CHARMm, CVFF

electrostatic

or similar

AMBER, CFF, CHARMm,CVFF, ESFF, UFF



43

2

1

k 1 nχ χ0–( )cos+[ ]

k χ χ0–( )2

kχ2

kχ2

k χcos χ0cos–( )2

Aij

rij9

------Bij

rij6

------– ε 2 r*/r( )9 3 r*/r( )6–[ ]

Aij

rij12

-------Bij

rij6

------– ε r*/r( )12 2 r*/r( )6–[ ]

qiqj

εrij---------



quadratic bond–bond

CFF, CVFFb

quadratic bond–angle

CFF, CVFF

angle–angle CFF, CVFF

end bond–torsion

CFF

center bond–tor-sion

CFF

angle–torsion CFF



k r r0–( ) r ′ r ′0–( )

k r r0–( ) θ θ0–( )

k θ θ0–( ) θ′ θ′0–( )

b b0–( ) k1 φcos k2 2φcos k3 3φcos+ +[ ]

b ′ b ′0–( ) k1 φcos k2 2φcos k3 3φcos+ +[ ]

θ θ0–( ) k1 φcos k2 2φcos k3 3φcos+ +[ ]

AMBER atom types


AMBER atom types

Standard AMBER forcefield

aMajor forcefields, for the purposes of this table, are AMBER, CFF, CHARMm, CVFF, ESFF, UFF.bAMBER, CHARMm, ESFF, and UFF contain no cross terms.

angle–angle–tor-sion

CFF, CVFF

improper out-of-plane–-out-of-plane, improper

CVFF

Table 25. Atom types—AMBER (Page 1 of 4)

gener-al class

atom typea description

hydrogen typesH amide or imino hydrogen

HC explicit hydrogen attached to carbonHO hydrogen on hydroxyl oxygenHS hydrogen attached to sulfurHW hydrogen in waterH2 amino hydrogen in NH2

H3 hydrogen of lysine or arginine (positively charged) all-atom carbon typesb



k φcos θ θ0–( ) θ′ θ′0–( )

k 1 2χcos–[ ] 1 2/ 1 2χ′cos–[ ] 1 2/



C sp2 carbonyl carbon and aromatic carbon with hydroxyl substitu-ent in tyrosine

CA sp2 aromatic carbon in 6-membered ring with 1 substituentCB sp2 aromatic carbon at junction between 5- and 6-membered

ringsCC sp2 aromatic carbon in 5-membered ring with 1 substituent and

next to a nitrogenCK sp2 aromatic carbon in 5-membered ring between 2 nitrogens

and bonded to 1 hydrogen (in purine)CM sp2 same as CJ but one substituentCN sp2 aromatic junction carbon in between 5- and 6-membered

ringsCQ sp2 carbon in 6-membered ring of purine between 2 NC nitro-

gens and bonded to 1 hydrogenCR sp2 aromatic carbon in 5-membered ring between 2 nitrogens

and bonded to 1 H (in his)CT sp3 carbon with 4 explicit substituentsCV sp2 aromatic carbon in 5-membered ring bonded to 1 N and

bonded to an explicit hydrogenCW sp2 aromatic carbon in 5-membered ring bonded to 1 N–H and

bonded to an explicit hydrogenC* sp2 aromatic carbon in 5-membered ring with 1 substituent

united carbon typesc

CD sp2 aromatic carbon in 6-membered ring with 1 hydrogenCE sp2 aromatic carbon in 5-membered ring between 2 nitrogens

with 1 hydrogen (in purines)CF sp2 aromatic carbon in 5-membered ring next to a nitrogen with-

out a hydrogenCG sp2 aromatic carbon in 5-membered ring next to an N–HCH sp3 carbon with 1 hydrogenCI sp2 carbon in 6-membered ring of purines between 2 NC nitro-

gensCJ sp2 carbon in pyrimidine at positions 5 or 6 (more pure double

bond than aromatic with 1 hydrogen)CP sp2 aromatic carbon in 5-membered ring between 2 nitrogens

with one hydrogen (in his)C2 sp3 carbon with 2 hydrogensC3 sp3 carbon with 3 hydrogens


gener-al class


AMBER atom types


nitrogen typesN sp2 nitrogen in amide group

NA sp2 nitrogen in 5-membered ring with hydrogen attachedNB sp2 nitrogen in 5-membered ring with lone pairsNC sp2 nitrogen in 6-membered ring with lone pairsNT sp3 nitrogen with 3 substituentsN2 sp2 nitrogen in base NH2 group or arginine NH2

N3 sp3 nitrogen with 4 substituentsN* sp2 nitrogen in purine or pyrimidine with alkyl group attached

oxygen typesO carbonyl oxygen

OH alcohol oxygenOS ether or ester oxygenOW water oxygenO2 carboxyl or phosphate nonbonded oxygen

sulfur typesS sulfur in disulfide linkage or methionine

SH sulfur in cystinephosphorus

P phosphorus in phosphate groupion types

CU copper ion (Cu+2)CØ calcium ion (Ca+2)

I iodine ion (I–)IM chlorine ion (Cl–)

MG magnesium ion (Mg+2)QC cesium ion (Cs+)QK potassium ion (K+)QL lithium ion (Li+)QN sodium ion (Na+)QR rubidium ion (Rb+)


gener-al class




Homan’s carbohydrate forcefield

aFrom Weiner et al. (1984) and Weiner et al. (1986).bNon-hydrogen-containing carbons.cUnited-atom carbons with implicit inclusion of hydrogens.

otherLP lone pair

Table 26. Atom types—Homans

general class atom type descriptioncarbohydrate-hydrogen types

AH α anomeric hydrogenBH β anomeric hydrogenHT sp3 hydrogenHY hydroxyl hydrogen

carbohydrate-carbon typesAC α anomeric carbonBC β anomeric carbonCS sp3 carbon in sugar ring

carbohydrate-oxygen typesOA α anomeric oxygenOB β anomeric oxygenOE ring oxygenOT hydroxyl oxygen


gener-al class


CFF91 atom types


CFF91 atom types

Table 27. Atom types—CFF91 (Page 1 of 4)

general classatom type description

hydrogen typesdw deuterium in heavy water (equiv. to h*)h hydrogen bonded to C or S

hc hydrogen bonded to C (equiv. to h)hi hydrogen in charged imidazole ringhn hydrogen bonded to N (equiv. to h*)ho hydrogen bonded to O (equiv. to h*)hp hydrogen bonded to P (equiv. to h)hs hydrogen bonded to S (equiv. to h)hw hydrogen in water (equiv. to h*)h* polar hydrogen bonded to N or Oh+ charged hydrogen (in cation)

carbon typesc generic sp3 carbon

ca general amino acid alpha carbon (sp3) (equiv. to c)cg sp3 alpha carbon in glycine (equiv. to c)ci sp2 aromatic carbon in charged imidazole ring (his+)

(equiv. to cp)co sp3 carbon in acetal (equiv. to c)

coh sp3 carbon in acetal with hydrogen (equiv. to c)cp sp2 aromatic carboncr carbon in guanidinium group (HN=C(NH2)2) (arg)cs sp2 carbon in 5-membered ring next to S (equiv. to cp)ct sp carbon involved in triple bondc1 sp3 carbon bonded to 1 H, 3 heavy atoms (equiv. to c)c2 sp3 carbon bonded to 2 H’s, 2 heavy atoms (equiv. to

c)c3 sp3 carbon in methyl (CH3) group (equiv. to c)c5 sp2 aromatic carbon in 5-membered ring (equiv. to cp)

c3h sp3 carbon in 3-membered ring with hydrogens (equiv. to c)

c3m sp3 carbon in 3-membered ring (equiv. to c)



c4h sp3 carbon in 4-membered ring with hydrogens (equiv. to c)

c4m sp3 carbon in 4-membered ring (equiv. to c)c′ sp2 carbon in carbonyl (C=O) group in amidec" carbon in carbonyl group, not amide (equiv. to c*)c* carbon in carbonyl group, not amidec– carbon in carboxylate (COO–) groupc+ carbon in guanidinium groupc= nonaromatic end doubly bonded carbon

c=1 nonaromatic, next-to-end doubly bonded carbonc=2 nonaromatic doubly bonded carbon

nitrogen typesn sp2 amide nitrogen

na sp3 amine nitrogennb sp2 nitrogen in aromatic amine (equiv. to nn)nh sp2 nitrogen in 5- or 6-membered ring, bonded to

hydrogennho sp2 nitrogen in 6-membered ring, next to a carbonyl

group and with a hydrogen (equiv. to nh)nh+ protonated nitrogen in 6-membered ringni sp2 nitrogen in charged imidazole ring (his+) (equiv. to

nh)nn sp2 nitrogen in aromatic aminenp sp2 nitrogen in 5- or 6-membered ring, not bonded to

hydrogennpc sp2 nitrogen in 5- or 6-membered ring, bonded to a

heavy atom (equiv. to nh)nr sp2 nitrogen in guanidinium group (HN=C(NH2)2)nt sp nitrogen involved in triple bondnz sp nitrogen in N2

n1 sp2 nitrogen in charged arginine (equiv. to nr)n2 sp2 nitrogen in guanidinium group (HN=C(NH2)2)

(equiv. to nr)n4 sp3 nitrogen in protonated amine (equiv. to n+)

n3m sp3 nitrogen in 3-membered ring (equiv. to na)n3n sp2 nitrogen in 3-membered ring (equiv. to n)n4m sp3 nitrogen in 4-membered ring (equiv. to na)



CFF91 atom types


n4n sp2 nitrogen in 4-membered ring (equiv. to n)n+ protonated amine nitrogenn= nonaromatic end doubly bonded nitrogenn=1 nonaromatic, next-to-end doubly bonded nitrogenn=2 nonaromatic doubly bonded nitrogen

oxygen typeso sp3 oxygen in alcohol, ether, acid, or ester group

oc sp3 oxygen in ether or acetal (equiv. to o)oe sp3 oxygen in ester (equiv. to o)oh oxygen bonded to H (equiv. to o)op oxygen in aromatic ring (e.g., furan)o3e sp3 oxygen in 3-membered ring (equiv. to o)o4e sp3 oxygen in 4-membered ring (equiv. to o)o′ oxygen in carbonyl (C=O) groupo* oxygen in watero– oxygen in carboxylate (COO–) group

sulfur typess sp3 sulfur in sulfide, disulfide, or thiol group

sc sp3 sulfur in methionine (C–S–C) group (equiv. to s)sh sulfur in sulfhydryl (SH) group (equiv. to s)sp sulfur in aromatic ring (e.g., thiophene)s1 sulfur involved in S–S disulfide bond (equiv. to s)

s3e sulfur in 3-membered ring (equiv. to s)s4e sulfur in 4-membered ring (equiv. to s)s′ sulfur in thioketone (>C=S) groups– partial-double sulfur bonded to something that is

bonded to another partial-double oxygen or sulfurphosphorus

p general phosphorous atomhalogen types

br bromine bonded to a carboncl chlorine bonded to a carbonf fluorine bonded to a carboni covalently bound iodine

ion typesBr bromide ion





CHARMm atom types

ca+ calcium ion (Ca2+)Cl chloride ionNa sodium ion

argonar argon atom

siliconsi silicon atom

otherlp lone pairnu null atom for relative free energy

Table 28. Atom types—CHARMm (Page 1 of 4)


hydrogen typesH hydrogen bonding hydrogen (neutral group)

HA aliphatic or aromatic hydrogenHC hydrogen bonding hydrogen (charged group)

HMU mu-bonded hydrogen for metals and boron-hydrideHO hydrogen on an alcohol oxygenHT TIPS3P water-model hydrogen

carbon typesC carbonyl or guanidinium carbonC3 carbonyl carbon in 3-membered aliphatic ringC4 carbonyl carbon in 4-membered aliphatic ring

C5R aromatic carbon in 5-membered ringC5RP for aryl-aryl bond between C5R ringsC5RQ for second aryl-aryl bond between C5RP rings (ortho)C6R aromatic carbon in a 6-membered ring

C6RP for aryl-aryl bond between C6R rings C6RQ carbon of C6RP type ortho to C6RP pair



CHARMm atom types


CF1 carbon with one fluorineCF2 carbon with two fluorinesCF3 carbons with three fluorinesCM carbon in carbon monoxide or other triply bonded carbonCP3 carbon on nitrogen in proline ring

CPH1 CG and CD2 carbons in histidine ringCPH2 CE1 carbon in histidine ringCQ66 third adjacent pair of CR66 types in fused rings

CT aliphatic carbon (tetrahedral)CT3 carbon in 3-membered aliphatic ring, usually tetrahedralCT4 carbon in 4-membered aliphatic ring, usually tetrahedral

CUA1 carbon in double bond, first pairCUA2 carbon in double bond, second conjugated pairCUA3 carbon in double bond, third conjugated pair CUY1 carbon in triple bond, first pairCUY2 carbon in triple bond, second conjugated pair

extended-atom carbon typesC5RE extended aromatic carbon in 5-membered ringC6RE extended aromatic carbon in 6-membered ringCH1E extended-atom carbon with one hydrogenCH2E extended-atom carbon with two hydrogensCH3E extended-atom carbon with three hydrogensCR55 aromatic carbon-merged 5-membered ringsCR56 aromatic carbon-merged 5- or 6-membered rings CR66 aromatic carbon-merged 6-membered ringsCS66 second adjacent pair of CR66 types in fused rings

nitrogen typesN nitrogen: planar-valence of 3, i.e., nitrile, etc.

N3 nitrogen in a 3-membered ringN5R nitrogen in a 5-membered aromatic ring

N5RP for aryl-aryl bond between 5-membered rings N6R nitrogen in a 6-membered aromatic ring

N6RP for aryl-aryl bond between 6-membered ringsNC charged guanidinium-type nitrogen

NC2 for neutral guanidinium group - Arg sidechain





NO2 nitrogen in nitro or related groupNP nitrogen in peptide, amide, or related group

NR1 protonated nitrogen in neutral histidine ringNR2 unprotonated nitrogen in neutral histidine ringNR3 nitrogens in charged histidine ring

NR55 N at fused bond between two 5-membered aromaticsNR56 N at fused bond between 5- and 6-membered arylsNR66 N at fused bond between two 6-membered aromatics

NT nitrogen (tetrahedral), i.e., amine, etc.NX proline nitrogen or similar

oxygen typesO carbonyl oxygen for amide or related structures

O2M oxygen in Si-O-Al or Al-O-Al bondO5R oxygen in 5-membered aromatic ring-radicals, etc.O6R oxygen in 6-membered aromatic ring-radicals, etc.OA carbonyl oxygen for aldehydes or related

OAC carbonyl oxygen for acids or relatedOC charged oxygenOE ether oxygen / acetal oxygen

OH2 ST2 water-model oxygenOK carbonyl oxygen for ketones or related OM oxygen in carbon monoxide or other triply bonded oxygenOS ester oxygen

OSH massless O for zeolites or related cage compoundsOSI oxygen in Si-O-Si bondOT hydroxyl oxygen (tetrahedral) or ionizable acid

OW TIP3P water-model oxygensulfur types

S5R sulfur in a 5-membered aromatic ringS6R sulfur in a 6-membered aromatic ring SE thioether sulfur

SH1E extended-atom sulfur with one hydrogenSK thioketone sulfur

SO1 sulfur bonded to one oxygenSO2 sulfur bonded to two oxygens



CVFF atom types


CVFF atom types

SO3 sulfur bonded to three oxygensSO4 sulfur bonded to four oxygensST sulfur, general: usually tetrahedral

phosphorus typesP6R phosphorous in aromatic 6-membered ringPO3 phosphorous bonded to three oxygensPO4 phosphorous bonded to four oxygensPT phosphorous, general: usually tetrahedral

PUA1 double-bonded phosphorousPUY1 triple-bonded phosphorus

otherLP ST2 lone pair

Table 29. Atom types—CVFF (Page 1 of 4)


hydrogen typesd general deuterium (equiv. to h)

dw deuterium in heavy water (equiv. to h*)h generic hydrogen bonded to C, Si, or H

hc hydrogen bonded to C (equiv. to h)hi hydrogen in charged imidazole ring (equiv. to hn)hn hydrogen bonded to Nho hydrogen bonded to Ohp hydrogen bonded to P (equiv. to h)hs hydrogen bonded to Shw hydrogen in water (equiv. to h*)h* hydrogen in waterh+ charged hydrogen in cation (equiv. to hn)

hscp hydrogen in SPC water model





htip hydrogen in TIP3P water modelcarbon types

c generic sp3 carbonca general amino acid alpha carbon (sp3) (equiv. to cg)cg sp3 alpha carbon in glycineci sp2 aromatic carbon in charged imidazole ring (his+)cn sp3 carbon bonded to N (equiv. to cg)co sp3 carbon in acetal (equiv. to c)

coh sp3 carbon in acetal with hydrogen (equiv. to cg)cp sp2 aromatic carbon (partial double bonds)cr carbon in guanidinium group (HN=C(NH2)2) (arg)cs sp2 carbon in 5-membered ring next to Sct sp carbon involved in triple bondc1 sp3 carbon bonded to 1 H, 3 heavy atoms (equiv. to

cg)c2 sp3 carbon bonded to 2 H’s, 2 heavy atoms (equiv. to

cg)c3 sp3 carbon in methyl (CH3) group (equiv. to cg)c5 sp2 aromatic carbon in 5-membered ring

c3h sp3 carbon in 3-membered ring with hydrogens (equiv. to cg)

c3m sp3 carbon in 3-membered ring (equiv. to c)c4h sp3 carbon in 4-membered ring with hydrogens (equiv.

to cg)c4m sp3 carbon in 4-membered ring (equiv. to c)

c′ sp2 carbon in carbonyl (C=O) group of amidec" carbon in carbonyl group, not amide (equiv. to c′)c* carbon in carbonyl group, not amide (equiv. to c′)c– carbon in charged carboxylate (COO-) group (equiv.

to c′)c+ carbon in guanidinium group (equiv. to cr)c= nonaromatic end doubly bonded carbonc=1 nonaromatic, next-to-end doubly bonded carbonc=2 nonaromatic doubly bonded carbon

nitrogen typesn generic sp2 nitrogen in amide



CVFF atom types


na sp3 nitrogen in amine (equiv. to n3)nb sp2 nitrogen in aromatic amine (equiv. to n3)nh sp2 nitrogen in 5- or 6-membered ring, with hydrogen

attached (equiv. to np)nho sp2 nitrogen in 6-membered ring, next to a carbonyl

group and with a hydrogen (equiv. to np)nh+ protonated nitrogen in 6-membered ringni sp2 nitrogen in charged imidazole ring (his+)nn sp2 nitrogen in aromatic amine (equiv. to n3)np sp2 nitrogen in 5- or 6-membered ring

npc sp2 nitrogen in 5- or 6-membered ring, bonded to a heavy atom (equiv. to np)

nr sp2 nitrogen in guanidinium group (HN=C(NH2)2)nt sp nitrogen involved in triple bondnz sp nitrogen in N2

n1 sp2 nitrogen in charged argininen2 sp2 nitrogen in guanidinium group (HN=C(NH2)2)n3 sp3 nitrogen with 3 substituentsn4 sp3 nitrogen in protonated amine

n3m sp3 nitrogen in 3-membered ring (equiv. to n3)n3n sp2 nitrogen in 3-membered ring (equiv. to n)n4m sp3 nitrogen in 4-membered ring (equiv. to n3)n4n sp2 nitrogen in 4-membered ring (equiv. to n)n+ sp3 nitrogen in protonated amine (equiv. to n4)n= nonaromatic end doubly bonded nitrogen

n=1 nonaromatic, next-to-end doubly bonded nitrogenn=2 nonaromatic doubly bonded nitrogen

oxygen typeso generic sp3 oxygen

oc sp3 oxygen in ether or acetal (equiv. to o)oe sp3 oxygen in ester (equiv. to o)oh oxygen bonded to Hop sp2 aromatic oxygen in 5-membered ringo3e sp3 oxygen in 3-membered ring (equiv. to o)o4e sp3 oxygen in 4-membered ring (equiv. to o)o′ oxygen in carbonyl (C=O) group





o* oxygen in watero– oxygen in charged carboxylate (COO–) group

oscp oxygen in SPC water modelotip oxygen in TIP3P water model

sulfur typess sp3 sulfur

sc sp3 sulfur in methionine (C–S–C) group (equiv. to s)sh sulfur in sulfhydryl (SH) groupsp sulfur in aromatic ring, e.g., thiophenes1 sulfur involved in S–S disulfide bond (equiv. to s)

s3e sulfur in 3-membered ring (equiv. to s)s4e sulfur in 4-membered ring (equiv. to s)s′ sulfur in thioketone (>C=S) groups– partial-double sulfur bonded to something that is

bonded to another partial-double oxygen or sulfurphosphorus

p general phosphorous atomhalogen types

br bromine bonded to a carboncl chlorine bonded to a carbonf fluorine bonded to a carboni covalently bound iodine

ion typesBr bromide ion

ca+ calcium ion (Ca2+)Cl chloride ionNa sodium ion

argonar argon atom

siliconsi silicon atom

otherlp lone pairnu null atom for relative free energy calculations



CVFF_aug atom types


CVFF_aug atom types

As in CVFF (Table 29) with these additional atom types:

Table 30. Atom types—CVFF_aug (Page 1 of 3)

elementatom type charge description

Si sz 2.4 tetrahedral silicon in a zeolite or silicateO oz -1.2 oxygen in a zeolite or silicateAl az 1.4 tetrahedral aluminum atom in zeolitesP pz 3.4 phosphorous atom in zeolites

Ga ga 1.4 gallium atom in zeolitesGe ge 2.4 germanium atom in zeolitesTi tioc 1.6 titanium (octahedral) in zeolitesTi titd 2.4 titanium (tetrahedral) in zeolitesLi li+ 1.0 lithium ion in zeolites

Na na+ 1.0 sodium ion in zeolitesK k+ 1.0 potassium ion in zeolites

Rb rb+ 1.0 rubidium ion in zeolitesCs cs+ 1.0 cesium ion in zeolitesMg mg2+ 2.0 magnesium ion in zeolitesCa ca2+ 2.0 calcium ion in zeolitesBa ba2+ 2.0 barium ion in zeolitesCu cu2+ 2.0 copper(II) ion in zeolitesF f- -1.0 fluoride ion in zeolitesCl cl- -1.0 chloride ion in zeolitesBr br- -1.0 bromide ion in zeolitesI i- -1.0 iodide ion in zeolitesS so4 2.8 sulfur in sulfate ion to be used with oz

Si sy 4.0 tetrahedral silicon atom in clays O oy -2.0 oxygen atom in claysAl ay 3.0 octahedral aluminum atom in claysAl ayt 3.0 tetrahedral aluminum atom to be used with oyNa nac+ 1.0 sodium ion in claysMg mg2c 2.0 octahedral magnesium ion in clays



Fe fe2c 2.0 octahedral Fe(II) ion in claysMn mn4c 4.0 manganese (IV) ion to be used with oyMn mn3c 3.0 manganese (III) ion to be used with oyCo co2c 2.0 cobalt (II) ion to be used with oyNi ni2c 2.0 nickel (II) ion to be used with oyLi lic+ 1.0 lithium ion to be used with oy

Pd pd2+ 2.0 palladium(II)Ti ti4c 4.0 titanium (octahedral) to be used with oySr sr2c 2.0 strontium ion to be used with oy

Ca ca2c 2.0 calcium ion to be used with oyCl cly- -1.0 chloride ion to be used with oy H hocl 1.0 hydrogen in hydroxyl group in clays P py 5.0 phosphorous atom to be used with oy V vy 4.0 tetrahedral vanadium to be used with oy N nh4+ 1.0 united-atom type for ammonium ion to be used with oy S so4y 6.0 sulfur in sulfate ion to be used with oyLi lioh 1.0 lithium ion in water to be used with o*

Na naoh 1.0 sodium ion in water to be used with o* K koh -1.0 potassium ion in water to be used with o* F foh -1.0 fluoride ion in water to be used with o*Cl cloh -1.0 chloride ion in water to be used with o*Be beoh 0.0 beryllium (II) in water to be used with o*Al al 0.0 aluminum metalNa Na 0.0 sodium metalPt Pt 0.0 platinum metalPd Pd 0.0 palladium metalAu Au 0.0 gold metalAg Ag 0.0 silver metalSn Sn 0.0 tin metal K K 0.0 potassium metalLi Li 0.0 lithium metal

Mo Mn 0.0 molybdenum metalFe Fe 0.0 iron metal W W 0.0 tungsten metalNi Ni 0.0 nickel metal



ESFF atom types


ESFF atom types

The atom types in ESFF for the first three rows of the periodic table are listed in Table 31. For the atom types that are not listed in Table 31, please refer to the $BIOSYM_LIBRARY/esff.frc file.

Atom-typing rules in ESFF The names of the atom types for metals are based on the symmetry of the metal complex and on both the oxidation state and coordi-nation number of the metal. For example, Ag024t indicates an Ag that has an oxidation level of 2+, is 4-coordinated, and has tetrahe-dral symmetry. The following table lists the abbreviations used for the symmetry types:

Some metals with differing oxidation numbers and symmetries may be handled with the same parameters. Here, a generic metal atom type is used.

The oxidation number of a metal is determined according to:

Cr Cr 0.0 chromium metalCu Cu 0.0 copper metalPb Pb 0.0 lead metal



symbol symmetry

l C2v s D4h t Td o Oh p D5h h D2h, D3h d D∞



Eq. 136

where Qt is the total charge on the complex, and the sums over Fqi and Fqj are the sums of formal charges on atoms not bonded to metals and bonded to metals, respectively. Nm is the number of metal atoms in the complex, and Nb is the number of metal atoms bonded to the jth ligand atom.

Nox Qt1

Nm------- Fqi∑–

Fqj

Nb--------∑–=

Table 31 Atom types for the first three rowsa of the periodic table—ESFF (Page 1 of 4)

gener-al class

atom type description

hydrogen typesdw deuterium in heavy waterh generic hydrogenhi hydrogen in charged imidazole ring (equiv. to h*)hw hydrogen in water (equiv. to h*)h* hydrogen bonded to nitrogen, oxygenh+ charged hydrogen in cations

carbon typesc generic sp3 carbonca general amino acid alpha carbon (sp3) (equiv. to c)cg sp3 alpha carbon in glycine (equiv. to c)ci carbon in charged imidazole ring (equiv. to cp)co sp3 carbon in acetals (equiv. to c)coh sp3 carbon in acetals with hydrogen (equiv. to c)cp sp2 aromatic carbon with partial double bondcr c in neutral arginine (equiv. to c=)cs sp2 aromatic carbon in 5-membered ring next to S (equiv. to cp)ct sp carbon involved in a triple bondct3 sp carbon involved in COc1 sp3 carbon with 1 H 3 heavies (equiv. to c)c2 sp3 carbon with 2 H's, 2 heavies (equiv. to c)c3 sp3 carbon with 3 H's, 1 heavy (equiv. to c)c5 sp2 aromatic carbon in 5-membered ringc5p sp2 aromatic carbon in 5-membered big pi ringc' carbon in carbonyl (C=O) group

ESFF atom types


c- c in charged carboxylatec+ c in guanidinium group (equiv. to c=)c= generic sp2 carbon

nitrogen typesn generic sp2 nitrogen (in amides)na sp3 nitrogen in aminesnb sp2 nitrogen in aromatic aminesnh sp2 (3 [sp2] 2 [p]) nitrogen in 5-membered ringnho sp2 nitrogen in 6-membered ringni nitrogen in charged imidazole ring (equiv. to nh)no sp2 nitrogen in oxides of nitrogennp sp2 nitrogen in 5-membered ringnt sp nitrogen involved in a triple bondnt2 central nitrogen involved in azide groupnz sp nitrogen in N2n1 sp2 nitrogen in charged arginine (equiv. to n=)n2 sp2 nitrogen (NH2) in guanidinium group (HN=C(NH2)2) (equiv. to n=)n4 sp3 nitrogen with 4 substituents (equiv. to n+)n+ sp3 nitrogen in protonated aminesn= sp2 nitrogen in neutral arginine (double bond)

oxygen typeso generic sp3 oxygen in alcohol, ether, or acid groupoa sp3 oxygen in ester or acidoc sp3 oxygen in ether or acetalsE (equiv. to o)oh oxygen bonded to hydrogen (equiv. to o)op sp2 aromatic in 5-membered ringos oxygen bonded to two siliconsot oxygen with hybridization spo1 oxygen bonded to oxygeno' oxygen having a single double bondo* oxygen in watero- double bonded oxygen in charged carboxylate COO– (equiv. to o')

sulfur typess sp3 sulfursp sulfur in an aromatic ring (e.g., thiophene)


gener-al class




s1 sp3 sulfur involved in (S-S) group of disulfidess2d sulfur with oxidation number 4, two double sigma bonds3d sulfur with oxidation number 4, three sigma bond, (C3v)s4d sulfur with oxidation number 6, four sigma bond, (Td)s4l sulfur with coordination number 4 (C2v)s5l sulfur with coordination number 5 (D4h, C2v)s5t sulfur with coordination number 5 (D3h)s6 sulfur with coordination number 6 (D4h, D2h)s6o sulfur with coordination number 6 (Oh)s' S in thioketone groups- double bonded sulfur in charged phosphate PSS– or PSO– (equiv. to s')

phosphorus typesp general phosphorous atomp4d phosphorous atom with oxidation number 5 and 4 sigma bonds (CTd)p4l phosphorous atom with oxidation number 5 and 3 sigma bonds (C2v)p5l phosphorous atom with oxidation number 5 and 3 sigma bonds (D4h, C2v)p5t phosphorous atom with oxidation number 5 and 3 sigma bonds (D3h)p53 phosphorous atom with oxidation number 5 and 3 sigma bonds (planar)p6 phosphorous atom with oxidation number 5 and 3 sigma bonds (D4h, D2h)p6o phosphorous atom with oxidation number 5 and 3 sigma bonds (Oh)p' sp2 phosphorous atom

other second-row elementsb boron sp3 atombt boron sp atomb' boron sp2 atomBe berillium atomBe+ berillium cationBe+2 berillium cationf fluorine atomF fluorine anionLi lithium atom with s orbitals involved in bondingLi+ lithium ionne neon atom

other third-row elementsar argon atom


gener-al class


ESFF atom types


aPlease see $BIOSYM_LIBRARY/esff.frc for heavier elements.

Al aluminum atomAl033 aluminum atom with coordination number 3Al034 aluminum atom with coordination number 4Al035 aluminum atom with coordination number 5Al035s aluminum atom with coordination number 5 (D4h)Al035t aluminum atom with coordination number 5 (D3h)Al036 aluminum atom with coordination number 6 (D4h, D2h)Al036o aluminum atom with coordination number 6 (Oh)cl chlorine atomCl chlorine ioncl' chlorine atom in oxo acidhe helium atomMg magnesium atomMg025 magnesium atom with 5 coordinationsMg025s magnesium atom with 5 coordinations (D4h)Mg025t magnesium atom with 5 coordinations (D3h)Mg026 magnesium atom with 6 coordinations (D4h, D2h)Mg026

omagnesium atom with 6 coordinations

Mg+ magnesium +1 cation (Oh)Mg+2 magnesium +2 cationNa sodium atomNa+ sodium ionsi silicon atomsi4l silicon atom (D3h, C2v)si5l silicon atom (D4h)si5t silicon atom (D3h)si6 silicon atom (D4h, D2h)si6o silicon atom (Oh)si' sp2 silicon atom


gener-al class




PCFF—additional atom types

This table lists those atom types included in PCFF in addition to the CFF91 atom types listed in Table 27.

Table 32. Additional atom types—PCFFa (Page 1 of 2)


hydrogen typeshn2 amino hydrogenho2 hydroxyl hydrogen

carbonyl functional groups (C and O)c_0 carbonyl carbon of aldehydes, ketonesc_1 carbonyl carbon of acid, ester, amide c_2 carbonyl carbon of carbamate, ureacz carbonyl carbon of carbonateo= oxygen double bonded to O, C, S, N, Po_1 oxygen in carbonyl groupo_2 ester oxygenoo oxygen in carbonyl group, carbonate onlyoz ester oxygen in carbonate

phosphorus typesp= phosphazene phosphorous atom

silicon-related typessi silicon atom

sio siloxane siliconhsi silane hydrogenosi siloxane oxygen

noble gas typeshe helium ar argonkr kryptonne neon xe xenon

metal atoms and halogen ionsAg silver metalAl aluminium metal



aPCFF also includes all the atom types listed for CFF91 (page 291).

Au gold metalBr bromine ionCl chlorine ionCr chromium metalCu copper metalFe iron metalK potassium metalLi lithium metal

Mo molybdenum metalNa sodium metalNi nickel metalPb lead metalPd palladium metalPt platinum metalSn tin metalW tungsten metal

zeolite-related typesaz aluminium atom in zeolitesoss oxygen atom betweem two siliconsosh oxygen atom in terminal hydroxyl group on siliconoah oxygen atom in terminal hydroxyl group on aluminiumoas oxygen atom between aluminium and siliconob oxygen atom in bridging hydroxyl groupsz silicon atom in zeoliteshb hydrogen atom in bridging hydroxyl group

hoa hydrogen atom in terminal hydroxyl group on alumin-ium

hos hydrogen atom in terminal hydroxyl group on silicon

Table 32. Additional atom types—PCFFa (Page 2 of 2)



Aab initio, 11ABM4 integrator, 196

compared with Verlet velocity integrator, 204

absolute free energyalgorithm, 259calculation, 258conformational searching, 267constraining dynamics, 268convergence, 271dynamics running averages, 272errors, 269, 271example, 264ideal solid, 259λ intervals, number, 269reference state, 263, 265setting up, 267spring constants, 268, 269, 272

Adams–Bashforth–Moulton integrator, 196adiabatic compliance tensor, 210adiabatic compressibility, 210adiabatic ensembles, 205Alagona, G., 280Allen, M. P., 116, 210, 222, 227, 273Allinger, N. L., 69, 276, 279AMBER, 69

atom types, 54atom types for carbohydrates, 55characteristics, 22distance-dependent dielectric, 126functional form, 54hydrogen-bond term, 541–4 nonbond interactions, 123

amino acids, 24, 57, 60Andersen, H. C., 209, 227, 237, 273angles

constraints, 109, 237atom types

assigning, 79asterisks, 85

equivalences, 85forcefield parameter assignment, 84wildcarding, 85, 87wildcarding, precedence, 87X, 85

atomic positions, poorly defined, 101atomic velocities, 211atoms

fixing, 102forces on, 181types, 79, 283typing, 78

atom-type charge, 81Austin, N., 66, 280automatic atom types, 32, 60

BBash, P. A., 279Becker, J. M., 275Berendsen equation, 217Berendsen, H. J. C., 217, 224, 226, 273, 278, 279Beveridge, D. L., 180, 277BKS forcefield, 64bold type, meaning, 7bond hybridization, 50bond increments, 32, 60bonds

between symmetrically related objects, 120constraints, 109, 237

Born–Oppenheimer equation, 11Born, M., 11, 273Boyd, D. B., 274Brady, J. W., 275Brooks, B. R., 23, 56, 273Brooks, C. L., 255Brooks, C. L., III, 127, 234, 273Brown, D., 227, 273Brown, F. K., 279Bruccoleri, R. E., 273Brunger, A. T., 273

Index


.

buffer region, 134building models, 76bulk modulus, 211Burchart forcefield, 64

implementation, 64Burchart–Dreiding forcefield, 65Burchart–Universal forcefield, 65Burchart, E. de Vos, 64, 273

Ccalculation

dynamics, 238calculations

controlling, 15, 100, 231loops, 231minimization, 173

carboxylates, 136Carruthers, L. M., 277Cartesian and crystal axes, 114, 115Cartesian coordinates, 18, 114Casewit, C. J., 22, 41, 273, 274, 278Case, D. A., 280Catlow, C. R. A., 147, 274cell multipole method, 138

accuracy, 142and nonbond interactions, 143and system size, 142CPU time, 142derivation of, 139

Cerius2•Morphology module forcefields, 60CFF forcefields

availability, 27functional form, 29

CFF91atom types, 291, 308automatic parameter assignment, 87characteristics, 20functional form, 29, 32out-of-plane coordinate, 29

“CFF93”, 68CFF95

additional information, 32limitations, 32

charge groupscutoffs and, 135defined, 135

chargesassigning, 83

CHARMmcharacteristics, 22, 55functional form, 55

Cheetham, A. K., 277, 280chemical perturbations, 251chirality, 54Ciccotti, G., 278Clark, J. H. R., 227, 273classic harmonic oscillator, 13classical forcefields

availability, 52types, 52

Colwell, K. S., 273, 274, 278common structures, finding, 233compressibility, 211computation

costs, 102, 167efficiency, 128time, 118

conformationsearches, 232

conformational changesforcing, 103

conformational energies, 182conformational energy barriers, 231conformational searches, 191, 232conformations

comparing, 103searches, 191, 208stable, 154

consensus dynamics, 233Consistent Valence Forcefield, see CVFF and

forcefieldsconstraints, 109

creating, 109definition, 15, 98dynamics, 235

Coulombicinteractions, 29, 122terms, 18, 59

cross terms, 16, 29, 59, 177definition, 18importance of, 59

Cross, P. C., 280crystals

.


environment, 113, 149phase transitions, 227surface contributions, 149

cutoff distanceeffect on van der Waals energy, 129, 130switching function, 130

cutoffsdouble, 136methods, 127periodic systems, 116, 117variable names in versions 2.9.5 and 95.0,

132CVFF

atom types, 60automatic parameter assignment, 87characteristics, 23functional form, 58

DDauber-Osguthorpe, P., 57, 274Dauber, P., 275, 276Decius, J. C., 280Deem, M. W., 149, 274Demontis, P., 66, 274Denouden, C. J. J., 277density, periodic boundary conditions, 222diagonal terms, 58dielectric constant, 59, 126Dielectric parameter, 127diffusion coefficients, 191Ding, H. Q., 139, 142, 274DiNola, A., 273Dinur, U., 20, 274, 276, 277dipole–dipole interactions, 122disordered periodic systems, 137distance constraints, 109distance restraints, 106

potential form, 106, 108distorted structures, 177Dist_Dependent parameter, 127DNA, 23Dreiding forcefield

atom type naming, 51availability, 35characteristics, 22versions, 52

dynamics, 189ABM4, 196achieving equilibrium, 217algorithms, 193Andersen method, 220, 227animation, 238appropriate ensemble, 211artifacts, 201Berendsen method, 217, 226blowing up, 202canonical ensemble, 208colliding hydrogen atoms example, 199computing time, 235consensus, 233constant-energy, constant-volume ensem-

ble, 207constant-pressure, constant-enthalpy en-

semble, 209constant-temperature, constant-pressure

ensemble, 208constant-temperature, constant-stress en-

semble, 208constant-temperature, constant-volume

ensemble, 208constraints, 237constraints and restraints, 235continuing a run, 246data-collecting stage, 239definition, 12direct velocity scaling, 216double cutoffs and, 137energy conservation example, 204equilibration, 217, 239files, 247generating statistical ensembles, 191Hoover, 217impulse, 233initial velocities, 212integration errors, 199, 204integrators, 193, 194Langevin, 234lengths of run stages, 240limitations, 199liquid simulations, 227microcanonical ensemble, 207multiple timesteps, 136Nosé, 217Nosé–Hoover, 217Nosé–Hoover method and fictitious mass,

218, 219


.

Nosé–Hoover method and integrator, 220Nosé–Hoover method and relaxation time,

219Nosé–Hoover method and timestep, 219Nosé–Hoover thermostat, 217obtaining accurate fluctuations, 210Parrinello-Rahman method, 228periodic systems, 224preparing the system, 240pressure, 220, 226pressure and cell size, 227pressure control, 205, 208pressure, effect of cutoffs, 227quenched, 232RATTLE algorithm, 237reinitializing velocities, 247relaxation time, 217repeating a run, 193reproducibility, 193restarting a run, 246results, 238reviewing setup, 245Runge–Kutta-4, 197setting up, 239, 241setting up SHAKE or RATTLE constraints,

235SHAKE algorithm, 236simulated annealing, 232snapshots, 238specifying output, 244specifying run conditions, 242specifying the integrator, 194specifying the pressure- and stress-control

methods, 226specifying the temperature-control meth-

od, 215specifying the thermodynamic ensemble,

206specifying the timestep, 198specifying types of simulations, 230stability of numerical integration, 199stages, 242, 246starting a new run, 246starting the run, 245statistical ensembles, 205, 229stochastic boundary, 234stress control, 209target temperature, 216temperature control, 205, 208, 209, 214, 217temperature cycles, 232

temperature effects, 203theory, 192timestep, 197timestep length, 199, 236trajectories, 238trajectory, and absolute free energy calcula-

tion, 261true canonical ensemble, 217types, 229unit cell changes, 208, 209uses, 3, 189, 191Verlet velocity integrator, 195with minimization, 232

EEichinger, B. E., 278Einstein solid, 262elastic constant, 211electronic motion equation, 11electrostatic interactions, 54, 59energy

barriers, 233contributions of terms, 15

enthalpy, 209at absolute zero, 180binding, 180

entropy, 180, 183, 271equilibrium thermodynamic properties, 210Ermer, O., 4, 166, 274ESFF

angle rules, 38angle types, 37atom types, 304availability, 35bond energy, 36bond rules, 36characteristics, 21charges, 38electronegativity, 39energy expressions, 36functional form, 36hardness, 39ionization potential, 40out-of-plane term, 38periodic table coverage, 41torsion rules, 38torsion term, 38

.


van der Waals interactions, 40ethane, 255Ewald calculations

accuracy, 149and nonbond interactions, 149and system size, 149computational load, 147CPU time, 149dipole moment, 1452D periodicity, 150

Ewald, P. P., 143, 274Ewig, C. S., 20, 32, 275, 276explicit image model, 117, 118

Ffentanyl, 264

phi,psi map, 265, 267Field, M., 275files

.car, 257, 268

.cli, 268, 272

.cor, 268dynamics, 247forcefield, 85.frc, 32, 60.his, 270history, 270.inp, 257.mdf, 79, 257molecular data, 79.out, 270, 272output, 270.tot, 257, 270, 272

Fisher, J., 279Flannery, B. P., 278Fleischman, S. H., 255Fletcher, R., 165, 178, 274fluids, nonviscous, 221fluorescence depolarization rates, 238forcefields

aluminophosphates, 64AMBER, 53and modeling programs, 1atom type equivalences, 85atom types, 79automatic, 86automatic parameter assignment, 86

broadly applicable, 21, 35Cerius2•Morphology module, 66choosing, 19classical, 22components, 12crystal morphology, 66CVFF, 57definition, 4, 12editing, 90ESFF, 36functional forms, 12, 16general sequence of activities for using, 75glass, 61graphical and standalone-mode use, 75graphical molecular modeling interfaces,

75Homans AMBER, 22importance, 4internal coordinates, 18limitations, 15mechanical approach, limitations of, 15mechanical vs. quantum mechanical ap-

proach, 13MSXX, 63old, 67, 68parameters and atom types, 84purpose, 13quantum calculations and, 27quantum mechanical parameterization, 27reading parameters, 76rule-based, 21, 35second-generation, 20, 27selecting, 78silicas, 64sorption, 66sorption onto zeolite structures, 66special-purpose, 23standalone, 75summary table, 24terms, 12types, 20types of molecules, 23unsupported, 67, 68using, 75zeolites, 64

frictional coefficients, 234, 238


.

GGarofalini, S. H., 61, 274, 276, 278, 281Gaussian–Legendre quadrature method, 254Gelin, B. R., 277general multipole moments, 140generating Cartesian coordinates, 77Genest, M., 274Ghio, C., 280ghost

images, 118list regeneration, 118molecules, 117, 118

Giammona, A., 275glass forcefield, 60, 61

automated setup, 63functional form, 61

global minimum, 179, 231glycoproteins, 54Goddard, W. A., 43, 63, 83, 125, 146, 147, 149,

274, 276, 277, 278Greengard, L., 138, 142, 274Gunsteren, W. F., 165, 274

HHaak, J. R., 273Hagler, A. T., 20, 28, 32, 60, 125, 128, 180, 181,

274, 275, 276, 277, 279, 280Halgren, T. A., 21, 33, 34, 125, 275Hamiltonian, 259Harris, F., 150Harvey, S. C., 125, 275Ha, S. N., 54, 55, 275help, on-line, 6hemoglobin, 142Hessian matrix, 169, 177

computational cost, 167, 169definition, 166from second derivatives, 172mathematical form, 169preconditioning, 172properties, 169size, 167

heterogenous atom pairs, 124Hill, J.-R., 20, 33, 275Homans, S. W., 22, 53, 54, 275

Hoover, W. G., 217, 262, 275Hwang, J. K., 180, 276Hwang, M.-J., 20, 28, 32, 68, 276, 277hydrocarbons, 60hydrogen atoms, colliding, 200hydrogen bonds, 60hydrostatic pressure, 208, 221, 224, 228hypervalent molecules, 48

IIkels, K. G., 277image centering, 119inexact line searches, 169instantaneous kinetic temperature, 213instantaneous pressure function, 222interatomic distances, controlling, 106internal energy, 210International Tables for Crystallography, 114IR line widths, 238isothermal compressibility, 210, 240isothermal ensembles, 205

JJacucci, G., 180, 278Jorgensen, W. L., 255

KKao, J., 69, 276Karasawa, N., 63, 125, 146, 147, 149, 274, 276Karplus, M., 23, 55, 165, 273, 274, 276, 277kinetic energy, 210kinetic energy of nuclei, 183King, G., 280Kirkwood, J. G., 180, 276Kitson, D. H., 128, 276Klein, M. L., 274Knaebel, K. S., 277Kohler, A. E., 61, 276Kollman, P. A., 279, 280Kramer, G. J., 280

.


Llattices, periodic, 113Lee, M. A., 139, 142, 278Lennard–Jones terms, 18, 59Levitt, M., 162, 276Levy, R. M., 234, 276Lifson, S., 66, 162, 275, 276Liljefors T., 69, 276Lipkwowitz, K. B., 274Li, S., 276local minimum, 179

MMaple, J. R., 20, 28, 29, 32, 276, 277, 279Maxwell–Boltzmann

distribution, 212equation, 211factor, 257

Mayo, S. L., 22, 51, 68, 277McCammon, J. A., 180, 234, 276, 277, 280McGuire, R. F., 277McQuarrie, 220, 277methanol, 255Mezei, M., 180, 251, 277Mezei’s algorithm, 251Miller, G. W., 66, 277minimization, 153

algorithms, 155, 176constraints, 100convergence, 177, 178, 179derivatives, 156, 178efficiency, 155energy expression, 155energy zero, arbitrary, 180initial relaxation, 163iteration, defined, 160line search, 157, 158progressive, 101restraints, 100setting up, 173significance, 180strategies, 155, 173system size, 165, 167uses, 2

minimizers

advantages, 169, 170Broyden–Fletcher–Goldfarb–Shanno, 169comparative efficiency, 172conjugate gradients, 164, 177Davidon–Fletcher–Powell, 169differences among Newton methods, 168discontinuities in potential energy surface,

sensitivities to, 137limitations, 167Newton–Raphson, 166, 167, 177Newton–Raphson, algorithm, 168quasi-Newton-Raphson, 168steepest descents, 161, 163truncated Newton-Raphson, 170

minimum image model, 117, 225MMFF

availability, 27energy expression, 33

modeling incomplete systems, 101molecular dynamics, see dynamicsmolecular mechanics, 12molecular modeling program, difinition, 5Momany, F. A., 23, 55, 67, 277monomer definitions, 76Montgomery, 273Morphology/Lifson forcefield, 66Morphology/Momany forcefield, 67Morphology/Scheraga forcefields, 67Morphology/Williams forcefield, 67Morse potential, 58, 177

harmonic potential, compared with, 59MSI’s website, 6, 125MSXX, 63

Morse functional form, 64parameterization, 63

Mumby, S. J., 279Murcko, M. A., 55, 280

NNachbar, R. B., 21, 34, 275Naider, F., 275natural response functions, 210neighbor list, 134Némethy, G., 67, 277Newsam, J. M., 274Newton’s equation of motion, 11, 192


.

Nguyen, D. T., 280NMR relaxation times, 238nonbond interactions, 59

cell multipole method, 138cutoff distance and, 128, 129definition, 18Ewald calculations, 143system size and, 127

nonbond list, 134nonbond terms, 16non-hypervalent, 48nonpolar hydrogens, 54Norgett, M. J., 147, 274normal mode analysis, 179Nosé, S., 217, 219, 277Nowak, A. K., 277, 280nuclear motion equation, 11nucleic acids, 53number density, 238

Ooff-diagonal terms, see cross termsOlafson, B. D., 273, 277oligosaccharides, 54Oppenheimer, J. R., 11, 273Osguthorpe, D. J., 274

PParrinello, M., 209, 228, 277partition function, 259peptides, 23periodic boundary conditions, 224, 225

definition, 113periodic systems

disordered, 137dynamics, 208, 209, 222, 224Ewald calculations, 143

Peterson, B. K., 277Pettitt, B., 273Pickett, S. D., 66, 277polymer forcefield, 61polymers, 53polysaccharides, 53, 54polyvinylidene fluoride forcefield, 60, 63

Postma, J. P. M., 273, 279Post, M. F. M., 277potential energy surface, 10, 11, 16

discontinuities, 137empirical fit, 11, 12non-quadratic, 177saddle points, 186shape of, 183transition states, 186water molecule, 17

Potentials Forcefield, 81, 83Pottle, M. S., 277Powell, M. J. D., 165, 278pressure, 210

and correct statistical ensemble, 225calculation, 222changing, 226control, 225functional form, 220periodic boundary conditions, 222sign conventions, 222thermodynamic, 222units, 221virial theorem, 222

Press, W. H., 178, 197, 254, 278Profeta, S. Jr., 280proteins, 23, 53protonated amines, 136p1_4 parameter, 123

Qquantum effects, 180quantum mechanical probability function, 14quartz, 144Quirke, N., 180, 278

Rradius of gyration, 238Rahman, A., 228, 277Rappé, A. K., 22, 41, 42, 43, 83, 273, 274, 278rattle command, 109Ravimohan, C., 255Ray, J. R., 207, 210, 278reading models, 76Reeves, C. M., 165, 274

.


Ree, F. H., 262Reidl, D., 114, 278relative free energy

benchmark calculation, 255convergence, 254, 257degrees of freedom, 256drift, 258example, 255FDTI, 251, 253, 254finite difference thermodynamic integra-

tion, see FDTIfunctional form, 254perturbation method, 252results, 257setting up, 256thermodynamic cycle, 255

residue definitions, 76restraints, 16, 154

angle, 110definition, 15, 98distance, 106dynamics, 235inversion, 112torsion, 110

Rigby, D., 278, 279RNA, 23Roberts, V. A., 274Rokhlin, V. I., 138, 142, 274Rone, R., 23, 55, 277Rosenthal, A. B., 61, 278rotational barriers, heights, 182Runge–Kutta integrator, 197Ryckaert, J.–P., 236, 278

Ssaddle points, 178Sauer, J., 20, 33, 275Scale_Terms Parameters, 123Scheraga, H. A., 277Schmidt, K. E., 138, 142, 278Schrödinger equation, 10Set Parameters, 127setting up calculations, 77Sharon, R., 275shear stress, 221shielded dielectric function, 126

Shi, S., 279simulated annealing, 232simulation engine, definition, 4Singh, U. C., 180, 255, 279, 280Sinha, S. K., 274Skiff, W. M., 278small molecules, 53Smit, B., 277solvent, 113, 115Sorption Demontis forcefield, 66Sorption Pickett forcefield, 66Sorption Yashonath forcefield, 66Soules, T. F., 61, 279special-purpose forcefields

availability, 60characteristics, 61

specific heat, 210, 211, 240specific heat at constant pressure, 210spectral densities, 238spline switching nonbond cutoff, 133Sprague, J. T., 69, 279Stapleton M. R., 280States, D. J., 273statistical ensembles, 191, 205Stern, P. S., 275Stockfisch, T. P., 276, 277Straatsma, T. P., 180, 279stress, 221, 228, 229

control, 225sign conventions, 222tensor components, 208units, 221

stress–strain relationship, 209, 221, 228structural similarities, 233Sun, H., 20, 33, 278, 279surface charges, 149Sussman, F., 280Swaminathan, S., 273Swift, J. F. P., 277switching atom defined, 135switching function, effect on nonbond energy,

132symmetry relations, 119


.

TTai, J. C., 276, 279Tembe, B. L., 180, 280temperature, 211

and average kinetic energy, 212and correct statistical ensemble, 214calculation of, 213control, 220damping, 217distribution of atomic velocities, 211integration errors, 203nonperiodic system, 213nonzero, 189periodic system, 213

template atoms, 103template forcing, 103, 155

functional forms, 103tensile stress, 221Tesar, A. A., 61, 280tethering, 101, 105

functional forms, 105Teukolsky, S. A., 278Thacher, T. S., 277thermal expansion, 210, 211thermodynamic cycle, 180thermodynamic ensembles, 191thermodynamic response function, 207thermodynamic temperature, 213Thomas, J. M., 277, 280Tildesley, D. J., 116, 210, 222, 227, 273timestep length in dynamics, 197Tolleanaere, J. P., 264torsions

forcing, 110restraints, 110, 111

trajectoryanimation, 238contents, 238definition, 193length, 241uses, 238

transition states, 183

Uunited-atom representation, 54

Universal forcefieldatom type naming, 42availability, 35characteristics, 22charges, 43implementation, 41parameter generation, 42versions, 43

VVALBOND

characteristics, 22valence exclusions, 123valence interactions, 16van Beest, B. W. H., 64, 280van der Waals

combination rules, 124cutoff distance and, 129, 130Ewald calculations, 144interaction potential, 199interactions, 29, 54, 59

van Gunsteren, W. F., 273van Santen, R. A., 280Varshneya, A. K., 61, 279, 280Verlet leapfrog integrator, 194, 195Verlet velocity integrator, 194, 195

compared with ABM4 integrator, 204Verlet, L., 195, 280Vetterling, W. T., 278vibrational calculations, 185

computational costs, 187cross terms and, 187entropy, 187forcefield accuracy and, 187free energy, 182frequencies, 59, 167, 182, 185from potential energy surface, 184harmonic, 153imaginary frequencies, 186normal mode analysis, 167prerequisites, 185quality, 185quantum mechanical equation, 186uses, 182

virial, 210viscous fluids, 234volume, periodic boundary conditions, 222

.


WWaldman, M., 20, 125, 276, 280Warshel, A., 180, 276, 280Watanabe–Austin forcefield, 66Watanabe, K., 66, 280water, 57

constrained, 237fixed-geometry model, 109, 237SPC, 109, 237TIP3P, 109, 237

Weiner, P., 280Weiner, S. J., 22, 53, 69, 280, 290Wiberg, K. B., 55, 280Williams, D. E., 67, 280Wilson, E. B., 29, 183, 280Wolff, J., 274Wolynes, P. G., 277Woodcock, L. V., 61, 280

YYang, Y., 279Yan, L., 279Yashonath, S., 66, 274, 280Yuh, Y., 279

Zzeolite forcefield, 60, 61zero-point corrections, 180, 182, 186Zirl, D. M., 61, 274, 281


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