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