Ab initio structural optimization at finite temperatures based on anharmonic phonon theory: Application to the structural phase transitions of BaTiO$_3$

We formulate a first-principle scheme for structural optimization at finite temperature ($T$) based on the self-consistent phonon (SCP) theory, which accurately takes into account the effect of strong phonon anharmonicity. The $T$-dependence of the shape of the unit cell and internal atomic configuration is determined by minimizing the variational free energy in the SCP theory. At each optimization step, the interatomic force constants in the new structure are calculated without running additional electronic structure calculations, which makes the method dramatically efficient. We demonstrate that the thermal expansion of silicon and the three-step structural phase transitions in BaTiO$_3$ and its pressure-temperature ($p$-$T$) phase diagram are successfully reproduced. The present formalism will open the way to the non-empirical prediction of physical properties at finite $T$ of materials having a complex structural phase diagram.