Spin-polarized Energy Density Method from Spin-Density Functional Theory

The energy density method is generalized to include spin polarization with the full formalism derived based on spin-density functional theory, which aims at decomposing the total energy into well-defined atomic energies. The method involves two steps: (1) decomposing the total energy into spin-polarized energy density functions in real space, and (2) integrating these energy densities over chosen gauge-invariant volumes for uniquely defined atomic energies, whose summation over all the atoms restores the DFT total energy up to a constant difference. This method is numerically implemented into the Vienna ab initio simulation package for the projector augmented-wave method, and is showcased with two applications. In the first application, we model the paramagnetic face-centered cubic Fe using spin special quasirandom structures; the spin energies are fit to spin cluster expansions and a deep neural network. In the second application, we calculate the atomic energy distributions of dilute magnetic semiconductor Ni-doped GaN with different dopant distances and spin configurations. This method extracts additional useful information for the study of magnetic systems with density functional theory.