Atomic Shell Theory Recast

Abstract

Conventional shell theory for the electronic configurations $l^N$ rests on the separation of the spin and orbital spaces according to the scheme $U(4l+2)\mathrm{}\supset \mathrm{}\left(2\right)\mathrm{}\times U(2l+1)\mathrm{}$. If we are prepared to abandon the total spin quantum number $S$, the more symmetrical reduction $U(4l+2)\mathrm{}\supset U(2l+1)\mathrm{}\times U(2l+1)\mathrm{}$ can be adopted, corresponding to the separation of electrons whose spins point up from those whose spins point down. The orbital structure of a representation [$\lambda$] of $U(2l+1)\mathrm{}$ can be carried over from ordinary shell theory; but the quantum numbers $L_A$ and $L_B$ in a typical coupled state $\left|\right(\left[{\lambda }_A\right]L_A\times \left[{\lambda }_B\right]L_B)L{M}_L{M}_S〉$ are only pseudo-orbital. The existence of two distinct spaces, each of dimension ($2l+1$), leads to many simplifications in the theory. States of the $f$ shell can be classified unambiguously. The spin-orbit splitting of terms near the Russell-Saunders limit is easy to calculate; no fractional parentage coefficients are necessary. What is especially remarkable is that $L_A$ and $L_B$ are sometimes quite good quantum numbers, and this makes it possible to give simple explanations for many regularities in the energy-level patterns for configurations of the type $f^N$.