Valence-dependent analytic bond-order potential for magnetic transition metals

We extend the analytic bond-order potentials for transition metals [Phys. Rev. B 74, 174117 (2006)] to include ferro, antiferro, and noncollinear magnetism and charge transfer. This is achieved by first deriving a suitable tight-binding model through the expansion of the spin-density energy functional to second order with respect to magnetic and charge fluctuations. The tight-binding model is then approximated locally by the bond-order potential expansion, where the variational property of the bond-order potential expansion allows us to derive analytic expressions for the forces and torques on the atoms. From the bond-order potentials we then extract a hierarchy of multispin interactions beyond the conventional Heisenberg model. The explicit valence dependence of the bond-order potentials enables us to characterize the magnetic properties of the 3$d$ transition metals and to reproduce the trend from antiferromagnetic spin ordering close to the center of the $d$ band through noncollinear spin configurations to ferromagnetic ordering toward the edges of the $d$ band. The analytic representation of the energy within the bond-order potentials is then further expanded in the form of a Ginzburg-Landau expansion, deriving the prefactors explicitly from tight-binding and bond-order potentials. Thus, in this paper we present a coherent simplification from fundamental to empirical models of magnetism through coarse graining the electronic structure from spin-density functional theory to tight binding to bond-order potentials to the Ginzburg-Landau expansion.