Computational Modelling of Materials

1
Computational Modelling of Materials
Recent Advances in Contemporary Atomistic
Simulation
Or Understanding the physical and chemical
properties of materials from an understanding of
the underlying atomic processes
http//secamlocal.ex.ac.uk/people/staff/ashm201/
Atomistic/ Lectures Introduction to computational
modelling and statistics 1 Potential
models 2 Density Functional (quantum) 1
3 Density Functional 2 4
2
Introduction
The increased power of computers have allowed a
rapid advance in the use of simulation techniques
for modelling the properties of materials.
Why do it?

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

But how?
The answer depends on the length and time-scale
3
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

4
Simulation of Forces
All the atomistic simulation techniques require
that the total interaction energy is evaluated
and are more efficient if the forces between
every atom is evaluated.

• Interatomic Potentials (Force fields)
• Parameterised equations describing forces -
  fast
• Empirical Derivation
• Non-empirical Derivation
• Quantum Mechanics
• direct solution of the Schrodinger Equation
  slowreliable?
• Semi-empirical
• Density functional approach
• Molecular Orbital approach

5
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

6
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.

Monte Carlo Genetic Algorithms
7
Monte Carlo

• 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.

8
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.

9
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!

10
Metropolis Monte Carlo
THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21,
NUMBER 6 JUNE, 1953

• 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

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Selection
U
Local Minima

• 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

Global Minimum
q
12
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

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Example of Use Template Design
Lewis, et al, Nature , 382, 604.
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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

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Computer Designed Template

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

Co-AlPO4 Preparations

• 170oC, 4hours
• Chabazitic structure
• NO competing phase

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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

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ProblemStructure 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 ?

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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)25
(ZnS)12
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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
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Alternative Approach Genetic Algorithms for
Cluster Geometry Optimisation

• GA procedure is for optimising a function,
  structure or process which depends on a large
  number of variables.
• Developed by computer scientists in the 1970s.
• Based on principals of natural evolution.
• Works through a combination of mating, mutation
  and natural selection.

Roy L. Johnston, University of Birmingham DALTON
T (22) 4193-4207 2003
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GA Definitions

• Chromosome a string of variables (genes)
  corresponding to a trial solution.
• Allele the value of a particular gene (i.e.
  variable).

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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).

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Selection Process

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

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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.

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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.

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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

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

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Ionic MgO Clusters

• Rigid 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.

Bij (Mg-O) 821.6 eV
?ij (Mg-O) 0.3242 Å
Bij (O-O) 22764 eV
?ij (O-O) 0.1490 Å
PHYS CHEM CHEM PHYS 3 (22) 5024-5034 2001
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Formal charges q ? 2
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Formal charges q ? 1
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Variation of Structure with Magnitude of Formal
Ion Charge q
(MgO)8
(MgO)9
(MgO)12
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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

33
Electrostatic Forces (Multipolar Forces)

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

ions
dipolar
quadrupole
octopole
This leads to electrostatic (Coulomb) forces
between the molecules. If we approximate the
charge distribution as a collection of discrete
charges qi,
where qi are charges in molecule 1 and qj are
those in molecule 2
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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. Glotzers lectures on
Computational Nanoscience of Soft Materials,
University of Michigan http//www.engin.umich.edu/
dept/cheme/people/glotzertch.html
35
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)
e.g. CO2
LJ Coulomb
taken from Prof. K. Gubbins lectures on Computer
simulation , NC State Univ http//chumba.che.ncsu.
edu/
36
Flexible molecules

• Total pair energy breaks into a sum of terms

Intramolecular only

• UvdW - van der Waals
• Uel - electrostatic
• Upol - polarization

• Ustr - stretch
• Ubend - bend
• Utors - torsion
• Ucross - cross

Mixed terms
See Leach 2nd ed., ch. 4 also, Gubbins and
Quirke, pp. 25-27, 28-33
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A Typical Force Field
taken from Dr. S. C. Glotzers lectures on
Computational Nanoscience of Soft Materials,
University of Michigan http//www.engin.umich.edu/
dept/cheme/people/glotzertch.html
38
A (More Complicated) Force Field
Analytic expression for the CFF 95 force field
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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 -gt Biosym -gt MSI -gt Accelrys)
  -gt COMPASS
• ? biomolecules -gt more general
• ? Dauber-Osguthorpe Hagler

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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/Rese
arch/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/
41
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.)

42
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

43
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

44
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) 6472and Che et al.,
  Theor. Chem. Acc. 102 (1999) 346.

where
45
Shell Model Potential
For example polar solids

• 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

46
Ewald Method

• Approach for calculating the Coulombic
  interaction energy
• Replace point charges (charge density delta
  functions) by Gaussians.
• Gives
• difference between Gaussians and delta functions
• Interacting Gaussians
• remove interaction of Gaussian with self

q
q
q
q
q
47
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

U 0.5 k d2
Y (shell charge)
Free ion polarisability a Y2/k
k (spring constant)
48
Partially covalent solids
For example work by S.H. Garofalini
Two-Body Term
Three-Body Term
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.
49
Issues when using Potential Models

• 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
  isnt.

• Leach, AR Molecular Modelling Principles and
  Applications 2nd Edn (2001) Pearson Prentice Hall

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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)

51
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 Baughmans 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)

52
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