Download Lecture 8 - Institute of Materials Science

Assorted Topics
[Partly based on Chapters 9 & 10, Sholl & Steckel]
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Molecular Dynamics
Monte Carlo – Revisited
Accuracy of Standard DFT
Some beyond-DFT methods
Multiscale Modeling
Final Thoughts
Molecular Dynamics
A bird’s eye view
In the absence of external forces and damping …
Time, t = 0
Time, t = dt
dx = v1dt
v1
v2
v2
v2dt
v1
In the presence of external forces …
v1
Time, t = 0
Time, t = dt
F1
dx = v1dt+0.5(F1/m)dt2
v1+(F1/m)dt
Molecular Dynamics (MD)
• External forces could be because of external
electric/magnetic fields, but mainly due to interatomic
interactions; in the latter case
– Fi = dE(R1,R2,…RN)/dRi
– E(R1,R2,…RN) may be obtained from DFT (ab initio MD) or
empirical potentials (classical MD)
• Average kinetic energy  defines temperature
– 1.5NkT = Si 0.5mivi2
• Time step has to be of the order of 10-15 seconds (fs) so
that trajectories during vibrations can be captured
Ensembles
• Microcanonical Ensemble
– E, N, V constant during the simulation (closed, isolated system)
– Temperature fluctuates with time
• Canonical Ensemble
– N, V, T constant during the simulation (system in contact with a
heat bath that maintains a constant temperature)
– At the end of every few time steps, velocity is rescaled so that
the temperature is maintained
• Molecular Statics
– If the velocity is rescaled to zero at the end of every time stem,
we reach the zero temperature ground state (this is the same as
geometry optimization)
Ab Initio MD
• Electrons are treated quantum mechanically, but
nuclear motion is treated classically (Newtonion
mechanics)
• At the present time, these simulations can
extend only up to nanoseconds (if that)
– If the time step is fs, this implies 106 iterations
• Classical MD can be extended to microsecondsmilliseconds as energy and force evaluations
are cheaper (but also less accurate)
Hyperdynamics
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Standard MD is impractical to solve long time (> microseconds) diffusion
problems, as as it may involve larger than 1010 time steps!
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A way to accelerate the dynamics is by making the potential wells at atomic
sites “shallow” (proposed by Art Voter); this artificially slows down the
atomic vibrations (hence timesteps can be large) by several orders of
magnitude while preserving the basics physics of diffusion
Standard Implementations
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Euler method
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Verlet method
– xi(t + dt) = xi(t) + vi(t) dt
– vi(t + dt) = vi(t) + (Fi/mi) dt
– Correct up to first order, and hence requires very small dt
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Taylor expand: xi(t + dt) = xi(t) + vi(t) dt + (1/2)(Fi/mi) dt2 + (1/6)(d3xi(t)/dt3) dt3 + O(dt4)
Note: xi(t - dt) = xi(t) - vi(t) dt + (1/2)(Fi/mi) dt2 - (1/6)(d3xi(t)/dt3) dt3 + O(dt4)
Add: xi(t + dt) = 2xi(t) - xi(t - dt) + (Fi/mi) dt2
Correct up to 3rd order!
To determine the position at the next time step, we need: current position, previous
position and current force (what do we do for the 2nd step of the simulation?)
Velocity still given by Euler formula
No dependence of position evolution on velocity  good for constant energy MD
simulations
If we want to perform constant T simulations, then we need to rescale velocities, and
hence the position and velocity evolution has to be coupled
How do we do this?
MD Example I: Liquids and Amorphous
Systems: InP
Lewis, De Vita & Car, PRB 57 1594 (1998)
• The melt & quench approach
Each plateau: “equilibration”
Radial Distribution Function: InP
• g(r) = r(r)/(4pr2ravg)
• P-P bonds found in the amorphous phase (not found in
crystalline phase)
Example II: Equilibrium geometries of
ultrasmall nanocrystals
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13-atom Pt cluster (particularly stable “magic size” cluster, and occurs in catalysts)
High T MD simulations, followed by “quenching” (i.e., standard geometry optimization)
of specific low-energy configurations
Several disordered clusters which we couldn’t have guessed based on symmetry, are
as stable as symmetric geometries; MD is thus useful in the identification low-energy
structures
Monte Carlo (MC) Revisited
The case of diffusivity
• MC methods are stochastic, and involves ensemble averages of
properties (in contrast to MD simulations which involve time
averages of properties once equilibrium is reached)
• Let us consider diffusion (in 1-d): For the case of vacancies, we will
attempt to relate the macroscopic diffusivity and measured activation
barriers for long range diffusion (Q) to atomic-level site-to-site
hopping barriers (E) of vacancies
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D = D0exp[-Q/kT]
<x2> = Dt (i.e., D is related to an ensemble average)
Probability of hopping, p = exp[-E/kT]
D, Q, < x2> are macroscopic quantities, while p and E are atomic-level
properties
– E may be calculated using DFT for a variety of situations, followed by
MC simulations of large scale diffusion vs. microstructure
MC Ensemble Averaging
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1-d lattice with a vacancy
We consider an ensemble of 8 lattices
At high T, p = 1
It is clear that <x2> is linearly related to t  at T = , D = 1
This simple scheme can be used to determine the macroscopic
diffusivity if the microscopic information is available
t=0
t=1
t=2
<x2> = 0
<x2> = 1
<x2> = 2
t =3
<x2> = 3
Density Functional Theory
Formidable problem  Manageable problem
True! But what to expect?
 (r1 , r2 ,..., rN , R1 , R2 ,..., RM )  E(r1 , r2 ,..., rN , R1 , R2 ,..., RM )
Density Functional Theory (DFT)
[W. Kohn, Chemistry Nobel Prize, 1999]
1-electron wave function
(function of 3 variables!)
 2 2

  Veff (r ) i (r )   i i (r )

2
m


occ
The “average” potential
seen by electron i
r(r)    i (r)
i
2
1-electron energy
(band structure energy)
Energy can be obtained
from r(r), or from i and i (i labels electrons)

DFT provides a practical prescription for computing the total ground state energy
When combined with statistical thermodynamics, DFT provides free energies too
Predictions of geometry
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Structural details
predicted typically to
within 1% of
experiments for a wide
variety of solids and
molecules
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Results from various
sources [collected in
Ramprasad, Shi and
Tang, in Physics and
Chemistry of
Nanodielectrics,
Dielectric Polymer
Nanocomposites
(Springer)]
Predictions of other basic properties
Band offsets
The Exchange-Correlation Functional
Perdew’s Ladder & the problem with choices!
• Unfortunately, a step up the ladder does not guarantee
greater accuracy!
GGA-OLYP
GGA+GW0GGA-OPBE
LDA+GW
B3LYP
GGA+GW
B1LYP
QMCKMLYP
HF HF-CI
O3LYP MPn
X3LYP
CCSD
LDA+X
Beyond-DFT
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(Not so) quick fixes
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Self-interaction correction (VH ≠ 0 for H2+ in DFT)
DFT + X (X: exact exchange; or is it?)
DFT + U (inspired by the previous 2 methods)
Beyond the 1-electron approximation
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GW (G: Green function; W: screened Coulomb interaction)
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Quantum Monte Carlo (QMC)
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Direct solution of the many-electron Schrodinger equation
Multi-dimensional integrals evaluated using MC methods
HF + Configuration Interaction (CI) & related strategies
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Replacing W by the bare Coulomb interaction results in the HF exchange potential
Largely used by chemists and has been applied only to molecules so far
The “gold standard”
Order-N methods
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Scaling:
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Standard DFT: N2-N3
HF: N4
GW, QMC: N4-N5
CI: N6
Linear scaling (or Order-N) methods take advantage of localization
Physics Today, April 2008
Modeling vs. Scales
Time
Years
Engineering
Design
Hours
Finite element
Analysis
(Continuum/classical)
Minutes
Seconds
Mesoscale
Modeling
(Semi-classical)
Microseconds
Molecular
Mechanics
Nanoseconds
Picoseconds
Quantum
Mechanics
Femtoseconds
Å
nm
mm
mm
m
Distance
e.g., density functional theory (DFT)
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If more than one “box” is involved in a computation  multi-scale modeling
Multi-scale modeling:
Two perspectives
• Implicit multi-scale approach
– Perform computations at a higher level, with parameters and
constants occurring in the equations determined at a more
accurate “lower” level of theory (e.g., solving diffusion equations
with diffusion coefficients determined using atomic level
computations)
– Not new!
• Explicit multi-scale approach
– Simultaneously perform different levels of computation for the
same system (e.g., atomic level computation in a small important
region plus finite element type computation further away)
– Very recent, and very difficult, approach
Implicit Multi-scale modeling:
Historical examples
• At any level of theory, the parameters found in equations
conceptualize events occurring at smaller scales not
directly treated by the theory
– Elasticity theory: sij = Cijklkl (elasticity theory assumes the
existence of the elastic constants Cijkl which contain the sum
total of relevant atomic level information)
– Electromagnetism: although classical electromagnetic theory
does not explicitly consider electrons, permeability (m),
permittivity () and conductivity (s) are input material properties,
whose existence is determined by electronic structure
– Kinetic theory of gases: PV = nRT
– Phase transformations, statistical thermodynamics
– Diffusion coefficients, kinetic rate constants (kMC)
– Etc., etc.
Explicit Multiscale Approach
Example: Crack propagation in silicon
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“Concurrent coupling of length scales: Methodology and applications”, Broughton et
al, Phys. Rev. B 60, 2391 (1999)
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Silicon was chosen in this study due to
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The “gold standard” in the area of multi-scale computation
Its industrial relevance
The availability of robust & well-tested empirical potentials (for molecular dynamics
simulations)
The accuracy of the tight-binding method for Si (which is computationally more efficient than
fully first principles methods)
Same approach can be used for other materials (metals, organics, ceramics, etc.)
Combined tight-binding, molecular dynamics & finite element method
Uses parallel processing (especially for the tight-binding part)
The multi-scale framework
• Tight-binding at the
crack tip
• Semi-empirical
potentials (StillingerWebber) + molecular
dynamics in the
neighborhood of the
crack tip (to model
plastic behavior)
• Finite element
method elsewhere
(elasticity theory)
“Hand-shaking”
Handling the boundary regions
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Within each region, a Si atom behaves like
a Si atom
But, at the boundary, we need to have a
silicon-hydrogen hybrid (silogen)
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Within the MD region, we have a collection of Si
atoms
At the boundary between the MD and the finite
element (FE) regions, the FE mesh is equivalent to
the the Si unit cell, but away from the boundary, the
mesh sizes can be large
Results
• Can reliably predict the stress fields in materials
• Provides a framework for the modeling of the physical or
mechanical breakdown of materials
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All length/time scales treated simultaneously (example of explicit multi-scale
modeling)
This may not be the most desired approach
– Perhaps can break the problem into pieces that can be handled separately
Final thoughts …
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Several types of computational methods at various length & time scales
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Challenges
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Geometric complexity; Chemical complexity; Disorder; Multiple phenomena; Models of reality
DFT
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Need to use judgment in determining the right type of method or model to study a given problem
No computational method should be used as a “black box”
Typical scaling: 1980s: few-several atoms; 1990s: few-several 10s of atoms; 2000s: few-several 100s of
atoms; future: ???
Future directions: Materials Design using existing methods; ab initio thermodynamics; beyond-DFT methods
Regarding DFT predictions …
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DFT is in general complementary to experiments
Can provide insights, trends and guidance
Sometimes, computation may be the only option
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… Hypothesis non fingo …
[Isaac Newton]
Science is primarily an experimental investigation, and so experiments are the ultimate authority
in determining the validity of computational theory!
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“The underlying physical laws necessary for the mathematical theory of a large part of physics and
the whole of chemistry are thus completely known, and the difficulty is only that the exact
application of these laws lead to equations much too complicated to be soluble.”
[Paul Dirac]