http://secamlocal.ex.ac.uk/people/staff/ashm201/Atomistic/

LecturesIntroduction to computational modelling and statistics 1Potential models 2Density Functional (quantum) 1 3Density Functional 2 4

Or: Understanding the physical and chemical properties of materials from an understanding of the underlying atomic processes

Computational Modelling of Materials

Recent Advances in Contemporary Atomistic Simulation

The increased power of computers have allowed a rapid advance in the use of simulation techniques for modelling the properties of materials.

Introduction

•Interpret of experiment•Extrapolate experimental data •Empirical Search•Prediction of New Effects

Why do it?

But how?

The answer depends on the length and time-scale

Atomistic Simulation: Choices

Which Technique? Energy Minimisation Molecular Dynamics Monte Carlo Genetic Algorithms

How do you calculate the forces? Interatomic Potentials Quantum Mechanics

What Conditions? Select Ensemble Select Periodic Boundary Conditions

Simulation of Forces

• Interatomic Potentials (Force fields)– Parameterised equations describing forces - fast– Empirical Derivation– Non-empirical Derivation

• Quantum Mechanics– direct solution of the Schrodinger Equation – slow+reliable?– Semi-empirical– Density functional approach– Molecular Orbital approach

All the atomistic simulation techniques require that the total interaction energy is evaluated and are more efficient if theforces between every atom is evaluated.

Simulation Techniques:• Energy Minimisation

– Calculate Lowest Energy Structure

– Gives structural, mechanical and dielectric properties

• Molecular Dynamics– Calculates the effect of Temperature

– Gives dynamics e.g. diffusivity

• Monte Carlo– Calculates a range of structures

– Gives the thermally averaged properties

• Genetic Algorithms– Calculates a range of structures

– Efficient search for global minimum

Atomistic simulation - Dynamics Summary

• Molecular Dynamics can provide reliable models– Effect of Temperature– Time evolution of system

• Highly suited to liquids and molecular systems• Calculate dynamical properties, e.g. diffusivity

• PROVIDED– Reliable potential models

• Molecular Dynamics– Robust and reliable for solids and their surfaces

• BUT– Takes a long time to search configurational space– Does not easily allow atoms to pass over large energy

barriers– Can use constrained methods – but usually need to know

where the atom/molecule needs to go.

MonteCarlo +GeneticAlgorithms}

• In the widest sense of the term, Monte Carlo (MC) simulations mean any simulation (not even necessarily a computer simulation) which utilizes random numbers in the simulation algorithm.

• The term “Monte Carlo” comes from the famous casinos in Monte Carlo.

• Another closely related term is stochastic simulations, which means the same thing as Monte Carlo simulations.

Monte Carlo

• Metropolis MC– A simulation algorithm, central to which is the formula which

determines whether a process should happen or not. Originally used for simulating atom systems in an NVT thermodynamic ensemble, but nowadays generalized to many other problems.

• Simulated annealing– The Metropolis MC idea generalized to optimization, i.e. finding

minima or maxima in a system. This can be used in a very wide range of problems, many of which have nothing to do with materials.

• Thermodynamic MC– MC when used to determine thermodynamic properties, usually of

atomic systems.

Monte Carlo

• Kinetic MC, KMC– MC used to simulate activated processes, i.e.

processes which occur with an exponential probability– e−Ea/kT

• Quantum Monte Carlo, QMC– A sophisticated electronic structure calculation method.– Diffusional Monte Carlo (stochastic projector

technique, which solves the imaginary time-dependent Schroedinger equation). In theory DMC is exact!

Monte Carlo

– The approach is to calculate energy Ei then

– randomly move an atom or molecule to give a new energy, Ej

– Then decide whether to accept or reject move– Can easily extract Thermodynamic properties

within NVT -Canonical Ensemble

Metropolis Monte Carlo

kTEiAeQ

A /1

kTEieQ /

THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 6 JUNE, 1953

Selection

• Metropolis Monte Carlo– If the new energy is lower (i.e. a more stable

structure) then accept the move– If the new energy is higher (less stable) then

• generate a random number between 0 and 1

• calculate: Pij = exp(-(Ej -Ei)/kT)

– only accept the move if, Pij is higher than the random number.

– This enables the system to focus on the important configurations

qGlobal Minimum

Local Minima

U

Example of Use: TiO2

• Particularly powerful when used with energy minimisation

• Prediction of crystal structure without prior knowledge of atom positions– Freeman etal J.Materials Chem, 1993, 3, 531– used Monte Carlo to select a number of likely

structures– followed by energy minimisation of each

candidate to locate the precise atom positions

– Successfully found all the phases of TiO2

Lewis, et al, Nature , 382, 604.

Example of Use: Template Design

Predicted New Template for Levyne ZEBEDDE suggests 1,2-

dimethylcyclohexane as a template for LEV

Using 2-methylcyclohexylamine, a LEV structured CoAlPO (DAF-4) is formed

Barratt et al, Chem Commun,1996, 2001

Computer Designed Template

•Bi-cyclohexane motif•Amine derivative•4-piperidino piperidine

Co-AlPO4 Preparations•170oC, 4hours•Chabazitic structure•NO competing phase

Problems with Monte Carlo• The major problem is that computer

resources– A lot of configurations need to be sampled to

obtain reasonable statistics– A lot of configurations need to be sampled to

ensure that you have found the global minimum

– Hence need to keep rejection rate down– Has no ‘memory’ of good solutions

Problem:Structure of Clusters and Nuclei

• Clusters span a wide range of particle sizes – from molecular (well separated, quantized states) to micro-crystalline (quasi-continuous states). – How do properties change as they grow ?

• Clusters constitute new materials (nanoparticles) which may have properties that are distinct from those of discrete molecules or bulk matter.– New chemistry ?

Nucleation of Zinc SulphideS.H. Gomez, E. Spano, C.R.A. Catlow

• Generating Nuclei via molecular dynamics– Start with individual atoms are monitor how and they

assemble.

• ZnS– In the bulk both ions 4-fold coordinated– But get 3-fold coordinated clusters.

(ZnS)12(ZnS)25

Comparison of Stability

• Although still small – show continued stability of ‘bubble’ structures

• (ZnS)47 Bulk like cluster (300 kJ/mol less stable)

Bubble-like Bulk-like

CHEM COMMUN (7): 864-865 APR 7 2004 + J AM CHEM SOC 127 (8): 2580-2590 MAR 2 2005

• GA procedure is for optimising a function, structure or process which depends on a large number of variables.

• Developed by computer scientists in the 1970’s.

• Based on principals of natural evolution.

• Works through a combination of mating, mutation and “natural selection”.

Roy L. Johnston, University of BirminghamDALTON T (22): 4193-4207 2003

Alternative Approach: Genetic Algorithms for Cluster Geometry Optimisation

GA Definitions

• Chromosome – a string of variables (genes) corresponding to a trial solution.

• Allele – the value of a particular gene (i.e. variable).

B D A C A

chromosome

gene

allele

GA Approach

• Take a Population – the set of trial solutions.

• Measure of the quality of each member of the population - Fitness (usually by calculating the total interaction energy)

• Proceed with mating - the overall process of selecting strings (parents) and exchanging their genes to produce new strings (offspring).

Selection Process

• Roulette Wheel Selection: parents are chosen with a probability proportional to their fitness:

jj

ii

f

f P

Generating new structures• Crossover – the process of exchanging genes between

chromosomes.

• Some offspring will be fitter than their parents.

• Due to crossover the GA effectively explores the parameter space in parallel.

+

+

Single Point Crossover

parents

offspring

Possible Problem• It is possible to get stagnation – where certain

structures can appear to be ‘frozen-in’. • Overcome by introducing new genetic material

which ensures population diversity – preventing in-breeding and stagnation.

• Mutation – randomly changing certain genes in selected members of the population.

Single Point Mutation

Some Other Applications of GAs• Protein folding

G.A. Cox, T. V. Mortimer-Jones, R. P. Taylor and R. L. Johnston, Theor. Chem. Acc. 112, 163-178 (2004).

• Crystal structure solution K.D.M. Harris, R.L. Johnston and B.M. Kariuki, Acta Cryst. A 54, 632-645 (1998).

• Spectral deconvolution • Conformational analysis

I

2

GA

A variety of GAs have now been written for cluster geometry optimization.

The Birmingham Cluster GA Roy L. Johnston

• Apply “cut and paste” crossover operator

• One new cluster generated from each mating operation.

• Perform energy minimisation using BFGS algorithm

 • Mutation achieved by

randomly moving a fraction ( N/3) of atoms. Mutation probability: 

Pmute = 0.1

• The mutation operator acts on the offspring.

 

Ionic MgO ClustersRigid Ion Model

• First term – long-range electrostatic Coulomb energy.

• Second term – short-range repulsive Born-Mayer potential, which reflects the short range repulsive energy due to overlap of the ions.

PHYS CHEM CHEM PHYS 3 (22): 5024-5034 2001

ijijijij

jiij /rexpB

r

qqV

Bij (Mg-O) 821.6 eV

ij (Mg-O) 0.3242 Å

Bij (O-O) 22764 eV

ij (O-O) 0.1490 Å

Formal charges q = 2

Formal charges q = 1

(MgO)9

(MgO)8

Variation of Structure with Magnitude of Formal Ion Charge q

(MgO)12

Conclusions – GA• The GA is an efficient technique for

searching for global minima –a variety of potentials (LJ, Morse, Ionic, MM, Gupta, TB, EAM …) have been studied.

• As with Monte Carlo the chief problem is the time taken to investigate the different possible structures

• When particles become much bigger, e.g. beyond 10nm, most efficient is Molecular Dynamics– Care needed in generating structures

Electrostatic Forces (Multipolar Forces)

• Most molecules have an uneven distribution of charge, e.g.

+ -

K+ Cl-

ions

H-F

+ -

dipolar quadrupole

- -++

O = C = O++++

- -

- -

Cl

Cl

Cl Cl

C

octopole

This leads to electrostatic (Coulomb) forces between the molecules. If we approximate the charge distribution as a collection of discrete

charges qi,

i j ij

jielec r

qqu 12

where qi are charges in molecule 1 and qj are those in molecule 2

Potential Models (Force Fields)

•Potential models rely on Born-Oppenheimer, ignore electronic motions and calculate the energy of a system as a function of nuclear positions only

• Potential models rely on: – Relatively “simple” expressions that capture the essentials of the interatomic and intermolecular interactions. Such as stretching of bonds, the opening and closing of angles, rotations about bonds, etc. – Transferability: the ability to apply a given form for a potential model to many materials by tweaking parameters (e.g. MgO vs CeO2)

taken from Dr. S. C. Glotzer’s lectures on Computational Nanoscience of Soft Materials, University of Michiganhttp://www.engin.umich.edu/dept/cheme/people/glotzertch.html

Composite Pair Potentials for Small Molecules

• For small molecules (e.g. Ar, N2, CO2) many neglect molecular flexibility and treat the molecule as rigid.

• Commonly used models include:

- Lennard-Jones (12,6)

612

4rσ

rσ

εru

e.g. CO2

a b c de

f

LJ + Coulomb

r

qqruru LJ

taken from Prof. K. Gubbins lectures on Computer simulation , NC State Univ http://chumba.che.ncsu.edu/

Flexible molecules

• Total pair energy breaks into a sum of terms

( )Nstr bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr - stretch

• Ubend - bend

• Utors - torsion

• Ucross - cross

• UvdW - van der Waals

• Uel - electrostatic

• Upol - polarization

See Leach 2nd ed., ch. 4; also, Gubbins and Quirke, pp. 25-27, 28-33

Mixed terms

A Typical Force Field

taken from Dr. S. C. Glotzer’s lectures on Computational Nanoscience of Soft Materials, University of Michiganhttp://www.engin.umich.edu/dept/cheme/people/glotzertch.html

A (More Complicated) Force Field

Analytic expression for the CFF 95 force field

Some Commonly Used Models

• There are many different Potentials in the literature, particularly for organics. In most cases, they are developed to treat a particular class of systems.

• Some commonly used FFs are: (in blue: original systems studied; in red, some useful references and/or websites) - MM2, MM3 and MM4 (N. L. Allinger et al.) → small organic molecules → http://europa.chem.uga.edu/index.html - MMFF (Merck Molecular Force Field, proposed by T. A. Halgren) → biomolecules → T.A. Halgren, J. Comput. Chem. 17, 490 (1996) - AMBER (Assisted Model Building with Energy Refinement, by P. A. Kollman et al.) → biomolecules → http://www.amber.ucsf.edu/amber/amber.html- CVFF (A. Hagler -> Biosym -> MSI -> Accelrys) -> COMPASS → biomolecules -> more general → Dauber-Osguthorpe & Hagler

Some Commonly Used Models

- OPLS (Optimized Potentials for Liquid Simulation, W. L. Jorgensen et al) → organic liquids → W. Damm, A. Frontera, J. Tirado-Rives, W.L. Jorgensen, J. Comput. Chem. 18, 1955 (1997); http://zarbi.chem.yale.edu/ - CHARMM (Chemistry at HARvard Macromolecular Mechanics, by M. Karplus and coworkers) → biomolecules → http://www.charmm.org/ - ECEPP (Empirical Conformational Energy Program for Peptides, by H. A. Scheraga et al.) → biomolecules → http://www.tc.cornell.edu/Research/Biomed/CompBiologyTools/eceppak/ http://www.chem.cornell.edu/has5/ - GROMOS (GROningen MOlecular Simulation, by W. F. van Gunsteren and coworkers) → biomolecules → http://www.igc.ethz.ch/gromos/

Other Models

• There are also potential models, such as • MOMEC (P. Comba and T. W. Hambley) and• SHAPES (V. S. Allured et al) that were developed for

transition metal complexes

• There are also models developed with the purpose of treating the full periodic table, such as

• UFF (Universal Force Field, by A. K. Rappe et al.),• RFF (Reaction Force Field, by A. K. Rappe et al.) and• DREIDING (by S. L. Mayo et al.)

Problems: Unlike-Atom Interactions(non-bonding)

• “Mixing rules” give the potential parameters for interactions of atoms that are not the same type

– no ambiguity for Coulomb interaction

– for effective potentials (e.g., LJ) it is not clear what to do

• Lorentz-Berthelot is a widely used choice

0( )

4i jq q

U rr

112 1 22

12 1 2

( )

Problems: Unlike-Atom Interactions(bonding)

• Conservation of equilibrium bond distance and energy. On altering for example, charge, adjust short range parameters to maintain distance and energy.

• Issue for simple force fields

– Bond energy: U = 0.5 k (r AB – r 0 AB)2 If new bond is approx the equilibrium bind

length then the energy of reaction about 0 energy.

• Treatment is a very weak link in quantitative applications of molecular simulation

A

B

+ A

B

A

B

A

B

More Potentials for Solids

Even for polar/ionic solids there are a vast array of models, (e.g. see refs by Bush, Catlow, de Leeuw, Dove, Gale, Lewis, Jackson, Parker and Woodley) that are based on the shell model and for models based on three body potentials see refs by (S. Garofalini et al.)There other models for metals (Finnis and Sinclair) [Phil.Mag. A 50 (1984) 45; for an improvement see Phil. Mag. A 56 (1987) 15].

Semiconductors [Tersoff, Phys. Rev. Lett. 56 (1986) 632] extended by Brenner [D. W. Brenner, Phys. Rev. B 42 (1990) 9458] for conjugated systems, see further extensions [Stuart et al., J. Chem. Phys. 112 (2000) 6472]and [Che et al., Theor. Chem. Acc. 102 (1999) 346].

where

Shell Model Potential

• Electrostatic– despite simple expression (q1q2/r12) it has poor convergence - use

methods by Ewald, Parry and Madelung etc.

• Short-range

– includes repulsion + dispersion A12exp(-r12/p12) - C12/r126

– where A, p and C are needed for each pair of atoms

• Electronic polarisability– Via Shell model– specify shell charge and spring constant

• Angle dependent forces– For polyanions

For example: polar solids

Ewald Method• Approach for calculating the

Coulombic interaction energy– Replace point charges (charge

density – delta functions) by Gaussians.

– Gives 1. difference between Gaussians

and delta functions2. Interacting Gaussians 3. remove interaction of Gaussian

with self

q q

q q

q

Shell Model – many body forces• Valence electrons Massless shell

• distorted by electric field, size of distortion dependent on strength of spring, i.e. variable polarisability

• For quadrupolar distortions see work by P.A. Madden etal

• Shell charge remains symmetric

Y (shell charge)

k (spring constant)

U = 0.5 k 2

Free ion polarisability = Y2/k

Partially covalent solids

For example: work by S.H. Garofalini

Vij(2 ) ij exp

rij

ij

ziz je2

rij

erfcrij

ij

Vij

CSF

Va

b r cijCSF ij

x

ijx

ij ijx

x

11

6

exp( ( ))

V V Vtot ij jik ( ) ( )2 3

Two-Body Term Three-Body Term

r rij ij r rik ik

jik jik cos 13

2

jik cos jik 13 cos jik sin jik 2

Tetrahedral

B. P. Feuston and S. H. Garofalini, J. Chem. Phys., 89 (1988) 5818 (note error in Table I, where beta headings are mixed) R. G. Newell, B. P. Feuston, and S. H. Garofalini, J. Materials Research, 4 (1989) 434.S. Blonski and S. H. Garofalini, Surf. Sci. 295 (1993) 263.

• The main problem in fitting a general model is to ensure its transferability while using a reasonable number of parameters; in order to be useful the model has to be able to predict correctly properties for compounds that fall outside the set used to fit the parameters • How different models are linked together is still an area of debate – are the results meaningful?

• When using a potential model, it is important to know what is being included and how, and what isn’t.

Issues when using Potential Models

• Leach, AR Molecular Modelling: Principles and Applications; 2nd Edn

(2001) Pearson Prentice Hall

Derivation of parameters

• Empirical fitting – to crystal structure, elastic and dielectric constants– problems with

• validation (must not use all exptal data) e.g. ir and raman• interatomic separations far from those used in fitting e.g. at high

temperatures and pressures

– overcome with….

• Non-empirical fitting– to electronic structure calculations– problems with

• incomplete description of forces e.g. dispersion• open shell atoms (e.g. transition metals)

Exercise, Friday 23rd, unmarked• Download GULP from module website• Rename example17 to input.txt• Copy Suttonchen.lib• Run gulp.exe• Compare the lattice parameters and elastic constants

with experimental values• Auxetic ? (check in Baughman’s paper)• Modify input to simulate a fake, Centred Cubic phase

of Ni. What happens ? Can you compare stabilities ?• Try with other metals from Suttonchen.lib (especially

auxetic question)

Reference Books• M. P. Allen, D. Tildesley: Computer simulation of Liquids

(Oxford University Press, Oxford,1989)– the classical simulation textbook– statistical mechanics approach

• D. Frenkel, B. Smit: Understanding Molecular Simulation: From Algorithms to Applications, 2nd edition (Academic Press, 2001)– book home page (http://molsim.chem.uva.nl/frenkel_smit/) has

exercises

• R. Phillips: Crystals, defects and microstructure : modeling across scales (Cambridge University Press, 2001)– textbook on computational methods in materials research in general;

from atomistic to elastic continuum– includes chapter on interaction models