  We present a theory of electronic excitation energies and optical absorption
spectra which incorporates energy-level renormalization and phonon-assisted
optical absorption within a unified framework. Using time-independent
perturbation theory we show how the standard approaches for studying vibronic
effects in molecules and those for addressing electron-phonon interactions in
solids correspond to slightly different choices for the non-interacting
Hamiltonian. Our present approach naturally leads to the Allen-Heine theory of
temperature-dependent energy levels, the Franck-Condon principle, the
Herzberg-Teller effect and to phonon-assisted optical absorption in indirect
band gap materials. In addition, our theory predicts sub-gap phonon-assisted
optical absorption in direct gap materials, as well as an exponential edge
which we tentatively assign to the Urbach tail. We also consider a
semiclassical approach to the calculation of optical absorption spectra which
simultaneously captures energy-level renormalization and phonon-assisted
transitions and is especially suited to first-principles electronic structure
calculations. We demonstrate this approach by calculating the phonon-assisted
optical absorption spectrum of bulk silicon.
