Anisotropy and magnetism in the LSDA plus U method

Consequences of anisotropy (variation in orbital occupation) and magnetism, and their coupling, are analyzed for local-spin-density approximation (LSDA) plus interaction term U (LSDA+U) functionals, with both the commonly used ones as well as less commonly applied functionals. After reviewing and extending some earlier observations for an isotropic interaction, the anisotropies are examined more fully and related to use with the local-density approximation or with the LSDA. The total energies of all possible integer configurations of an open f shell are presented for three functionals, where some differences are found to be dramatic. Differences between how the commonly used "around mean-field" (AMF) and "fully localized limit" (FLL) functionals perform are traced to such differences. The LSDA+U interaction term, when applied self-consistently, usually enhances spin magnetic moments and orbital polarization, and the double-counting terms of both functionals provide an opposing moderating tendency ("suppressing the magnetic moment"). The AMF double-counting term gives magnetic states a significantly larger energy penalty than does the FLL counterpart.