First-principles excitons in periodic systems with Gaussian density fitting and Ewald potential functions

Excitons, namely, neutral excitations in a system of electrons arising from the electron-hole interaction, are often essential to explain optical measurements in materials. They are governed by the Bethe-Salpeter equation, which can be cast into a matrix form that is formally analogous to the one for electrons at the mean-field level. However, constructing the corresponding excitonic Hamiltonian in practice is challenging, specially from a computational perspective if one wishes to surpass effective models. Methods that enable such calculations from the different density-functional theory frameworks currently available are, therefore, convenient. In this work, we present an approach to solve the BSE starting from a self-consistent calculation in any nonmetallic solid by evaluating the electron-hole kernel in real space with a localized, Gaussian-type basis. It is based on the Gaussian density fitting or resolution of the identity approximation to reduce the initial quartic scaling in the basis dimension, in combination with the use of Ewald-type potential functions to automatically sum the conditionally convergent lattice series. The method inherits the features of the localized basis, in particular the reduced computational requirements due to the small and tunable Hilbert-space dimension, the natural adaptation to lower-dimensional systems or the possibility to evaluate optical matrix elements exactly. As an illustration of the computational implementation, we provide examples of exciton spectra and optical absorption in some paradigmatic 2D and 3D materials where the single-particle approximation fails qualitatively.