Landau free energy of ferroelectric crystals by thermodynamic integration

Using Molecular Dynamics simulations based on the effective hamiltonian developed by Zhong, Vanderbilt and Rabe [Phys. Rev. Lett. {\bf 73}, 1861 (1994)] (and fitted on first-principles calculations only), the technique of the thermodynamic integration is applied to barium titanate. It allows to compute the difference of free energy between macroscopic states with different polarizations, from the thermal averages of the forces acting on the local modes. This is achieved by performing molecular dynamics under the constraint of fixed polarization. The Landau free energy is thus interpreted as a potential of mean force. The thermodynamic integration directly gives access (numerically) to the Landau free energy of barium titanate as a function of $\vec P$, without any assumption on its analytical form. This technique, mainly used in computational chemistry, allows to make a direct connection between phenomenological theories and atomic-scale simulations. It is applied and validated to the case of BaTiO$_3$ under fixed volume. The results are compared to available phenomenological potentials of the litterature.