Establishing Quantum Monte Carlo and Hybrid Density Functional Theory as Benchmarking Tools for Complex Solids

Quantum mechanics provides an exact description of microscopic matter, but predictions require a solution of the fundamental many-electron Schrödinger equation. Since an exact solution of Schrödinger’s equation is intractable, several numerical methods have been developed to obtain approximate solutions. Currently, the two most successful methods are density functional theory (DFT) and quantum Monte Carlo (QMC). DFT is an exact theory which, which states that ground-state properties of a material can be obtained based on functionals of charge density alone. QMC is stochastic method which explicitly solves the many-body equation. In practice, the DFT method has drawbacks due to the fact that the exchange-correlation functional is not known. A large number of approximate exchange-correlation functionals have been produced to accommodate for this deficiency. Conceptual systematic improvements known as “Jacob’s Ladder” of functional approximations have been made to the standard local density approximation (LDA) and generalized gradient approximation (GGA). The traditional functionals have many known failures, such as failing to predict band gaps, silicon defect energies, and silica phase transitions. The newer generation functionals including meta-GGAs and hybrid functionals, such as the screened hybrid, HSE, have been developed to try to improve the flaws of lower-rung functionals. Overall, approximate functionals have generally had much success, but all functionals unpredictably vary in the quality and consistency of their predictions. Often, a failure of one type of DFT functional can be fixed by simply identifying another DFT functional that best describes the system under study. Identifying the best functional for the job is a challenging task, particularly if there is no experimental measurement to ii compare against. Higher accuracy methods, such as QMC, which are vastly more computationally expensive, can be used to benchmark DFT functionals and identify those which work best for a material when experiment is lacking. If no DFT functional can perform adequately, then it is important to show more rigorous methods are capable of handling the task. QMC is high accuracy alternative to DFT, but QMC is too computationally expensive to replace DFT. Hybrid DFT functionals appear to be a good compromise between QMC and standard DFT. Not many large scale computations have been done to test the feasibility or benchmark capability of either QMC or hybrid DFT for complex materials. This thesis presents three applications expanding the scope of QMC and hybrid DFT to large, scale complex materials. QMC computes accurate formation energies for single-, di-, and trisilicon-self-interstitials. QMC combined with phonon energies from DFT provide the most accurate equations of state, phase boundaries, and elastic properties available for silica. The HSE DFT functional is shown to reproduce QMC results for both silicon defects and high pressure silica phases, establishing its benchmark accuracy compared to other functionals. Standard DFT is still the most efficient and useful for general computation. However, this thesis shows that QMC and hybrid DFT calculations can aid and evaluate shortcomings associated the exchange-correlation potential in DFT by offering a route to benchmark and improve reliability of standard, more efficient DFT predictions.