  Recently, Zacharias et al [Phys. Rev. Lett. 115, 177401 (2015)] developed a
new ab initio theory of temperature-dependent optical absorption spectra and
band gaps in semiconductors and insulators. In that work the zero-point
renormalization and the temperature dependence were obtained by sampling the
nuclear wavefunctions using a stochastic approach. In the present work, we show
that the stochastic sampling can be replaced by fully deterministic supercell
calculations based on a single optimal configuration of the atomic positions.
We demonstrate that a single calculation is able to capture the
temperature-dependent band gap renormalization including quantum nuclear
effects in direct and indirect-gap semiconductors, as well as phonon-assisted
optical absorption in indirect-gap semiconductors. In order to demonstrate this
methodology we calculate from first principles the temperature-dependent
optical absorption spectra and the renormalization of direct and indirect band
gaps in Si, C, and GaAs, and we obtain good agreement with experiment and with
previous calculations. In this work we also establish the formal connection
between the Williams-Lax theory of optical transitions and the related theories
of indirect absorption by Hall, Bardeen, and Blatt, and of
temperature-dependent band structures by Allen and Heine. Furthermore, we
identify an additional band gap renormalization that arises in the case of
degenerate band extrema, and which has not been considered in previous work.
The present methodology enables systematic ab initio calculations of optical
absorption spectra at finite temperature, including both direct and indirect
transitions. This feature will be useful for high-throughput calculations of
optical properties at finite temperature, and for calculating
temperature-dependent optical properties using high-level theories such as GW
and Bethe-Salpeter approaches.
