  In polar insulators where longitudinal and transverse optical phonon modes
differ substantially, the electron-phonon coupling affects the energy-band
structure primarily through the long-range Fr\"ohlich contribution to the Fan
term. This diagram has the same structure as the $GW$ self-energy where $W$
originates from the electron part of the screened coulomb interaction. The two
can be conveniently combined by combining electron and lattice contributions to
the polarizability. Both contributions are nonanalytic at the origin, and
diverge as $1/q^2$ so that the predominant contribution comes from a small
region around $q{=}0$. Here we adopt a simple estimate for the Fr\"ohlich
contribution by assuming that the entire phonon part can be attributed to a
small 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 correction
is shown to agree with Fr\"ohlich's simple estimate $-\alpha_P\omega_L/2$ of
the polaron effect.
