  The free energy of barium titanate is computed around the Curie temperature
as a function of polarization from the first-principles derived effective
hamiltonian of Zhong, Vanderbilt and Rabe [Phys. Rev. Lett. 73, 1861 (1994)],
through Molecular Dynamics simulations coupled to the method of the
Thermodynamic Integration. The main feature of this approach is to calculate
the gradient of the free energy in the 3-D space (Px,Py,Pz) from the thermal
averages of the forces acting on the local modes, that are obtained by
Molecular Dynamics under the constraint of fixed P. A careful analysis of the
states of constrained polarization is performed at T=280 K (~ 15-17 K below Tc)
especially at low order parameter. These states are found reasonably
homogeneous for small supercell size (L=12 and L=16), until inhomogeneous
states are observed at low order parameter for large supercells (L=20).
However, for reasonable supercell sizes (L=12), the free energy curves obtained
are in very good agreement with phenomenological Landau potentials of the
litterature. Moreover, the free energy obtained is quite insensitive to the
supercell size from L=12 to L=16 at T=280 K, suggesting that interfacial
contributions, if any, are negligible at these sizes around Tc. The method
allows a numerical estimation of the free energy barrier separating the
paraelectric from the ferroelectric phase at Tc. However, our tests evidence
phase separation at low temperature and low order parameter, in agreement with
the results of Tr\"oster et al [Phys. Rev. B 72 (2005), 094103].
