How to choose efficiently the size of the Bethe-Salpeter equation Hamiltonian for accurate exciton calculations on supercells

The Bethe-Salpeter Equation (BSE) is the workhorse method to study excitons in materials. The BSE Hamiltonian size, which depends on how many valence-to-conduction band transitions are considered, needs to be chosen to be sufficiently large to converge excitons' energies and wavefunctions but should be minimized to make calculations tractable, as BSE calculations are expensive and scale with the number of atoms as $\mathcal{O}(N_{\rm{atoms}}^6)$. In particular, in the case of supercell (SC) calculations composed of $N_{\rm{rep}}$ replicas of a primitive cell (PC), a natural choice to build this BSE Hamiltonian is to include all transitions derived from PC calculations by zone folding. However, this leads to a very large BSE Hamiltonian, as the number of matrix elements in it is $(N_k N_c N_v)^2$, where $N_k$ is the number of $k$-points and $N_{c(v)}$ is the number of conduction (valence) states used. When creating a SC, the number of $k$-points decreases by a factor $N_{\rm{rep}}$ but both the number of conduction and valence states increase by the same factor, therefore the number of matrix elements in the BSE Hamiltonian increases by a factor $N_{\rm{rep}}^2$, making exactly corresponding calculations prohibitive. Here, we provide a workflow to decide how many transitions are necessary to achieve comparable results, based on only PC results. With our method, we show that to converge the first exciton binding energy of a LiF SC composed of 64 PCs, to an energy tolerance of 0.15 eV, we only need 12\% of the valence-to-conduction matrix elements that result from zone folding with a minimal set of bands. As an example, we use the number of bands from our method to obtain the absorption spectrum of LiF with a V$_k$-like defect. The procedure in our work helps in evaluating excitonic properties in large SC calculations.