  Calculations of electronic and optical properties of solids at finite
temperature including electron-phonon interactions and quantum zero-point
renormalization have enjoyed considerable progress during the past few years.
Among the emerging methodologies in this area, we recently proposed a new
approach to compute optical spectra at finite temperature including
phonon-assisted quantum processes via a single supercell calculation [Zacharias
and Giustino, Phys. Rev. B 94, 075125 (2016)]. In the present work we
considerably expand the scope of our previous theory starting from a compact
reciprocal space formulation, and we demonstrate that this improved approach
provides accurate temperature-dependent band structures in three-dimensional
and two-dimensional materials, using a special set of atomic displacements in a
single supercell calculation. We also demonstrate that our special displacement
reproduces the thermal ellipsoids obtained from X-ray crystallography, and
yields accurate thermal averages of the mean-square atomic displacements. At a
more fundamental level, we show that the special displacement represents an
exact single-point approximant of an imaginary-time Feynman's path integral for
the lattice dynamics. This enhanced version of the special displacement method
enables non-perturbative, robust, and straightforward ab initio calculations of
the electronic and optical properties of solids at finite temperature, and can
easily be used as a post-processing step to any electronic structure code.
Given its simplicity and numerical stability, the present development is suited
for high-throughput calculations of band structures, quasiparticle corrections,
optical spectra, and transport coefficients at finite temperature.
