BSE and time dependent DFT beyond the Tamm-Dancoff approximation: diagonalization versus time evolution

The Casida equation for time dependent DFT and the Bethe Salpether equation for many body response functions are tools commonly used to calculate optical properties in solids. Here an efficient algorithm to go beyond the commonly used Tamm-Dancoff approximation for periodic systems is discussed. The algorithm uses ideas from quantum chemistry and inversion symmetry to recast the Casida/BSE equation in a quadratic form. The final equation can be solve efficiently using standard linear algebra packages. For time dependent DFT, we compare this algorithm to time evolution, finding perfect agreement between spectra predicted from the Casida equation and the one obtained from the Fourier transformation of the polarization. Performance wise, we find that diagonalization of the Casida/BSE equation is extremely inefficient compared to the time evolution algorithms for density functionals, however, for hybrid functionals, the repeated application of the Fock operator at each time step, makes time evolution prohibitively expensive.
We finally discuss recent applications of the VASP BSE code to systems of current interest. This includes excitonic binding energy in MAPbI3 (hybrid organic inorganic perovskites).