Anisotropic temperature-dependent lattice parameters and elastic constants from first principles

We present an efficient implementation of the Zero Static Internal Stress Approximation (ZSISA) within the Quasi-Harmonic Approximation framework to compute anisotropic thermal expansion and elastic constants from first principles. By replacing the costly multidimensional minimization with a gradient-based method that leverages second-order derivatives of the vibrational free energy, the number of required phonon band structure calculations is significantly reduced: only six are needed for hexagonal, trigonal, and tetragonal systems, and 10–28 for lower-symmetry systems to determine the temperature dependence of lattice parameters and thermal expansion. This approach enables accurate modeling of anisotropic thermal expansion while substantially lowering computational cost compared to standard ZSISA method. The implementation is validated on a range of materials with symmetries from cubic to triclinic and is extended to compute temperature-dependent elastic constants with only a few additional phonon band structure calculations.