Density-functional formulation of the generalized pseudopotential theory. III. Transition-metal interatomic potentials

The first-principles, density-functional version of the generalized pseudopotential theory (GPT) developed in papers I and II of this series Phys. Rev. B 16, 2537 (1977) ; 26, 1754 (1982) for empty- and filled-d-band metals is here extended to pure transition metals with partially filled d bands. The present focus is on a rigorous, real-space expansion of the bulk total energy in terms of widely transferable, structure-independent interatomic potentials, including both central-force pair interactions and angular-force triplet and quadruplet interactions. To accomplish this expansion, a specialized set of starting equations is derived from the basic local-density formalism for a pure metal, including refined expansions for the exchange-correlation terms and a simplified yet accurate representation of the cohesive energy. The parent pseudo-Green's-function formalism of the GPT is then used to develop these equations in a plane-wave, localized-d-state basis. In this basis, the cohesive energy divides quite naturally into a large volume component and a smaller structural component. The volume component, which includes all one-ion intra-atomic energy contributions, already gives a good description of the cohesion in lowest order. The structural component is expanded in terms of weak interatomic matrix elements and gives rise to a multi-ion series which establishes the interatomic potentials. Special attention is focused on the dominant d-electron contributions to this series and complete formal results for the two-ion, three-ion, and four-ion d-state potentials (v₂^d, v₃^d, and v₄^d) are derived. In addition, a simplified model is used to demonstrate that while v₃^d can be of comparable importance to v₂^d, v₄^d is inherently small and the series is rapidly convergent beyond three-ion interactions. Analytic model forms are also derived for v₂^d and v₃^d in the case of canonical d bands. In this limit, v₂^d is purely attractive and varies with interatomic distance as r^-10, while v₃^d is weak and attractive for almost empty or filled d bands and maximum in strength and repulsive for half-filled d bands. Full first-principles expressions are then developed for the total two-ion and three-ion potentials and implemented for all 20 3d and 4d transition metals. The first-principles potentials qualitatively display all of the trends predicted by the model results, but they also reflect additional effects, including long-range hybridization tails which must be suitably screened in real-space calculations. Finally, illustrative application of the first-principles potentials is made to the calculation of the 100 phonon spectrum for V and Cr, where the importance of three-ion angular forces is explicitly demonstrated.