Understanding the valency of rare earths from first-principles theory

The rare-earth metals have high magnetic moments and a diverse range of magnetic structures1. Their magnetic properties are determined by the occupancy of the strongly localized 4f electronic shells, while the outer s–d electrons determine the bonding and other electronic properties2. Most of the rare-earth atoms are divalent, but generally become trivalent in the metallic state. In some materials, the energy difference between these valence states is small and, by changing some external parameter (such as pressure), a transition from one to the other occurs. But the mechanism underlying this transition and the reason for the differing valence states are not well understood. Here we report first-principles electronic-structure calculations that enable us to determine both the valency and the lattice size as a function of atomic number, and hence understand the valence transitions. We find that there are two types of f electrons: localized core-like f electrons that determine the valency, and delocalized band-like f electrons that are formed through hybridization with the s–d bands and which participate in bonding. The latter are found only in the trivalent systems; if their number exceeds a certain threshold, it becomes energetically favourable for these electrons to localize, causing a transition to a divalent ground state.