Monte Carlo Methods: Basics Charusita Chakravarty Department of Chemistry Indian Institute of Technology Delhi

Flavours of Monte Carlo Metropolis Monte Carlo Methods: Generation of multidimensional probability distributions Multidimensional integration Projector Monte Carlo Methods: Solution of partial differential equations in many-dimensions Green’s function Monte Carlo Matrix Inversion techniques Other uses of stochastic simulation techniques See J. S. Liu, Monte Carlo Strategies in Scientific Computing

Quantum Many-Body Systems electrons in atoms, molecules and solids bosonic superfluid: liquid helium semiclassical systems: liquid H2,water Quark-Gluon plasma References: 1. J. W. Negele and H. Orland, Quantum Many-Particle Systems 2. W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, Rev. Mod. Phys., 73, 33 (2001) 3. D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method, Phys. Rev. Lett., 45, 567 (1980) 4. D. M. Ceperley, Rev. Mod. Phys.,67, 279 (1995) 5. C. Chakravarty, Int. Rev. Phys. Chem., 16, 421 (1997) 6. B. L. Hammond, W. A. Lester and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry

Atomic and Molecular Systems Born-Oppenheimer approximation: Adiabatic separation of electronic and nuclear motion Electronic motion: Energy scales e.g chemical binding energies (1-5 eV) Excited state occupancy at temperatures of interest (< 1000K) is negligible. Aim is to obtain many-body ground state wavefunction Diffusion and Variational Monte Carlo Nuclear/Atomic Motion Energy scales: rotations and vibrations (0.1eV) Finite-temperature methods: Path-integral Monte Carlo Lattice/Continuum Hamiltonians

Organization Lattice Hamiltonians Basics of Monte Carlo Methods Continuum systems: interacting electrons and/or atoms Diffusion Monte Carlo Path integral Monte Carlo Lattice Hamiltonians

Integration: Grid-Based Stochastic Define a set of grid points and associated weights Error Computational efficiency will grow exponentially with dimensionality Stochastic Sample points randomly and uniformly in the interval [a:b] Error: Central Limit Theorem Computational efficiency

Importance Sampling: Reducing the Variance Sampling distribution p(x) must be finite and non-negative

Uniform Distributions Linear Congruential Generator

Non-uniform Distributions: von Neumann Rejection Method Convenient Sampling Distribution,f(x) Sample xi from the distribution f(xi) and compute p(xi). Sample a random number x uniformly distributed between 0 and 1. 3. If p(xi)/f(xi) > x, accept xi; otherwise reject xi. Desired distribution, p(x) xi

What is a Random Walk? A random walk is a set of probabilistic rules which allow for the motion of the state point of the system through some conguration space; the moving state point is referred to as a walker. All random walks must have the following features: The available configuration space and attributes of walker must be defined. An ensemble of walkers is the probability distribution of the system at some point in time. Source distribution at time t=0 must be defined. The system's kinetics will be determined by the transition matrix, T, which is a set of probabilistic rules which govern a single move of the walker from its current position to some neighbouring position.The transition matrix must satisfy the requirements:

Markov Chains

Metropolis Algorithm

Classical Monte Carlo Generating a set of configurations, {xi, i=1,N} distributed according to their Boltzmann weights, P(xi)=(exp(-bV(xi)) P(x)=(exp(-bV(x)) xold xnew x’new

Generating the Boltzmann Distribution Current Configuration Reject new configuration Trial Configuration No Compute ratio of Boltzmann weights Yes Is x < w? Accept new configuration Yes Is w> 1? Generate uniform random no. x between 0 and 1 No

Random Walks and Differential Equations One-dimensional Diffusion Equation Random Walks