Approximations in first-principles volumetric thermal expansion determination

In the realm of thermal expansion determination, the quasiharmonic approximation (QHA) stands as a widely embraced method for discerning minima of free energies across diverse temperatures such that the temperature dependence of lattice parameters as well as internal atomic positions can be determined. However, this methodology often imposes substantial computational demand, necessitating numerous costly calculations of full phonon spectra in a possibly many-dimensional geometry parameter space. Focusing on the volumetric thermal expansion only, the volume-constrained zero static internal stress approximation (v-ZSISA) within QHA allows one to limit significantly the number of phonon spectra determinations to typically less than 10. The linear Gr & uuml;neisen approach goes even further with only two phonon spectra determinations to find the volumetric thermal expansion, but a deterioration of the accuracy of the computed thermal expansion is observed, except at low temperatures. We streamline this process by introducing further intermediate approximations between the linear Gr & uuml;neisen and the v-ZSISA-QHA, corresponding to different orders of the Taylor expansion. The minimal number of phonon spectra calculations that is needed to maintain precise outcomes is investigated. The different approximations are tested on a representative set of 12 materials. For the majority of materials, three full phonon spectra, corresponding to quadratic order, is enough to determine the thermal expansion in reasonable agreement with the v-ZSISA-QHA method up to 800 K. Near perfect agreement is obtained with five phonon spectra. This study paves the way to multidimensional generalizations, beyond the volumetric case, with the expectation of much bigger benefits.