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

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