The development of new technology for computing and renewable energy sources requires new insight into the physics governing state-of-the-art materials for these applications. To optimize the search for transistors and solar cells to supplant silicon, it is desirable to have them investigated in advance of their large-scale manufacture. One potentially fruitful avenue of investigation is the study of the electronic and optical properties of materials containing heavy atoms. Such atoms have a large spin-orbit coupling, which can be responsible for relatively exotic physics. Topological insulator materials such as $\Bi2Se3$ may have utility in the development of, for example, spin-tronics, in which information may be conveyed without the need for transporting electrical charge.

The details of charges moving through a material, as well as a material absorbing light, require a suitable theoretical treatment. Within the purview of the quantum theory of solids, the technique of many-body perturbation theory gives researchers access to the means of calculating one-particle and two-particle excited states, the exact scenario needed to understand charged excitations and optical absorption, respectively.

In this work, we further extend the ability of the many-body perturbation theory software package of BerkeleyGW to allow for more accurate description of solids containing heavy atoms. Namely, we investigate the properties of materials with large spin-orbit coupling by allowing for the treatment of two-component spinor wavefunctions. In the introduction, we review the physics of one- and two-particle excitations, entirely within a formalism allowing for the presence of spin-orbit coupling. In Chapter 2, we further discuss the implementation of spinor wavefunction functionality in BerkeleyGW and provide many test calculations using materials with varying strengths of spin-orbit coupling, with varying geometries, and including the metallic system of bulk gold. In Chapter 3, we present a calculation of the quasiparticle bandstructure of $\beta$-HgS as a further benchmark material, for which there requires elucidation of the bandstructure topology. We find very close agreement with experiment for both the effective mass and band gap. In Chapter 4, we present the bandstructure of the prototypical topological insulator $\Bi2Se3$, finding significant qualitative differences in the bandstructure from a quasiparticle calculation and the more readily available description from Density Functional Theory (DFT). Namely, we find that, in agreement with experiment, the conduction and valence bands are both nearly parabolic, in contrast to the well-known camel-back feature in the valence band of previous DFT calculations. Finally, in Chapter 5, we use DFT calculations to determine the ground-state geometry of Ir dimers adsorbed to graphene and confirm this geometry, that of a horizontal dimer across the so-called bridge sites, by comparing the resulting density of states to that measured by experiment. We find both have a strong central peak near the graphene Dirac point energy.