Solving the Bethe-Salpeter Equation on a Subspace: Approximations and Consequences for Band-Edge and Core-Level Excitons in Quasi Low-dimensional Materials*

It is known that environment can dramatically renormalize the quasiparticle energy gap and exciton binding energies in quasi low-dimensional materials, but the effect of environmental screening on the energy splitting of the spin-singlet and spin-triplet exciton states is less explored. The renormalization of the exciton binding energy arises from additional environmental screening of the attractive direct Coulomb term in the kernel of the Bethe-Salpeter equation (BSE). The repulsive exchange interaction responsible for the singlet-triplet slitting is in principle unscreened, though it has been argued that in practical calculations using a subspace of the full Hilbert space the exchange interaction should be modified and effectively screened by states outside of the chosen subspace, the “S” approximation. We explore the accuracy of the S approximation for different systems, including molecules, heterostructures and core level excitations. We show, additionally, that the S approximation is actually exact in the limit of small exciton binding energies provided that a screening consistent with the Tamm-Dancoff approximation is employed.