  Convergence with respect to the size of the k-points sampling-grid of the
Brillouin zone is the main bottleneck in the calculation of optical spectra of
periodic crystals via the Bethe-Salpeter equation (BSE). We tackle this
challenge by proposing a double grid approach to k-sampling compatible with the
effective Lanczos-based Haydock iterative solution. Our method relies on a
coarse k-grid that drives the computational cost, while a dense k-grid is
responsible for capturing excitonic effects, albeit in an approximated way.
Importantly, the fine k-grid requires minimal extra computation due to the
simplicity of our approach, which also makes the latter straightforward to
implement. We performed tests on bulk Si, bulk GaAs and monolayer MoS2, all of
which produced spectra in good agreement with data reported elsewhere. This
framework has the potential of enabling the calculation of optical spectra in
semiconducting systems where the efficiency of the Haydock scheme alone is not
enough to achieve a computationally tractable solution of the BSE, e.g.,
large-scale systems with very stringent k-sampling requirements for achieving
convergence.
