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 ^ ⁿ Vu , where n is the number of electrons and u = s,p,d,... 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 Lz eigenstate with maximal eigenvalue is also an L² eigenstate (equivalently for Sz and S ²), as well as the traversal of LS eigenstates using the lowering operators L- and S-. Iterative application to the remaining states in ^ ⁿVu leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed n and increasing subshell number u yields run time O(u ^3n-2). A symbolic computer algebra implementation is available online.