Lattice polarization contribution in the quasi-particle self-consistent GW approach.

The lattice polarization contribution to the screening of the screened Coulomb interaction $W$ of the $GW$ approach should in principle be included and is expected to occur only in the limit ${\bf q}\rightarrow 0$ and for frequencies well below the gap. This effect has been proposed by Botti and Marques [PRL,{\bf 110}, 226404(2013)] to be important for large gap insulators with strong-LO-TO splitting. We have implemented this effect in the QSGW approach using a muffin-tin-orbital basis set, for crystals of arbitrary symmetry. The ${\bf q}\rightarrow0$ contribution to the self-energy requires a careful treatment because of the integrable divergence of the Coulomb interaction, which is here done by means of the offset-$\Gamma$ approach. In the limit ${\bf q}\rightarrow0$, the ratio of the total (lattice + electronic) macroscopic dielectric constant to the electronic contribution only can be represented by a generalized Lyddane-Sachs-Teller relation. We will discuss results for ionic crystals, like MgO, NaCl, and others with large LO-TO splittings SrTiO$_3$ and V$_2$O$_5$. We find the effect to be somewhat smaller than previously proposed and sensitive to the q-point sampling. The question to be addressed is how fast this effect decays as function of $q$ near $q=0$.