Analytic representation of multi-ion interatomic potentials in transition metals

The first-principles, density-functional version of the generalized pseudopotential theory (GPT), previously developed for empty- and filled-d-band metals, recently has been extended to pure transition metals with partially filled d bands Phys. Rev. B 38, 3199 (1988). Within this formalism, a rigorous real-space expansion of the bulk total energy has been obtained in terms of widely transferable, structure-independent interatomic potentials, including both central-force pair interactions and angular-force triplet and quadruplet interactions. In the central transition metals, the three- and four-ion potentials, v₃ and v₄, are essential to a proper description of materials properties, but are necessarily multidimensional functions which cannot be easily tabulated for application purposes. We develop here a simplified version of the theory, the model GPT, in which these potentials can be expressed analytically while retaining the most important physics of the full first-principles treatment. The analytic treatment of v₃ and v₄ is made possible because of three simplifying features in the central transition metals. First, due to the nonspherical nature of the Fermi surface in such metals, the long-range sp-d hybridization tails of the first-principles potentials destructively interfere in total-energy calculations and thus can be dropped at the outset without major consequences. Second, the direct d-d contributions to the potentials are short ranged and need only be retained to fourth order in interatomic d-state matrix elements to obtain a good representation of the d-band-structure energy. Third, the d bands are canonical in nature, with the interatomic matrix elements well approximated by simple forms, so that all remaining low-order d-state contributions can be evaluated analytically. This leads to a description of v₃ and v₄ in terms of universal short-range radial and angular functions. The model GPT is made quantitatively accurate for real materials by allowing the coefficient of each d-state contribution to be adjusted to match first-principles calculations and/or experimental data. In this manner, one can achieve a set of potentials which simultaneously yield a good description of cohesion, vacancy formation, structural phase stability, elastic constants, and phonons, as is demonstrated for the representative case of molybdenum. More generally, the analytic potentials are suitable for widescale applications and permit for the first time the use of the transition-metal GPT in molecular-dynamics and Monte Carlo simulations.