  The electron-phonon interaction contribution to the electronic energies is
included in density functional total energy calculations with ab initio
pseudopotentials via the Allen formalism [Phys. Rev. B 18, 5217 (1978)] to
obtain temperature dependent electronic structure of diamond and silicon. This
method allows us to obtain the thermally-averaged ab initio electronic
structure in a straightforward and computationally inexpensive way. Our
investigations on the finite temperature electronic structure of diamond and
silicon lead to a new criterion, temperature transferability, which is required
in the ab initio pseudopotentials for temperature dependent studies. The
temperature transferability of the Troullier-Martins pseudopotentials used in
this work is strongly dependent on the cut-off radius and the inclusion of the
unbound 3d$^0$ state. The finite temperature indirect band gaps are highly
sensitive to the choice of cut-off radius used in the pseudopotentials. The
finite temperature band structures and density of states show that thermal
vibrations affect the electron energies throughout the valence and conduction
band. We compare our results on the band gap shifts with that due to the
Debye-Waller term in the Allen-Heine theory and discuss the observed
differences in the zero point and high temperature band gap shifts. Although,
the electron energy shifts in the highest occupied valence band and lowest
unoccupied conduction band enable to obtain the changes in the indirect and
direct band gaps at finite temperatures, the shifts in other electronic levels
with temperature enable investigations into the finite temperature valence
charge distribution in the bonding region. Thus, we demonstrate that the Allen
theory provides a simple and theoretically justified formalism to obtain finite
temperature valence electron charge densities that go beyond the rigid
pseudo-atom approximation.
