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

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(N6), 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(N5), 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 21Ag 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.

Many-Body Green’s Function Theory for Electronic Excitations in Complex Chemical Systems

“…Most BSE calculations rely on the static screening approximation to the BSE interaction kernel within which the BSE spectrum is purely composed of single excitations . It is proposed that the dynamical screening part of the BSE kernel contains contributions from double excitations and even higher-order excitations. − However, approximations employed in the dynamical BSE kernel sometimes result in spurious excitations; the accuracy of the dynamical BSE kernel for double excitations is also not very satisfactory . (ii) BSE suffers from the triplet instability problem which also plagues TDDFT. ,,,,− Benchmark calculations on organic molecules demonstrate that BSE yields significantly too low triplet excitation energies although the singlet excited states can be accurately described by BSE. , How to cure the triplet instability problem and give excitation energies of the singlets and triplets with the same accuracy remains an open question for BSE.…”

“…178−182 However, approximations employed in the dynamical BSE kernel sometimes result in spurious excitations; the accuracy of the dynamical BSE kernel for double excitations is also not very satisfactory. 182 (ii) BSE suffers from the triplet instability problem which also plagues TDDFT. 7,9,85,86,183−185 Benchmark calculations on organic molecules demonstrate that BSE yields significantly too low triplet excitation energies although the singlet excited states can be accurately described by BSE.…”

A 'moment-conserving' reformulation of GW theory

“…Such upfolded representations have been considered previously in diagrammatic theories, in a recasting of GF2 theory in terms of its moments 45,46,50 as well as more recently to GW amongst others 47,[51][52][53][54][55] . For 'exact' G 0 W 0 , this auxiliary space (i.e.…”

“…Future work will explore other analogous approaches where (A − B)(A + B) can be decomposed in this way, for applicability to e.g. the Bethe-Salpeter equation or other RPA variants with (screened) exchange contributions 54,74,75 . From this low-rank decomposition and the recursive definition of Eqs.…”

Constructing “Full-Frequency” Spectra via Moment Constraints for Coupled Cluster Green’s Functions

“…This allows for the full CC Green’s function including off-diagonal elements to be probed for all frequencies, without requiring a priori frequency grid definitions on which the function is resolved. Furthermore, this Green’s function and self-energy can be directly obtained as a series of specific energies and spectral weights of all poles in the η → 0 + limit, in a fashion similar to recent reformulations of “frequency-free” GW, GF2, and other correlated Greens function methods. ,,− Furthermore, in contrast to some other approaches, this CC Green’s function is not solved for one frequency at a time, nor resolved as a state-specific expansion of successive IPs and EAs.…”