M\"obius-topological auxiliary function for $f$ electrons

$f$-electron systems exhibit a subtle interplay between strong spin--orbit coupling and crystal-field effects, producing complex energy landscapes that are computationally demanding. We introduce auxiliary functions, constructed by extending hydrogen-like wave functions through a modification of the Legendre function. These functions often possess a M\"obius-like topology, satisfying $\psi(\varphi) = -\psi(\varphi + 2\pi)$, while their squared modulus respects inversion symmetry. By aligning $|\psi|^2$ with the symmetry of the crystal field, they allow rapid determination of eigenstate structures without the need for elaborate calculations. The agreement with established results indicates that these functions capture the essential physics while offering considerable computational simplification.