Valence-dependent analytic bond-order potential for transition metals

An analytic interatomic bond-order potential is derived that depends explicitly on the valence of the transition-metal element. It generalizes the second-moment Finnis-Sinclair and fourth-moment Carlsson potentials to include higher moments. We find that the sixth-moment approximation predicts not only the structural trend from hcp→bcc→hcp→fcc that is observed across the nonmagnetic $4d$ and $5d$ transition-metal series, but also the different ferromagnetic moments of the bcc, fcc, and hcp phases of the $3d$ transition-metal iron. An analytic expression for the force is obtained and proved to converge to the Hellmann-Feynman force as higher moments are included.