Combining orbital and real space quantum Monte Carlo methods

Abstract

Quantum Monte Carlo (QMC) methods are wave-function-based approaches based on random sampling that provide a very direct treatment of many-body problems and serve as benchmarks against which other techniques may be compared. This thesis is concerned with the development of QMC methods for the calculation of highly-accurate electronic energies. In particular, it focusses on combining QMC methods, namely continuum methods such as variational and diffusion Monte Carlo (VMC and DMC) with full configuration interaction quantum Monte Carlo (FCIQMC). The principal aim of this thesis is to show that thanks to their complementary nature, these methods can be used in combination to overcome specific weaknesses that each methodology has by itself. The thesis begins with introductory chapters which describe the various QMC methods as they are used in the literature. Following this, three different methodologies are explored, which have applicabilities to different types of systems. In the first method, we present a new technique to obtain very accurate total and relative energies by extrapolating VMC and DMC energies, with the help of FCIQMC-generated trial wave functions combined with a Jastrow factor and an optional backflow transformation. We find that the VMC and DMC energies are smooth functions of the sum of the squared coefficients in the FCIQMC wave function, and that quadratic extrapolations of the VMC and DMC energies intersect within uncertainty of the exact total energy. With adequate statistical treatment of quasi-random fluctuations, the extrapolated intersection with polynomials of order two (XSPOT) method is shown to yield results in agreement with benchmark-quality total and relative energies for various molecular systems. Following this, we combine the DMC and transcorrelated FCIQMC (TC-FCIQMC) methods, along with a finite-size extrapolation framework to calculate the thermodynamic limit of the exact correlation energy of the high-density spin-unpolarized uniform electron gas. The thermodynamic limit obtained at rs = 0.5 is within uncertainty of the most recent QMC correlation energy from the literature. In the final method, we introduce a new extrapolation scheme which eliminates the energy bias stemming from the initiator approximation in the FCIQMC method. We find that extrapolating the FCIQMC energy linearly in n^(−1/3) to the infinite-walker limit yields results in excellent agreement with the benchmark full configuration interaction (FCI) limits from the semistochastic heat-bath configuration interaction (SHCI) method. The method is applied to a series of diatomics and the results are accurate, with FCI limits obtained from FCIQMC differing by 0.4 mHa or less from their SHCI counterparts throughout. The more complex chromium atom and dimer are also successfully studied using this method.