In this dissertation, we apply second-order many-body perturbation theory (MBPT) to calculate the single-particle (charged) excitations of solids using localized atom-centered orbital basis set. The single-particle excitation energies that shape the band structure of the solid are defined as the differences of total energies of the neutral and charged systems. It was shown (J.-Q. Sun and R. J. Bartlett: J. Chem. Phys. 107, 5058 (1997)) that a practical approximation can be obtained by applying MBPT to the energy differences. This method can be used for the calculation of the correction to the charged excitation energies with a possibility to systematically improve accuracy by including higher orders of perturbation, without calculating neutral and charged systems separately. We implement and apply this method for corrections up to second order for the calculation of the band gaps and band structure of three-dimensional solids. We find that the second-order correction is not sufficient, and there is the need to include higher-order terms. For this reason, we employ the Dyson equation, which provides an explicit summation of classes of perturbation terms (described by certain diagrams) up to infinite order. We use the Dyson equation with the self-energy that is produced by the second-order MBPT to calculate band gaps of a series of prototypical semiconductors. We find a significant improvement of the calculated band gaps, compared to the second-order correction, which is just the first iteration of the Dyson equation. A significant difficulty for the application of MBPT to solids is the slow convergence of the reciprocal-space integrals, caused by the singularity of the Coulomb potential. The main approach used until now to overcome the slow convergence is by extrapolation to an infinitely dense k-point grid. We demonstrate that this approach leads to very large errors. The reason is that the converged single-particle excitation energy value is very sensitive to the extrapolation function that defines the asymptotic behavior of the value as a function of the number of k-points in the first Brillouin zone. To solve this problem, we have examined the singularity and the contribution to the slow convergence of the energies analytically. The analysis is done for a localized basis set, but it can be generalized for any basis set. Due to this analysis, the contribution of the singularity to the slow convergence is now known and can be tackled by either a reliable extrapolation, since the law of extrapolation is now known, or by correcting the value using a generalization of the Gygi-Baldereschi method, as proposed in this thesis. This way, we calculate single-particle energies that are converged in reciprocal space.
