  Consequences of anisotropy (variation of orbital occupation) and magnetism,
and their coupling, are analyzed for LSDA+U functionals, 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
(LDA) or local spin density (LSDA) approximations. 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 in
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, 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.
