Anomalous propagators and the particle-particle channel: Bethe-Salpeter equation

TL;DR

This work develops a particle-particle Bethe-Salpeter equation (pp-BSE) by leveraging pairing fields and anomalous Green's functions to access double ionization potentials and double electron affinities. It derives a pp-BSE kernel structurally analogous to the standard electron-hole BSE and implements several kernels (first- and second-order Coulomb, GW, and T-matrix) with both static and dynamic variants, benchmarking their performance against high-accuracy CIPSI-DIP-EOM-CCSD references for valence and core DIPs. The results show that a static GW kernel within the TDA already delivers competitive accuracy, and adding a perturbative dynamic correction further improves agreement with reference data, approaching DIP-EOM-CCSD quality for many cases, though core-hole states remain challenging due to limited orbital relaxation in a linear-response framework. The study highlights the potential of pp-BSE kernels for efficiently treating two-body excitations and suggests avenues for future work on DEAs, core-level states, and extensions to three-body propagators.

Abstract

The Bethe-Salpeter equation has been extensively employed to compute the two-body electron-hole propagator and its poles which correspond to the neutral excitation energies of the system. Through a different time-ordering, the two-body Green's function can also describe the propagation of two electrons or two holes. The corresponding poles are the double ionization potentials and double electron affinities of the system. In this work, a Bethe-Salpeter equation for the two-body particle-particle propagator is derived within the linear-response formalism using a pairing field and anomalous propagators. This framework allows us to compute kernels corresponding to different self-energy approximations ($GW$, $T$-matrix, and second-Born) as in the usual electron-hole case. The performance of these various kernels is gauged for singlet and triplet valence double ionization potentials using a set of 23 small molecules. The description of double core hole states is also analyzed.

Figures (8)

• Figure 1: Diagrammatic representation of the eh propagator $L$ (top) and pp propagator $K$ (bottom) at the RPA level. The dashed lines represent the Coulomb interaction and the solid lines with arrows denote the one-body propagator. The first and second-order exchange terms have not been represented but should be included in $K^\RPA$.
• Figure 2: Diagrammatic representation of the eh-BSE, as defined in Eq. \ref{['eq:ehBSE']}.
• Figure 3: Diagrammatic representation of the pp-BSE [see Eq. \ref{['eq:ppBSE']}]. The rightmost $K_0$ has been replaced by its first term [see Eq. \ref{['eq:K0']}] using the antisymmetry of the kernel.
• Figure 4: Diagrammatic representation of the two direct second-order terms contained in $\Sigma^\ee$. The dashed lines represent the Coulomb interaction, the solid lines with arrows denote the one-body propagator while the double-arrowed propagators represent $G^\hh$ and $G^\ee$.
• Figure 5: Histogram of the errors (with respect to FCI) for the singlet and triplet principal DIP of 23 small molecules computed in the aug-cc-pVTZ basis set at the pp-RPA level using various one-body energies: HF, $GW$, $T$-matrix, and GF(2).