  Time-Dependent Density Functional Theory (TDDFT) has been currently
established as a computationally cheaper, yet effective, alternative to the
Many-Body Perturbation Theory (MBPT) for calculating the optical properties of
solids. Within the Linear Response formalism, the optical absorption spectra
are in good agreement with experiments, as well as the direct determination of
the exciton binding energies. However, the family of exchange-correlation
kernels known as long-range corrected (LRC) kernels that correctly capture
excitonic features have difficulties simultaneously producing good-looking
spectra and accurate exciton binding energies. More recently, this discrepancy
has been partially overcome by a hybrid-TDDFT approach. We show that the key
resides in the numerical treatment of the long-range Coulomb singular term. We
carefully study the effect of this term, both in the pure-TDDFT and hybrid
approach using a Wigner-Seitz truncated kernel. We find that computing this
term presents technical difficulties that are hard to overcome in both
approaches, and that points to the need for a better description of the
electron-hole interaction.
