Efficient algorithm for many-electron angular momentum and spin diagonalization on atomic subshells

We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the n-fold antisymmetrized tensor product wedgeⁿ V_u, where n is the number of electrons and u = s, p, d,dots denotes the atomic subshell. This is an essential step for dimension reduction in configuration-interaction (CI) methods applied to atomic many-electron quantum systems. The algorithm relies on the observation that each L_z eigenstate with maximal eigenvalue is also an boldsymbol{L}² eigenstate equivalently for $S_z$ and $boldsymbol{S}²$, as well as the traversal of LS eigenstates using the lowering operators L_- and S_-. Iterative application to the remaining states in wedgeⁿ V_u leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed n and increasing subshell number u yields run time mathcal{O}\(u^3n-2\). A symbolic computer algebra implementation is available online.