UCLA

Electronic structure theory seeks to describe the behavior of electrons in atomic and molecular systems.Due to the intractable nature of solving the molecular Schr\o dinger's equation, approximations are made. The main challenge is to create methods that are accurate enough to gain insight while also being efficient enough to run calculations in a reasonable amount of time. In this balancing act, many strategies have been developed to allow for electronic structure calculations of large systems. Much progress has been made from calculating the states of isolated one-electron systems to now being able to simulate dynamic processes in large extended systems. This dissertation seeks to contribute to the development of novel methods to enable more efficient large-scale electronic structure calculations. A major theme of the dissertation is the use of stochastic techniques to reduce the computational scaling of methods.

Chapter 2 discusses these techniques and highlights the improvement in computational scaling when implemented with density functional theory (DFT) and many-body perturbation theory within the GW approximation.Many improvements to stochastic DFT (sDFT) have been made over the years, incorporating techniques such as embedding to reduce the required number of statistical samples. Chapter 3 continues in the same line of work and introduces the concept of tempering and its application in sDFT. The core idea of tempering is to rewrite the electronic density into the sum of a cheaper "warm" term and a smaller more expensive "cold" term. This results in a significant reduction in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort.

Chapter 4 discusses the gapped filtering method and its application in the stochastic GW (sGW) approximation.In gapped-filtering, a short Chebyshev expansion accurately represents the density-matrix operator. The method optimizes the Chebyshev coefficients to give the correct density matrix at all energies except within the gapped region where there are no eigenstates. Gapped filtering reduces the number of required terms in the Chebyshev expansion compared to traditional expansion methods, as long as one knows or can efficiently determine the HOMO and LUMO positions such as in sGW. Another direction in this dissertation is laying the foundations to implement the projector augmented wave (PAW) method into stochastic quantum methods. Compared to norm-conserving pseudopotentials (NCPP), PAW has the advantage of lower kinetic energy cutoffs and larger grid spacing at the cost of having to solve for non-orthogonal wavefunctions. Orthogonal PAW (OPAW) was earlier developed with DFT to allow the use of PAW when orthogonal wavefunctions are desired. To make OPAW viable for post-DFT stochastic methods, time-dependent wavefunctions are required. For this purpose, chapters 5 and 6 detail OPAW and its implementation in the real-time time-dependent (TD) DFT framework.