QSym²: A Quantum Symbolic Symmetry Analysis Program for Electronic Structure

Symmetry provides a powerful machinery to classify, interpret, and understand quantum-mechanical theories and results. However, most contemporary quantum chemistry packages lack the ability to handle degeneracy and symmetry breaking effects, especially in non-Abelian groups, and they are not able to characterize symmetry in the presence of external magnetic or electric fields. In this article, a program written in Rust entitled QSym² that makes use of group and representation theories to provide symmetry analysis for a wide range of quantum-chemical calculations is introduced. With its ability to generate character tables symbolically on-the-fly and by making use of a generic symmetry-orbit-based representation analysis method formulated in this work, QSym² is able to address all of these shortcomings. To illustrate these capabilities of QSym², four sets of case studies are examined in detail in this article: (i) high-symmetry C₈₄H₆₄, C₆₀, and B₉^- to demonstrate the analysis of degenerate molecular orbitals (MOs); (ii) octahedral Fe(CN)₆^3- to demonstrate the analysis of symmetry-broken determinants and MOs; (iii) linear hydrogen fluoride in a magnetic field to demonstrate the analysis of magnetic symmetry; and (iv) equilateral H₃⁺ to demonstrate the analysis of density symmetries.