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
Potential models
Density Functional (quantum) 1
Density Functional 2
1
2
3
4
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
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
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 – slow+reliable?
Semi-empirical
Density functional approach
Molecular Orbital approach
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.
Monte
Carlo
+
Genetic
Algorithms
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.
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!
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
1
A   Ae Ei / kT
Q
Q   e Ei / kT
Selection
U
• Metropolis Monte Carlo
Local Minima
Global Minimum
– 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
q
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
Example of Use: Template Design
Lewis, et al, Nature , 382, 604.
Predicted New Template for Levyne
 ZEBEDDE suggests 1,2dimethylcyclohexane as a
template for LEV
 Using 2methylcyclohexylamine, 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 Sulphide
S.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
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 1970’s.
• 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
GA Definitions
• Chromosome – a string of variables (genes)
corresponding to a trial solution.
• Allele – the value of a particular gene (i.e.
variable).
gene
allele
B
D
A
C
A
chromosome
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:
Pi 
fi
 fj
j
Generating new structures
• Crossover – the process of exchanging genes
between chromosomes.
+
parents
+
offspring
Single Point
Crossover
• Some offspring will be fitter than their parents.
• Due to crossover the GA effectively explores the
parameter space in parallel.
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 inbreeding 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).
I
GA
2
• Spectral deconvolution
• Conformational analysis
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 Clusters
Rigid Ion Model
Vij 
qi q j
rij

 Bij exp  rij /  ij

• 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)
ij (Mg-O)
Bij (O-O)
ij (O-O)
821.6 eV
0.3242 Å
22764 eV
0.1490 Å
PHYS CHEM CHEM PHYS 3 (22): 5024-5034 2001
Formal charges q =  2
Formal charges q =  1
Variation of Structure with Magnitude of
Formal Ion Charge q
(MgO)8
(MgO)9
(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.
Cl
K+
Cl-
+
-
H-F
+
O=C=O
+
+
-
-
quadrupole
dipolar
Cl
-
Cl
++
++ C
-
ions
-
Cl
octopole
This leads to electrostatic (Coulomb) forces between the molecules.
If we approximate the charge distribution as a collection of discrete
charges qi,
uelec 12   
i
j
qi q j
rij
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 Michigan
http://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)
 σ 12  σ 6 
u r   4ε      
r 
 r 
e.g. CO2
LJ + Coulomb
a b c
d
e
f
u r    u r   
LJ


taken from Prof. K. Gubbins lectures on Computer simulation , NC State Univ
http://chumba.che.ncsu.edu/


q q
r
Flexible molecules
• Total pair energy breaks into a sum of terms
U (r N )  U str  U bend  U tors  U cross  U vdW  U el  U pol
Intramolecular only
• Ustr - stretch
• UvdW - van der Waals
• Ubend - bend
• Uel - electrostatic
• Utors - torsion
• Upol - polarization
• Ucross - cross Mixed terms
See Leach 2nd ed., ch. 4; also, Gubbins and Quirke, pp. 25-27, 28-33
A Typical Force Field
taken from Dr. S. C. Glotzer’s lectures on Computational Nanoscience of Soft Materials, University of Michigan
http://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
qi q j
– no ambiguity for Coulomb interaction U (r )  4 r
0
– for effective potentials (e.g., LJ) it is not clear
what to do
• Lorentz-Berthelot is a widely used choice
 12  12 ( 1   2 )
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
B
A
B
B
B
+
A
A
A
– 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
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].
where
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].
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
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 functions
2. Interacting Gaussians
3. remove interaction of Gaussian
with self
qq
qq
q
Shell Model – many body forces
• Valence electrons
Massless shell
U = 0.5 k d2
Y (shell charge)
Free ion polarisability
 = Y2/k
k (spring constant)
• 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
Partially covalent solids
For example: work by S.H. Garofalini
Vtot  Vij( 2)  Vjik( 3)
Two-Body Term
Three-Body Term
r  z z e 2
r 
(2 )
ij
i
j

 ij  VijCSF
Vij  ij exp
erfc
  
 
rij
 ij 
 ij 
6
CSF
ij
V

aijx
x
x
1

exp(
b
(
r

c
x 1
ij ij
ij ))
rik  rik

rij  rij
 jik  cos jik  1 3



2
Tetrahedral
 ji k  cos ji k  13 cos ji k sin  ji k

2
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.
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 isn’t.
• 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