Anisotropy and magnetism in the $\text{LSDA}+\text{U}$ 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 $\left(\text{LSDA}+\text{U}\right)$ 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 $\text{LSDA}+\text{U}$ 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.

Figure 1

(Color online) Angular momentum values of $S_z$, $L_z$, and $J_z$ of the lowest-energy state for (a) AMF (Fl-S) and (b) FLL, with spin-orbit coupling. Parameter values are $U=8$, $J=1$, and $I=0.75$. The AMF (Fl-S) curves do not follow Hund’s rules because the Stoner parameter is too small. FLL follows Hund’s rules exactly with these parameters.

Figure 2

(Color online) Shown here is $\Delta E^{\text{FLL}}$ plotted vs $\Delta E^{\text{Fl-S}}$ for each of the 3432 configurations of $N=7$ electrons using $U=8,J=1,I=0.75$, all in eV. The ordering of states is shown for Fl-S by counting from left to right and for FLL by counting from bottom to top. Open squares show values for $U=7$ and $J=0$.

Figure 3

(Color online) Scatter plot of all energies $\Delta E$ for all states in the (a) AMF (Fl-S), (b) FLL, and (c) Fl-nS double-counting schemes for $U=8$, $J=1$, and $I=0.75$ (FLL and AMF only). Spin-orbit is neglected here. For AMF, low-spin states (black and red circles) appear as lowest-energy configurations for all $N$, but this is not the case for FLL or Fl-nS. The dashed lines indicate the mean energy over configuration for each $N$; note that the variation with $N$ is much less for FLL than for the other two functionals.

Figure 4

(Color online) Shell-Stoner integrals for the $3d$ and $4f$ atoms. For explanations, see text.