Stochastic Parameter Optimization for Empirical Molecular Potentials

1. Stochastic Parameter Optimization for Empirical Molecular Potentials • function optimization • simulated annealing • tight binding parameters

2. Motivation • simulate dynamics of atomic structures • derive total energy and forces acting on atoms • empirical potentials + fit parameters to experiment soft spheres: only distance dependent • quantum mechanics: electrons dominate bonding • millions of atoms: approximate electronic degree of freedom • semi-empirical: capture QM origin of bonding tight binding: provides directional bonding • fit simulated properties to experimental ones • more approximations: more parameters to adjust • BOP4 potential : 11 parameters [material/compound] automatic fit procedure providing one or more good parameter sets

3. 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

4. Deterministic Methods (downhill only) • 1D Golden Section Search • higher dimensions: • Steepest Descent • Conjugate Gradient • Variable Metric • downhill simplex (no derivative)

5. Monte Carlo • statistical physics: access ensemble averages • magnetization 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?

6. 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

7. Simulated Annealing propose new state accept reject update TopList lower T in intervals • 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

8. Traveling Salesman • visit all cities: combinatorial problem • minimize salesman’s way • different cost for crossing the river: minimize salesman’s cost equal weight: smuggler: river penalty:

9. 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

10. 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:

11. 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)=11 • strong repulsion at and strong attraction at equilibrium at

12. Fitting BOP4 • cost-function: equilibrium values of • bulk modulus • rem. elastic constants • lattice parameter • cohesive energy • lattice parameter for graphitic and -tin phase for diamond phase

16. distinguish btw truly different sets and slight variation from same local minimum • T-dependent criterion: • „distance in vector space“

17. 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