Full-frequency dynamical Bethe-Salpeter equation without frequency and a study of double excitations

TL;DR

This paper introduces a reformulation of the full-frequency dynamical Bethe-Salpeter equation as a frequency-independent eigenvalue problem, enabling more efficient computation and access to double excitation states.

Contribution

The authors present an exact reformulation of the dynamical BSE as a static eigenvalue problem, reducing computational cost and allowing direct analysis of double excitations.

Findings

Reformulation reduces computational complexity from O(N^6) to O(N^5).

The method provides access to double excitation states absent in static BSE.

GW/BSE overestimates excitation energies and underestimates double excitation character.

Abstract

The Bethe-Salpeter equation (BSE) that results from the GW approximation to the self-energy is a frequency-dependent (nonlinear) eigenvalue problem due to the dynamically screened Coulomb interaction between electrons and holes. The computational time required for a numerically exact treatment of this frequency dependence is $O(N^6)$, where $N$ is the system size. To avoid the common static screening approximation, we show that the full-frequency dynamical BSE can be exactly reformulated as a frequency-independent eigenvalue problem in an expanded space of single and double excitations. When combined with an iterative eigensolver and the density fitting approximation to the electron repulsion integrals, this reformulation yields a dynamical BSE algorithm whose computational time is $O(N^5)$, which we verify numerically. Furthermore, the reformulation provides direct access to excited states with dominant double excitation character, which are completely absent in the spectrum of the statically screened BSE. We study the $2^1A_\mathrm{g}$ state of butadiene, hexatriene, and octatetraene and find that GW/BSE overestimates the excitation energy by about 1.5-2 eV and significantly underestimates the double excitation character.