We discuss qualitatively the importance of the correlation energy in determining the ground state of a metal with an impurity atom. For a single, partly occupied impurity $d$-state orbital, the correlation energy acts to prevent the appearance of a nonvanishing ground-state spin, so that this simple nondegenerate model actually has a complicated structure. In one dimension, we show that this model of an impurity can never lead to a localized moment. In three dimensions, if we take linear combinations of Bloch functions transforming according to the irreducible representations of the point group of the impurity+crystal, we find that most of the new wave functions are entirely decoupled from the impurity, and only a small subset interacts with it. The noninteracting majority of states determine the Fermi level, which we therefore take to be fixed. The ground state of the band states interacting with the impurity states depends on the two-body Coulomb repulsion $U$, and we find that for sufficiently small $U$ the ground state has an even number of electrons with total spin $S=0\mathrm{}$. As $U$ is increased above a certain critical value, the ground state of the interacting subsystem changes to an odd number of electrons, having total spin $S=\frac{1}{2}$, and a localized moment is said to exist. The introduction of orbital degeneracy for the impurity $d$ state, and of Hund's rule matrix elements, makes the localized moment much stabler. The results are obtained by a combination of exact energy-level ordering theorems and a Green's-function calculation in the $t$-matrix approximation.

• Received 14 June 1965

DOI:https://doi.org/10.1103/PhysRev.140.A1412