  \ni We develop a simple method to study the zero-point and thermally
renormalized electron energy $\varepsilon_{\mathbf{k}n}(T)$ for $\mathbf{k}n$
the conduction band minimum or valence maximum in polar semiconductors. We use
the adiabatic approximation, including an imaginary broadening parameter
$i\delta$ to supress noise in the density-functional integrations. Fr\"{o}hlich
polaron methods provide analytic expressions for the contribution of the
problematic optical phonon mode. We use this to correct the renormalization
obtained from the adiabatic approximation. Test calculations are done for
zincblende GaN for an 18x18x18 integration grid. The Fr\"ohlich correction is
of order -0.02 eV for the zero-point energy shift of the conduction band
minimum, and +0.03 eV for the valence band maximum; the correction to
renormalization of the 3.28 eV gap is -0.05 eV, a significant fraction of the
total zero point renormalization of -0.15 eV.
