Quasi-Particle Groups in Atomic Shell Theory

Abstract

In 1968 some bizarre features of atomic shell theory were noticed. One such appeared when efforts were made to generalize the use of G₂ for atomic electrons. Racah had demonstrated the great utility of this exceptional Lie group in his classic paper on the Coulomb energies off electrons,¹ but it was known that no analogous group existed for electrons with higher ℓ. However, the G₂ classification of the states of the f shell could be accomplished by diagonalizing a two-electron operator belonging to the irreducible representation (irrep) (111) of S0(7). It can be shown that such an operator is necessarily a scalar with respect to G₂ too. For g electrons we can form an operator that is an S0(3) scalar and which also belongs to (1111) of S0(9). As such, it is an analog of the G₂ scalar although no actual analog of the group itself exists. When the operator was diagonalized for the states of maximum multiplicity (maximum S) of g^N, many eigenvalue repetitions were noticed.² Moreover, several fractional parentage coefficients vanished for no obvious reason. Armstrong found that corresponding simplifications occurred in the h shell.³