Lattice constants from semilocal density functionals with zero-point phonon correction

In a standard Kohn-Sham density functional calculation, the total energy of a crystal at zero temperature is evaluated for a perfect static lattice of nuclei and minimized with respect to the lattice constant. Sometimes a zero-point vibrational energy, whose anharmonicity expands the minimizing or equilibrium lattice constant, is included in the calculation or (as here) is used to correct the experimental reference value for the lattice constant to that for a static lattice. A simple model for this correction, based on the Debye and Dugdale-MacDonald approximations, requires as input only readily available parameters of the equation of state, plus the experimental Debye temperature. However, particularly because of the rough Dugdale-MacDonald estimation of Gr\"uneisen parameters for diatomic solids, this simple model is found to overestimate the correction by about a factor of two for some solids in diamond and zinc-blende structures. Using the quasiharmonic phonon frequencies calculated from density functional perturbation theory gives a more accurate zero-point anharmonic expansion (ZPAE) correction. However, the error statistics for the lattice constants of various semilocal density functionals for the exchange--correlation energy are little changed by improving the ZPAE correction. The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) for solids and the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-GGA, the latter of which is implemented self-consistently here in the band-structure program BAND and applied to a test set of 58 solids, remain the most accurate of the functionals tested, with MAREs below 0.7$%$ for the lattice constants. The most positive and most negative revTPSS relative errors tend to occur for solids for which full nonlocality (missing from revTPSS) may be important.