Monte Carlo Methods: Application to Hydrogen Gas and Hard Spheres by Mark Douglas Dewing

Quantum Monte Carlo (QMC) methods are among the most accurate for computing ground state properties of quantum systems. The two major types of QMC we use are Vari-ational Monte Carlo (VMC), which evaluates integrals arising from the variational principle , and Diffusion Monte Carlo (DMC), which stochastically projects to the ground state from a trial wave function. These methods are applied to a system of boson hard spheres to get exact, infinite system size results for the ground state at several densities. The kinds of problems that can be simulated with Monte Carlo methods are expanded through the development of new algorithms for combining a QMC simulation with a classical Monte Carlo simulation, which we call Coupled Electronic-Ionic Monte Carlo (CEIMC). The new CEIMC method is applied to a system of molecular hydrogen at temperatures ranging from 2800K to 4500K and densities from 0.25 to 0.46 g/cm 3. VMC requires optimizing a parameterized wave function to find the minimum energy. We examine several techniques for optimizing VMC wave functions, focusing on the ability to optimize parameters appearing in the Slater determinant. Classical Monte Carlo simulations use an empirical interatomic potential to compute equilibrium properties of various states of matter. The CEIMC method replaces the empirical potential with a QMC calculation of the electronic energy. This is similar in spirit to the Car-Parrinello technique, which uses Density Functional Theory for the electrons and molecular dynamics for the nuclei. The challenges in constructing an efficient CEIMC simulation center mostly around the noisy results generated from the QMC computations of the electronic energy. We introduce two complementary techniques, one for tolerating the noise and the other for reducing it. The penalty method modifies the Metropolis acceptance ratio to tolerate noise without introducing a bias in the simulation of the nuclei. For reducing the noise, we introduce the two-sided energy difference method, which uses correlated sampling to compute the energy change associated with a trial move of the nuclear coordinates. Unlike the standard reweighting method, it remains stable as the energy difference increases. iii Acknowledgments First I would like to thank my advisor, David Ceperley, for supporting me in this research, for teaching these Monte Carlo methods to me, and for being available and patient when answering questions. I would also like to thank the graduate students and postdocs in the group who have helped me and provided useful and interesting discussions. And special …