A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.
A formally exact Bethe-Salpeter-like equation for the linear-response function is introduced with a kernel which depends only on the one frequency of the applied field. This is in contrast with the standard Bethe-Salpeter equation (BSE) which involves multiple-frequency integrals over the kernel and response functions. From the one-frequency kernel, known approximations are straightforwardly recovered. However, the present formalism lends itself to more powerful approximations. This is demonstrated with the exact analytical solution of the Hubbard molecule. Similarities and differences of the GW + BSE approach with the self-consistent random-phase approximation are also discussed.
Standard many-body perturbation theory for optical excitations combines the
GW approximation for quasiparticle energies with the Bethe-Salpeter equation (BSE) for the electron-hole response. Formally, the BSE kernel must equal the functional derivative of the self-energy,
K=δΣ/δG. In routine practice, however, the kernel is built from a statically screened direct interaction and a bare exchange interaction, which breaks this consistency. Using exact diagonalization of the extended Hubbard dimer, we construct a controlled benchmark that isolates the price of this inconsistency. We show that the internally inconsistent practical
GW+BSE construction frequently yields accurate optical gaps through accidental cancellation between an underestimated
GW gap and an oversized bare-exchange kernel, and we map the regions of parameter space where this accidental accuracy occurs. Upon switching on the nearest-neighbor interaction and binding the exciton, the practical construction remains the accurate one throughout the weak-binding regime, with cancellation optima at parts in
103, while the frozen-
W construction fails at every binding strength. Beyond a binding threshold both static constructions deteriorate, showing that deeply bound excitons require physics beyond any static kernel. Accidental accuracy is therefore a property of the weak-binding regime, and the systematic failure of the frozen-
W derivative kernel within the explored parameter space traces to its overscreening of the exchange channel. The accuracy of the practical kernel is an accidental cancellation of two independent errors, an underestimated quasiparticle gap and an oversized exchange term, not a hidden consistency. These results provide a controlled taxonomy of approximation failure in
GW+BSE and a precise diagnostic for when standard workflows can be trusted.