  An effective Hamiltonian for the ferroelectric transition in PbTiO3 is
constructed from first-principles density-functional-theory total-energy and
linear-response calculations through the use of a localized, symmetrized basis
set of ``lattice Wannier functions.'' Explicit parametrization of the polar
lattice Wannier functions is used for subspace projection, addressing the
issues of LO-TO splitting and coupling to the complementary subspace. In
contrast with ferroelectric BaTiO3 and KNbO3, we find significant involvement
of the Pb atom in the lattice instability. Monte Carlo simulations for this
Hamiltonian show a first-order cubic-tetragonal transition at 660 K. Resulting
temperature dependence of spontaneous polarization, c/a ratio and unit-cell
volume near the transition are in good agreement with experiment. Comparison of
Monte Carlo results with mean field theory analysis shows that both strain and
fluctuations are necessary to produce the first order character of this
transition.
