Fast optical absorption spectra calculations for periodic solid state systems

Abstract

We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe-Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe-Salpeter Hamiltonian scales nearly optimally as O(N k ) with respect to the number of samples in the Brillouin zone N k . In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales as O(N k log N k ). Therefore the optical absorption spectrum, as well as selected excitation energies can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the O(N 2 k ) and O(N 3 k ) scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with N k = 2197, it takes 6 hours to assemble the Bethe-Salpeter Hamiltonian and 4 hours to fully diagonalize the Hamiltonian using 169 cores when the brute force approach is used. The new method takes less than 3 minutes to set up the Hamiltonian and 24 minutes to compute the absorption spectrum on a single core.

LCAO-TDDFT-k- ω : spectroscopy in the optical limit

“…Computationally, a G 0 W 0 calculation will scale cubically with the number of plane waves [49], whereas constructing and applying the Bethe-Salpeter Hamiltonian will scale with the k-point sampling of the Brillouin zone (BZ), resulting in restrictive time and memory costs [50]. Although neither electronic screening nor excitonic binding are included in standard TDDFT calculations, these effects tend to compensate for each other in the calculated G 0 W 0 -BSE optical absorbance spectra, often leading to a fortuitous error cancellation with TDDFT.…”

Accelerating Time-Dependent Density Functional Theory and GW Calculations for Molecules and Nanoclusters with Symmetry Adapted Interpolative Separable Density Fitting

“…10 By far, the ISDF method has been applied and demonstrated promising performance in calculations involving RPA correlation energy, 10 hybrid functional theory, 11 and the Bethe−Salpeter equation. 12,13 Linear response time-dependent density functional theory (LR-TDDFT) 14,15 and the GW approximation 16−18 reducing the scaling 22−26 or prefactors 27−29 of GW calculations.…”

LCAO-TDDFT-$k$-$ω$: Spectroscopy in the Optical Limit

“…To describe the optical band gap, exciton binding, i.e., electronhole interactions, should also be included through the solution of the four-point Bethe-Salpeter equation 28 when calculating the macroscopic dielectric function. Computationally, a G 0 W 0 calculation will scale cubically with the number of plane waves 38 , whereas constructing and applying the Bethe-Salpeter Hamiltonian will scale with the k-point sampling of the Brillouin Zone (BZ), resulting in restrictive time and memory costs 39 . Although neither electronic screening nor excitonic binding are included in standard TDDFT calculations, these effects tend to compensate for each other in the calculated G 0 W 0 -BSE optical absorbance spectra, often leading to a fortuitous error cancellation with TDDFT.…”