Group Theory for Atomic Shells

Abstract

The basic elements of the theory of Lie groups and their irreducible representations (IRs) are described. The IRs are used to label the states of an atomic shell and also the components of operators of physical interest. Applications of the generalized Wigner-Eckart theorem lead to relations between matrix elements appearing in different electronic configurations. This is particularly useful in the f shell, where transformations among the seven orbital states of an f electron can be described by the unitary group U(7) and its sequential subgroups SO(7), G2, and SO(3) with respective IRs [λ], W, U, and L. Extensions to groups that involve electron spin S (like Sp(14)) are described, as are groups that do not conserve electron number. The most useful of the latter is the quasispin group whose generators Q connect states of identical W, U, L and seniority v in the f shell. The symmetries of products of objects (states or operators) that themselves possess symmetries are described by the technique of plethysms.