Development of the Bethe-Salpeter method considering second-order corrections for a GW electron-hole interaction kernel

We derive two second-order exchange terms in the $GW$ electron-hole interaction kernel. The contributions of these terms have been neglected in the conventional $GW +$ Bethe-Salpeter method, and we implement them in an all-electron mixed basis program. To reveal the effect of these terms, we apply them to 28 molecules of Thiel's benchmark set and compare the ${S}₁$ excitation energies with those obtained from the conventional $GW +$ Bethe-Salpeter method. In addition, using the exciton analysis method with exciton wave functions, we estimate the expectation values for each term in the $GW$ electron-hole interaction kernel. The contribution of the two second-order exchange terms is approximately 0.1--0.2 eV at the exciton states of the $n\ensuremath{\rightarrow}{\ensuremath{\pi}}^*$ transition; however, the contributions are smaller for the $\ensuremath{\pi}\ensuremath{\rightarrow}{\ensuremath{\pi}}^*$ and $\ensuremath{\pi}\ensuremath{\rightarrow}$ Rydberg transitions. Our findings reveal that the errors of the conventional $GW +$ Bethe-Salpeter method are potentially reduced by considering these terms; however, the extent of the corrections is insufficient for the underestimated excitation energies. We believe that our findings are a significant step towards advancing the conventional $GW +$ Bethe-Salpeter method.