First Principles Excitons in Periodic Systems with Gaussian Density Fitting and Ewald Potential Functions

In this work we present a method to solve the Bethe-Salpeter equation from first-principles in any non-metallic solid by employing Gaussian basis functions. The approach 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 adapts naturally to lower dimensional materials without range truncation in the Coulomb interaction, allows for rapidly convergent virtual states sums in the irreducible polarizability, and exact calculation of the velocity matrix elements for optical properties. Furthermore, it opens the way to optimize computation time by fine-tuning the fitting Gaussian basis and metric. As an illustration of the implementation, we provide examples of exciton spectra and optical absorption in some paradigmatic 2D and 3D materials, with good agreement to previous calculations based on plane-waves. The whole code is open-source, available in github as part of the XATU package.