Reactive Empirical Force FieldsJason Quennevillejasonq@lanl.gov

X-1: Solid Mechanics, EOS and Materials PropertiesApplied Physics Division

Los Alamos National Laboratory

Timothy C. Germann, Los AlamosAlejandro Strachan, PurdueAdri C. T. van Duin, Caltech

William A. Goddard III, CaltechAlexei A. Stuchebrukhov, UC Davis

2006 Summer School on Computational Materials ScienceJuly 31 - August 11, 2006 · University of Illinois

Motivation

Empirical force fields are used in biology, chemistry, physics and materials science to calculate the potential energy surface and atomic forces.

Most, like CHARMM and AMBER, assume the same atomic connectivity (molecular composition) throughout simulation.

No Chemistry!!!

Straightforward solution: ab initio or QM/MM (up to 300 atoms for QM system)

For materials simulation, we may want 10s of 1000s to millions of atoms and as much as a nanosecond of simulation time.

Need a more efficient method!!!

Empirical Force Fields

Empirical force fields contain potential energy functions for each atomic interaction in a molecular system.

Bond Stretch:

Bond Bending:

Bond Torsion:

Parameters can be taken from experiment (e.g., vibrational spectroscopy) or from ab initio quantum chemistry calculations.

2

stretch 0

1

2 RV k R R

2

bend 0

1

2V k

torsion 1 cos ; 1nV V s n s

Non-Bonded potentials give the intermolecular interactions:

Coulomb:

van der Waals:

Parameters obtained through ab initio quantum chemistry and liquid simulations.e.g. OPLS (optimized potentials for liquid simulations, W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc. 110, 1657 (1988). )

Empirical Force Fields

Lennard-Jones 12 6

ij ij

ij ij

A CV

R R

2

Coulombi j

ij

q q eV

R

12

6

; 4

; 4

ij i j i i i

ij i j i i i

A A A A

C CC C

Empirical Valence Bond (EVB)EVB attempts to combine empirical potential energy functions with valence bond

ideas to describe chemical reactions efficiently and accurately.

EVB Applications

Proton transport in aqueous acid (CPL, 284, 71 (’98); JPCB, 102, 5547 (’98))

Aqueous acid-base reactions (JPCA, 105, 2814 (‘01))

Enzyme catalysis (Warshel)

Nucleophilic substitution reactions

Good Introduction:

Computer Modeling of Chemical Reactions in Enzymes and Solutions, A. Warshel Wiley-Interscience (02/01/1997)

EVB: introduction

EVB starts with a NN potential energy matrix: N diabatic states (diagonal) N(N-1) couplings (off-diagonal)

Each diabatic state looks like a configuration in a standard non-reactive force field.

Off-diagonal coupling elements: interaction between each diabatic state and the N-1 remaining states.

Diagonalize V adiabatic states. The minimal value is the ground state.

coupling term

adiabatic ground state

diabatic states

Calculation of Forces

0 0

2 2n nmn n m

n n m

HF

RH V

a a aR R

Diagonalizing the NN EVB matrix yields the ground state as a linear combination of diabatic states.

If an is the set of corresponding coefficients, the forces can be calculated using the Hellman-Feynman theorem:

EVB: diagonal matrix elementsBecause we need to treat bond breaking and formation, the Bond Stretch should

be anharmonic:

Bending and Torsional potentials can be as before:

System-environment interactions treated with standard non-bonding potentials:

2

stretch 01 expeV D R R 2

stretch 0

1

2 RV k R R

2

bend 0

1

2V k torsion 1 cos ; 1nV V s n s

Lennard-Jones 12 6

ij ij

ij ij

A CV

R R

2

Coulombi j

ij

q q eV

R

12

6

; 4

; 4

ij i j i i i

ij i j i i i

A A A A

C CC C

Interaction between EVB States

System-system non-bonding interactions more complicated due to the potential for chemical reaction.

A functional form more flexible than Coulomb + Lennard-Jones is required.

The intermolecular interactions (part of the diagonal element) and the coupling terms (off-diagonal) must be parametrized together in order to describe the reaction correctly.

In the activated complex, the favorable interaction between the two states is controlled by the intermolecular interaction. It is normally written in terms of the distance between the two reactant centers.

The reaction barrier is controlled by coupling term. This term is generally a function of the reaction coordinate.

1.2 Å

1.2 Å

0.97 Å

0.97 Å

0.97 Å

0.97 Å

116°

118°

116°

118°

109°

114° 1.22 Å

1.22 Å

0.96 Å

0.96 Å

103°

103°

101°

Optimized geometries of the {H2O–H–OH2}+ (left) and {HO–H–OH}– (right) complexes, obtained from first principles (MP2/aug-cc-pVTZ).

Application: Proton Transfer in Water

H

HO O

H

HH

H

HO O

H

HH

H

HO O

H

HH

Diabatic States:

Adiabatic State:

EVB of H3O+–H2O Proton Transfer

2 2 2 2

2 2

2 2H O H O H O H OO-H

intra 01

2 2H O H OH-O-H0

1

1 exp

1

2

e ii

jj

V D R R

k

3 2 3

3

3 3

3

3 3

22 3 4H O /H O H O

inter damp 4O-O1 1

O-O,H OO-O O-O O-O0 0O-O,H O O-O,H O

O-H ,H OO-H O-H O-H0 0O-H ,H O O-H ,H O

2

exp exp /

exp exp / .

i j

i j i j

q q eeV c

R RR

D R R R L

D R R R L

+ + + +3 3 3 3

+ +3 3

3 2H O H O H O H OO-H

intra 01

3 2H O H OH-O-H

01

1 exp

1

2

e ii

jj

V D R R

k

H3O+–H2O Proton Transfer:Diagonal Elements

O O O O,X Y X Y

I JV P R Q R

O O (1) (1) O O O O (1) 2--H-- --H-- 0 --H--

(2) (2) O O O O (2) 2--H-- --H-- 0 --H--

, exp

exp

X Y X Y X Y

X Y X Y

X Y X Y X Y

X Y X Y X Y

P R V R R

V R R

O O O O O O--H-- tanh,0

11 tanh ,

2

X Y X Y X Y

X YQ R R R

H O O12

t X Y

R R R

H3O+–H2O Proton Transfer:Coupling Elements

H

H

O O

H

H

H

EVB vs Ab Initio for H3O+/H2O

2.4 Å

2.6 Å

2.8 Å

EVB

MP2/aug-cc-pVTZ

EVB Summary

Very good for systems with small number of possible reactions

Reaction barriers are treated explicitly

Offers an empirical description of chemical reactions

Gives mixing of diabatic states during reaction

Can be difficult to parametrize intermolecular potentials and couplings

Limitation on number of states due to diagonalization (cubic scaling)

ReaxFF

Bond-Order potential, developed at CalTech by Adri van Duin and Bill Goddard

Potential parametrized using ab initio calculations (B3LYP/6-31G**) on a “training set” of reactions

Why bond-order based?

non-reactive potentials have atom-types that define connectivity

Applications:

High Explosives, Propellants, Catalysis, Fuel Cells, Corrosion, Friction, etc.

HN NH N N + H2

Background References

Bond Order/Bond Length relationship Pauling, J. Am. Chem. Soc., 69, 542 (1947).

Reactive Empirical Bond Order (REBO) Johnston, Adv. Chem. Phys., 3, 131 (1960). Johnston, Parr, J. Am. Chem. Soc., 85, 2544 (1963).

Other Bond-Order Potentials Tersoff, Phys. Rev. Lett., 56, 632 (1986); Tersoff, Phys. Rev. Lett., 61, 2879 (1988). Brenner, Phys. Rev. B, 42, 9458 (1990). Brenner, et al, J. Phys.: Condens. Matter, 14, 783 (2002).

ReaxFF van Duin, Dasgupta, Lorant, Goddard, J. Phys. Chem. A, 105, 9396 (2001). Strachan, Kober, van Duin, Oxgaard, Goddard, J. Chem. Phys., 122, 054502 (2005). User Manual: http://www.wag.caltech.edu/home/duin/reax_um.pdf

ReaxFF allows for computationally efficient simulation of materials under realistic conditions, i.e. bond breaking and formation with accurate chemical energies.

Due to the chemistry, ReaxFF has a complicated potential energy function:

bond valency angle tooverpotential under penalty

lone pair conjugation

rsion

H-bond vdWaals Coulomb

E E E E

E E

E E E

E E E

ReaxFF Potential Energy Function

Charge equilibration: EEM (Mortier, et al, JACS, 108, 4315, (’86).)

H C C H

Example: Acetylene

Bond Order goes smoothly from 0 1 2 3 as C-C Bond Length shortens from large distance to 1.0 Å

Bond Order, Bond Energy

0 0 0

BO exp exp expij ij ijij

r r r

r r r

…not explicitly a function of bond distance be,1

bond e be,1BO exp 1 BO pij ijE D p

Bo

nd

Ord

er

C-C Distance / Å

Bond orders adjusted to get rid of unphysical bonds.

Bond Order Corrections

Over- and under-coordination of atoms must be avoided.

Energy penalty added to the potential energy function for the case where an atom has more bonds than its valence allows.

e.g., Carbon can’t have more than 4 bonds; Hydrogen no more than 1

If an atom is under-coordinated, the stabilization of bonding should be used if possible.

Bond Angles, Bond Torsion

Bond Angles and Torsions are intimately tied to the bond types.

With a bond order potential, angles and torsions must be written in terms of the bond order.

Angle and torsion energy terms must 0 as B.O. 0.

angle ( , , )ijk ij jlE f BO BO

torsion ( , , , )ijkl ij jk klE f BO BO BO

See J. Phys. Chem. A, 105, 9396 (2001) for full potential form.

CCH 120° CCH = 180°

bond angle torsion lone pair conjugation H-bond vdWaals CoulombE E E E E E E E E

Lone Pair Electrons, Conjugation

The creation or reaction of lone-pair electrons should be assigned an energy term.

Elone pair corresponds to an energy penalty for having too many lone pairs on an atom (i.e., overcoordination)

Conjugated systems should have added stabilization.

Econj has maximum contribution when successive bonds have bond-order values of 1.5.

bond angle torsion lone pair conjugation H-bond vdWaals CoulombE E E E E E E E E

implicit in AMBER/CHARMM-like potentialsthrough atom type

Hydrogen bonding extremely important in biological systems but also in many organic solids.

X – H Y

Hydrogen bonds are calculated between group X-H and Y, where X and Y are atoms known to form H-bonds (e.g., N, O)

The H-bond energy term is written in terms of the bond-order of X-H, the distance between H and Y, as well as the X-H-Y angle.

Can be an expensive part of the calculation because many acceptor (Y) atoms could be available for any given X-H group. All interactions must be calculated out to a cutoff distance (~10 Å) in order to remain consistent from timestep to timestep.

Hydrogen Bonding

angle XH Y H XH Y( , , ) E f BO R

- Short-range Pauli Repulsion- Long-range attraction (dispersion)- Coulomb forces

van der Waals and Coulomb terms are included for all atom pairs (whether bonded or non-bonded)!

This avoids changing the potential when chemistry occurs. Such alterations, which are natural in the EVB formalism, would be awkward in ReaxFF.

Shielding included for both Coulomb and van der Waals in order to avoid excessive interaction between atoms sharing bond and/or bond angle.

Non-Bonded Interactions

En

erg

y /

kca

l mo

l-1Interatomic Distance / Å

Charge Equilibration

The charge on an atom depends on the molecular species:

Atomic charges are adjusted with respect to connectivity and geometry.

Many QEq methods available. ReaxFF uses Electronegativity Equalization Method (EEM: Mortier, et al, JACS, 108, 4315, (’86).)

The desired charge distribution is that which minimizes,

Final Coulomb energy from screened potential – all atom-pairs calculated

Coulomb 1/33 3(1/ )

i j

ij ij

q qE C

r

HN NH N N + H2

212

i ji i i i

i j iij

q qE q J q

r

ReaxFF/Ab Initio ComparisonReaxFF can decribe a wide variety of chemical reactions.

Strachan, et al, JCP, 122, 054502 (’05).

e.g., unimolecular

decomposition of RDX

103,680 atoms(4320 molecules,

121215 unit cells)

256 processors

TATB

768 atoms(32 molecules,

224 unit cells) Interested in chemical reaction dynamics of high explosives (HE) under shock conditions

Want as big a system (105 to 106 atoms) as possible in order to study the spread of reactions, temperature distribution, carbon-clustering, etc.

Application: HE at High T, P

TATBN2

H2O

CO2

Tinitial = 1700 K

Tfinal = ~3200 K

Decomposition of TATB at High T

GRASP (General Reactive Atomistic Simulation Program) developed at Sandia National Lab by Aidan P. Thompson.

Objective: Parallel scalable MD code (C++) which enables implementation of a wide range of force field types, particularly reactive force fields, including ReaxFF.

CPU Time per timestep: serial code: 2.8 seconds parallel code: 0.6 seconds (32 CPUs)

System size limits: serial code: 5000 atoms parallel code: 500,000 atoms (510 CPUs)

ASC Flash: 2.0-GHz procs 8 GB memory per node

ReaxFF in Parallel

ReaxFF SummaryCan simulate chemistry for a wide range of materials significantly faster than ab initio and semi-empirical methods

Accuracy similar to semi-empirical methods

Hydrocarbons, CHNO explosives, silicon oxides, etc.

Main limitation is governed by the size of reaction training set

Used extensively for explosives under extreme conditions - many possible reactions

Simulation sizes up to a half million atoms

Tim

e/It

erat

ion

(sec

onds

)

Natoms