Density-functional theory and atomic multiplet levels

The multiplet splittings resulting from Hund’s rules effects for open-shell atomic $d^n$ configurations are investigated within density-functional theory. The calculated results are compared to experiment, Hartree-Fock, and Slater-Condon-Racah fits. Marked systematic errors are found in the local-spin-density (LSDA) and generalized-gradient (GGA) results arising from the failure of the LSDA/GGA single-particle Hamiltonian to commute with L, $M_L,$ and S, which are good quantum numbers for the exact wave functions. The shortcomings of LSDA/GGA for the middle of the transition-metal row ions are more severe than those seen in previous work that concentrated on open p shells, even for multiplet levels with single-determinant wave functions. A number of issues confronting density-functional-based theories are addressed; in particular, it is demonstrated that any exchange-correlation functional that depends on the charge and spin densities alone is incapable of describing multiplet effects correctly.