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

Influence of Fr\"ohlich polaron coupling on renormalized electron bands in polar semiconductors. Results for zincblende GaN