$G_0W_0$@HF and BSE methods in periodic systems from Hartree-Fock theory: gaussian orbital and density fitting approach
The $GW$ method for calculating quasi-particle energies of solids commonly begin from a DFT Hamiltonian and Kohn-Sham orbitals in a plane wave basis. Screening of the coulomb interaction is implemented using the inverse dielectric function in the random phase approximation (RPA). We present $G_0W_0$ calculations which begin from the Hartree-Fock method in a basis of gaussian orbitals. The screened coulomb interaction, $W$, is obtained using a $W$ = $v$ + $v\Pi v$ approach without invoking a plasmon pole approximation. The polarizability, $\Pi$, in $W$ is treated at the RPA level. RPA polarizabilities require solution of Bethe-Salpeter equations (BSE) for each unique $\textbf{Q}$ point. A strategy for obtaining self-energies which are converged with respect to number of virtual states is employed in which $G_0W_0$ yields the majority of the self-energy and the remaining part from high energy virtual levels is evaluated at second-order. The methods are evaluated by applying them to elemental semiconductors (C, Si) and oxides (MgO and anatase and rutile TiO$_2$). Common errors of HF theory applied to materials include overestimation of both the band gap and valence band widths. These are corrected in the approach employed here. Typically, the RPA screened interaction results in overestimation of band gaps while the $G_0W_0$ self-energy band width renormalization yields band widths for diamond and Si which are in good agreement with experiment. HF calculations are performed in gaussian orbital basis sets and $G_0W_0$ and BSE calculations are performed using density fitting with a coulomb metric.