Quantitative analysis of the first-principles effective Hamiltonian approach to ferroelectric perovskites

The various approximations used in the construction of a first-principles effective Hamiltonian for ${\mathrm{BaTiO}}_3,$ and their effects on the calculated transition temperatures, are discussed. An effective Hamiltonian for ${\mathrm{BaTiO}}_3$ is constructed not from first-principles calculations, but from the structural energetics of an atomistic shell model for ${\mathrm{BaTiO}}_3$ of Tinte et al. This allows the elimination of certain uncontrolled approximations that arise in the comparison of first-principles effective Hamiltonian results with experimental values and the quantification of errors associated with the selection of the effective Hamiltonian subspace and subsequent projection. The discrepancies in transition temperatures computed in classical simulations for this effective Hamiltonian and for the atomistic shell model are shown to be associated primarily with a poor description of the thermal expansion in the former case. This leads to specific proposals for refinements to the first-principles effective Hamiltonian method. Our results suggest that there are at least two significant sources of error in the effective-Hamiltonian treatment of ${\mathrm{BaTiO}}_3$ in the literature, i.e., the improper treatment of thermal expansion and the errors inherent in the first-principles approach itself.