Using a combination of first-principles and effective-Hamiltonian approaches, we map out the structure of BaTiO3 under epitaxial constraints applicable to growth on perovskite substrates. We obtain a phase diagram in temperature and misfit strain that is qualitatively different from that reported by Pertsev et al. [Phys. Rev. Lett. 80, 1988], who based their results on an empirical thermodynamic potential with parameters fitted at temperatures in the vicinity of the bulk phase transitions. In particular, we find a region of 'r phase' at low temperature where Pertsev et al. have reported an 'ac phase'. We expect our results to be relevant to thin epitaxial films of BaTiO3 at low temperatures and experimentally-achievable strains. PACS numbers: 77.55.+f, 77.80.Bh, 77.84.Dy, 81.05.Zx The perovskite oxide barium titanate (BaTiO 3 ) is a prototypical ferroelectric, an insulating solid whose macroscopic polarization can be reoriented by the application of an electric field [1]. In the perovskite ferroelectrics, it is well known both experimentally and theoretically that the polarization is also strongly coupled to strain [2], and thus that properties such as the ferroelectric transition temperature and polarization magnitude are quite sensitive to external stress.Experimentally, the properties of ferroelectrics in thin film form generally differ significantly from those in the bulk [3]. While many factors are expected to contribute to these differences, it has been shown that the properties of perovskite thin films are strongly influenced by the magnitude of the epitaxial strain resulting from latticematching the film to the substrate. For example, Yoneda et al.[4] used molecular-beam epitaxy (MBE) to grow BaTiO 3 (lattice constant of 4.00Å) on (001)-oriented SrTiO 3 (lattice constant of 3.91Å); they found that the ferroelectric transition temperature exceeds 600 • C, to be compared to the bulk Curie temperature of T C = 130 • C. Other studies have shown that the amount of strain in BaTiO 3 /SrTiO 3 superlattices on SrTiO 3 substrates strongly influences properties including the observed polarization, phase transition temperature, and dielectric constant [5,6,7,8].In a seminal paper, Pertsev, Zembilgotov and Tagantsev [9] introduced the concept of mapping the equilibrium structure of a ferroelectric perovskite material versus temperature and misfit strain, thus producing a "Pertsev phase diagram" (or Pertsev diagram) of the observable epitaxial phases. The effect of epitaxial strain is isolated from other aspects of thin-film geometry by computing the structure of the bulk material with homogeneous strain tensor constrained to match a given substrate with square surface symmetry [10]. In addition, short-circuit electrical boundary conditions are imposed, equivalent to ideal electrodes above and beneath the film [9]. Given the recognized importance of strain in determining the properties of thin-film ferroelectrics, Pertsev diagrams have proven to be of enormous interest to experimentalists seeking to interpret the ...