Solving the Bethe-Salpeter equation on a subspace: Approximations and consequences for low-dimensional materials

It is well known that the ambient environment can dramatically renormalize quasiparticle gap and exciton binding energies in low-dimensional materials, but effect of on energy splitting spin-singlet spin-triplet states less understood. A prominent renormalization optical strength (and hence spectrum) through additional screening direct Coulomb term describing attractive electron-hole interaction kernel Bethe-Salpeter equation (BSE). The repulsive exchange responsible for singlet-triplet slitting, other hand, unscreened within formal many-body perturbation theory. However, Loren Benedict argued practical calculations restricted to a subspace full Hilbert space, should be appropriately screened by outside subspace, so-called $S$ approximation \cite{Benedict2002}. Here, we systematically explore accuracy different confined systems, including molecule heterostructures semiconducting metallic layered materials. We show actually exact limit small (i.e., term) used significantly accelerate convergence with respect number empty states, provided particular effective consistent conventional Tamm-Dancoff employed. further find excitons largely unaffected external dielectric most quasi-two-dimensional