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### Abstract

An accurate description of the electronic structure of semiconductors and insulators is essential in materials discovery. However, the Schrodinger equation[124] of many-body systems, e.g., electrons in solids or molecules, cannot be solved exactly. Many approaches have been proposed to solve approximately the Schrodinger equation of interacting electrons, and new methods and algorithms are still being developed to improve the efficiency and accuracy of the calculations, and/or to incorporate new physics.This dissertation focuses on the developments of methods to study the electronic structure of solids, in particular electron-phonon interactions in semiconductors and insulators, using many-body perturbation theory (MBPT). We start with a brief review of existing methods to study electron-electron interactions in solids, including density functional theory (DFT)[63, 71] and post-DFT methods (GW approximation),[61, 65, 8] and methods to study electron-phonon interactions including density functional perturbation theory (DFPT).[10] Then we describe our developments to: (i) improve the efficiency of G0W0 calculations in Chapter 3, (ii) combine electron-electron and electron-phonon calculations in solids in Chapter 4, and (iii) compute electron-phonon interactions at the hybrid functional level of theory in Chapter 5.
First, we develop an approximation to increase the efficiency of the G0W0 calculations in molecules and heterogeneous systems. The G0W0 approximation predicts the electronic energy gap of materials, but at a higher computational cost compared to DFT. Starting from an existing implementation of the G0W0 method, where the dielectric function is represented using a low-rank approximation, [51] we present an algorithm to improve the efficiency of the calculations by solving an approximate form of the Sternheimer equation.[132] The method presented here speeds up the calculation by 50%, without significant loss of accuracy.
Then, we develop a method to effectively combine the calculation of electron-electron and electron-phonon interactions at the G0W0 level of theory in extended systems. Our method allows for the calculations of the electron-electron and electron-phonon interaction without the separate evaluation of screening effects as well as for calculations beyond the Allen-Heine-Cardona (AHC) formalism[4].
In the last part of the dissertation, we propose a method to compute electron-phonon interactions at the level of hybrid functionals based on density matrix perturbation theory (DMPT).

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