<i>Ab initio</i> elasticity at finite temperature and stress in ferroelectrics

Computing the temperature and stress dependence of the elastic constants from first principles in noncubic materials remains a challenging problem. Here, we circumvent the aforementioned challenge via the generalized quasiharmonic approximation (gQHA) with the irreducible derivative approach for computing strain-dependent phonons using finite difference, explicitly including dipole-quadrupole contributions. We showcase the gQHA in ferroelectric ${\mathrm{PbTiO}}_{3}$, computing all elastic constants and piezoelectric strain coefficients at finite temperature and stress. The gQHA overestimates the temperature dependence of the lattice parameters and elastic constant tensor, demonstrating the need for an explicit treatment of lattice anharmonicity as a function of strain.