Second-order Bethe-Salpeter kernels: the interplay of vertex-corrected quasiparticle energies and the screened interaction

We study second-order Bethe-Salpeter kernels generated by truncating an iterative formulation of Hedin's equations after the second cycle. Starting from the Hartree-Fock or the $GW$ self-energy, whose first cycles correspond to TDHF and $GW$+BSE, we obtain the second-cycle kernel by differentiating the first-cycle correlation self-energy with the interacting correlation function replaced by the non-interacting one, and evaluate it in the static limit of the effective kernel. Using fixed restricted Hartree-Fock orbitals, we benchmark 30 singlet and 28 triplet excitation energies for seven small molecules (six for the triplets) against the theoretical best estimates from QUEST v1.2. The second cycle lowers the mean absolute error (MAE) in both sequences. For the $GW$ sequence, the MAE decreases from 0.63 to 0.29 eV (singlets), on par with ADC(2) (0.30 eV), and from 0.43 to 0.38 eV (triplets). BSE$^{(2)}$@HF removes the triplet instabilities of TDHF and yields MAEs of 0.43 and 0.38 eV, or 0.31 and 0.30 eV when the Tamm-Dancoff approximation is applied in all cycles. Assigning the statically screened interaction $W_0$ to the exchange part of the $GW$ self-energy counts second-order screening twice in the second-cycle kernel and strongly blueshifts valence excitations. Without this double counting, dressing the exchange-type terms with $W_0$ lowers the singlet MAE from 0.37 to 0.29 eV but leaves the triplets nearly unchanged. Replacing second-order Green's function (GF2) quasiparticle energies by vertex-corrected ones in the same kernel raises excitation energies by about 0.7 eV, improving Rydberg but degrading valence excitations. A perturbative dynamical correction of the kernel blueshifts valence excitations by 0.3-0.4 eV and increases the errors of all full-matrix BSE$^{(2)}$ variants. The consistent improvement over the first cycle suggests that higher cycles are worth pursuing.

Source: arXiv:2610.41920v1 - http://arxiv.org/abs/2610.41920v1 PDF: https://arxiv.org/pdf/2610.41920v1 Original Link: http://arxiv.org/abs/2610.41920v1