Hund's rules

Abstract

We review the present state of our undertanding of Hund's first and second rules, their domains of validity, and of generalizations in cases where the rules in their original form are invalid. These exceptions occur mainly in atomic configurations with more than one open shell withl ≥ 1, but also in cases with large orbital angular momentaL. We present a derivation of thealternating rule, which is, in some sense, a generalization of Hund's first rule, and present new rules, which generalize Hund's second rule. The importance of the concept ofunnatural parity states for an understanding of these rules is stressed. It is demonstrated that the lowest singlet-triplet average energy for the sameL corresponds to a pair of unnatural parity states (theunnatural parity rule). For sufficiently smalll 1 andl 2, the lowest average energy is that of the pair of states with the maximum possibleL (maximum-L rule), though exceptions are found already for moderately large values ofl 1 andl 2. Our analysis indicates that the validity of Hund's second rule is to some extent an accidental consequence of the minimisation of an elementary function ofl 1,l 2, andL — which does not depend monotonically onL — over states of unnatural parity, and that a more general and more fundamental rule should be formulated in terms of this function. We also discuss the generalisation of Hund's rules to molecules, as well as violations of them, with particular emphasis on the inversion of Hund's first rule by spin-polarization in molecules.