Static and dynamic properties of PbTiO$_3$ at finite temperatures*

The ABO$_3$-type perovskite crystals are key to several important technological applications. To mention a few, electro-optics, waveguides, laser frequency doubling and high capacity computer memory cells. In this work, we develop a route to first-principles parametrization of effective Hamiltonian for ferroelectric ferovskites [1] which allows an accurate description of both static and dynamic properties of such materials. We use this method to examine softening of the transverse optical mode in both paraelectric and ferroelectric phases of PbTiO$_3$. The computed static and dynamic properties are in good agreement with the available theoretical and experimental data. Our study also predicts a crossover between a displacive to an order-disorder transition near the Curie point.\\[4pt] [1] W.~Zhong, D.~Vanderbilt, and K.~M.~Rabe, Phys. Rev. B {\bf 52}, 6301 (1995).