Correlated Methods for Excited States in Periodic Systems

Abstract

In this thesis we apply equation-of-motion coupled-cluster theory to obtain the ground-state and excited-state of three-dimensional solids. We show the problems theorists face in applying traditional electronic structure methods to solids and derive the Hamiltonian matrix elements for extended systems. The coupled-cluster due to their size-extensivity and accuracy have long been a promising candidate for use as a benchmark in the ground- and excited-states of solids. After reviewing some basic theory we apply these methods to various periodic systems. We first look at the jellium model with a Wigner-Seitz radius of $r_s = 4$, a model for metallic sodium and compare our findings with state-of-the-art $GW$ theory. Despite the relatively higher scaling of the coupled-cluster methods we were able to achieve modest system sizes of 114 electrons in 485 orbitals, without the use of symmetry. We compute the one-particle coupled-cluster Green's function and compare the experimental occupied bandwidth for metallic sodium against our calculated one. Moving on to \textit{ab initio} systems, we perform calculations of optical gaps and band structure of silicon and diamond using the same coupled-cluster methods. Using the k-point equation-of-motion equations provided, we are able to achieve system sizes of 256 electrons in 2,176 orbitals using a $4 \times 4 \times 4$ Monkhorst-Pack sampling of the Brillouin zone and polarized triple-zeta basis. These methods are then presented in the broader context of electronic structure methods provided in the PySCF framework, a software package providing the computational tools for studying the electronic structure of molecular and periodic systems. We conclude with forward directions and preliminary results for removing finite-size effects in correlated methods and pushing towards chemical accuracy through perturbative methods for excited states.