Local free energy in the paraelectric phase of barium titanate

A suitable definition for the local potential felt by local modes in the paraelectric phase of barium titanate is proposed. It is defined as the potential of mean force (free energy) acting on a given local mode u. This free energy F(u) is computed by constrained molecular dynamics coupled to the method of the thermodynamic integration. It is shown that this free-energy landscape is directly related to the density of probability of the local modes Pi(u) by Pi(u)=Pi(0)e(0)(-[F(u)-F)]/k(B)T. From T(c) to approximate to T(c)+400 K, the free-energy landscape obtained exhibits eight minima along the < 111 > directions, confirming the existence of off-center dipoles in the paraelectric phase of barium titanate in this range of temperature. Only above this temperature, the free energy has a single minimum for u=0. An analytical expression of F(u) is provided under the form of a fourth-order polynomial function in u(alpha).