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

2. 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

3. 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

4. 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

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

6. 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

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

8. Uniform Distributions
Linear
Congruential
Generator

9. 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

10. 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:

11. Markov Chains

12. Metropolis Algorithm

14. 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

15. 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

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