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Stochastic Parameter Optimization for Empirical Molecular Potentials
function optimization
simulated annealing
tight binding parameters
Motivation
simulate dynamics of atomic structuresderive total energy and forces acting on atoms
empirical potentials + fit parameters to experimentsoft spheres: only distance dependent
quantum mechanics: electrons dominate bondingmillions of atoms: approximate electronic degree of freedom
semi-empirical: capture QM origin of bonding
tight binding: provides directional bonding
fit simulated properties to experimental onesmore approximations: more parameters to adjust
BOP4 potential : 11 parameters [material/compound]
automatic fit procedure providing one or more good parameter sets
Optimization
find optimal solution to given problem such as:
economy: shortest itinerary between number of cities (traveling salesman)
engineering:
drug design/ circuit design
quantify the problem‘goodness’ of solution depends on parameters objective function
set of parameters state in vector space
goal: find best local minimum on Potential Energy Surface (PES)
cost function :recover exp. properties, some better than others
find point in 11-D continous space
Deterministic Methods (downhill only)
1D Golden Section Search
higher dimensions:Steepest DescentConjugate GradientVariable Metricdownhill simplex (no derivative)
Monte Carlo
statistical physics: access ensemble averagesmagnetization of Ising model
higher energy states less probable
trick: don’t weigh all possible states , but only representative subset
simple sampling: waste time on states, that don’t contribute
importance sampling: arithmetic mean
?how to judge importance without
prior knowledge of energy reference?
Metropolis Algorithm
judge upon relative energy-difference to previous state guarantee detailed balance of hopping between states
Metropolis-function: transition probability
Metropolis et al. (1953) : find optimal wiring (min. length) on chip
allow for uphill climbing: move to neighboring local minima
Simulated Annealing
in analogy to anneal process of metals:slower cooling: better crystalization (energetically lower state)
faster cooling: freezing small crystals (higher, local minimum)
Kirkpatrick et al. (1983) added T-schedule to Metropolis search
search parameter space at successively lower temperature (higher ) :T controls:
scale on which parameters are randomly changed:
prob. at which costly uphill moves are accepted:
find global minimum on PES for
logarithmic annealing (single crystal)
in practice: simulated quenching
with exponential cooling scheme
propose new state
accept reject
update TopList
lower Tin intervals
Traveling Salesman
visit all cities: combinatorial problemminimize salesman’s way
different cost for crossing the river:
minimize salesman’s cost
equal weight:
smuggler:
river penalty:
Variations of the Theme: Statistic Tunneling (ST)
simulated quenching is prone to freezing
process is trapped in a deep local (but not global) minimum, that is surrounded by higher intermediate states
-or-
very good (perhaps global) minimum is surrounded by higher states (on mountain top) and might never be found
transform PES:
‘tunnel’ through forbidden, higher regions
preserve/amplify lower lying regions
effectively raising T in higher regions
Tight Binding (TB) Parameters
molecular wavefunction is linear
combination of atomic wf.
replace hopping integral with parameter
angular dependence was given by Slater and Koster (1954) and is fitted to band structures of periodic systems
dynamic modeling needs continuous distance dependence
heuristic shape guided by radial solutions such as:
choice of dist. dep. is the integral part of TB
total energy:
Radial Dependence
repulsive potential and bond integral scale with same functional form
separate scaling parameter for -bonds and repulsive potential following common cut-off parameter #of parameters for s-p-bonded system:3x2(scaling)+1(cutoff)+3(screening)+1(promotion)=11strong repulsion at andstrong attraction at equilibrium at
Fitting BOP4
cost-function: equilibrium values ofbulk modulus
rem. elastic constants
lattice parameter
cohesive energy
lattice parameter for graphitic and -tin phase
for diamond phase
T-dependent criterion: „distance in vector space“
distinguish btw truly different sets and slight variation from same local minimum
Summary
Simulated annealing invaluable to handle our multi-variable optimization
drawback: may run to forbidden areas in parameters space many times, since only TopList and two current states are stored (blind search)
genetic algorithm: interchange subset of parameters btw good parameterization, once annealing process is finished/frozen
general strategy: locate various minima with SA at high T
refine once with SA at lower T
use variable metric method to find „bottom“ of local minima