# Symmetrization of Localized Molecular Orbitals

## TL;DR

This paper introduces a new algorithm to detect and enforce symmetries in localized molecular orbitals, significantly reducing the number of orbitals needed and enhancing computational efficiency in molecular simulations.

## Contribution

The paper presents a novel algorithm for symmetry detection and enforcement in localized orbitals, enabling substantial orbital compression and improved exploitation of molecular symmetries.

## Key findings

- Foster-Boys orbitals can be spanned by fewer symmetry-unique orbitals.
- The algorithm identified only 14 symmetry-unique orbitals for buckminsterfullerene.
- Potential for unprecedented speed-ups in local correlation methods.

## Abstract

We present a novel algorithm for (i) detecting approximate symmetries inherently present among spatially localized molecular orbitals and (ii) enforcing these in numerically exact manners by means of unitary optimization techniques. The vast potential of our algorithm to compress a full set of molecular orbitals into only a minimal set of symmetry-unique orbitals is demonstrated, starting from localized bases of either Pipek-Mezey or Foster-Boys orbitals. Comparisons of results based on either of these two localization procedures indicate that Foster-Boys molecular orbitals can be spanned by a smaller number of symmetry-unique orbitals on average, making these outstanding candidates for the exploitation of general, (non-)Abelian point-group symmetries in a range of local correlation methods. As an illustration of said compressibility, our algorithm is capable of identifying a mere 14 symmetry-unique orbitals for the buckminsterfullerene in the highly symmetric $I_h$ molecular point group, corresponding to only $1.7\%$ of its total 840 molecular orbitals in a standard double-$\zeta$ basis set. The present work thus marks an important advancement in the exploitation of point-group symmetry within local correlation methods, for which the appropriate adaption of symmetry uniqueness among orbitals has the potential to yield unprecedented speed-ups.