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

TL;DR

This paper introduces an efficient symbolic algorithm for calculating angular momentum and spin eigenspaces in atomic many-electron systems, improving the dimension reduction step in quantum chemistry computations.

Contribution

The paper presents a novel algorithm leveraging eigenstate properties and lowering operators for symbolic diagonalization in atomic subshells, with detailed complexity analysis and an available implementation.

Findings

Algorithm achieves polynomial complexity in subshell size.

Implementation available online for practical use.

Enhances efficiency of configuration-interaction methods.

Abstract

We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the $n$-fold antisymmetrized tensor product $\wedge^n V_u$, where $n$ is the number of electrons and $u = \mathrm{s}, \mathrm{p}, \mathrm{d},\dots$ 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 $\wedge^n V_u$ 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.