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In polar insulators where longitudinal and transverse optical phonon modesdiffer substantially, the electron-phonon coupling affects the energy-bandstructure primarily through the long-range Fr\"ohlich contribution to the Fanterm. This diagram has the same structure as the $GW$ self-energy where $W$originates from the electron part of the screened coulomb interaction. The twocan be conveniently combined by combining electron and lattice contributions tothe polarizability. Both contributions are nonanalytic at the origin, anddiverge as $1/q^2$ so that the predominant contribution comes from a smallregion around $q{=}0$. Here we adopt a simple estimate for the Fr\"ohlichcontribution by assuming that the entire phonon part can be attributed to asmall volume of $q$ near $q{=}0$. We estimate the magnitude for$\mathbf{q}{\rightarrow}0$ from a generalized Lyddane-Sachs-Teller relation,and the radius from the inverse of the polaron length scale. The gap correctionis shown to agree with Fr\"ohlich's simple estimate $-\alpha_P\omega_L/2$ ofthe polaron effect.

Lattice polarization effects on the screened Coulomb interaction $W$ of the GW approximation