QSym2: A Quantum Symbolic Symmetry Analysis Program for Electronic Structure

Abstract

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.

Figure 1

Equivalence between systems in external fields and systems with fictitious special atoms. (a) A single fictitious special atom of type e is placed at to represent a uniform electric field. (b) Two fictitious special atoms, one of type b+ and the other of type b–, are placed at R_com ± kB to represent a uniform magnetic field.

Figure 2

Two special cases involving a uniform external field where the original Beruski–Vidal algorithm needs to be modified. (a) A tetrahedral adamantane molecule placed in a uniform external magnetic field oriented along one of its C₃ axes. This illustrates a possible scenario in which a polyhedral SEA group (the six carbon atoms highlighted in orange) is arranged in a spherical top fashion (a regular octahedron). (b) A tetrahedral methane molecule placed in a uniform external magnetic or electric field oriented simultaneously parallel to one of the molecular mirror planes and perpendicular to another. This illustrates a possible scenario of the unitary group arising from the symmetric top rotational symmetry.

Figure 3

High-symmetry molecules whose ground-state KS MOs exhibit degeneracy.

Figure 4

Isosurface plot of the virtual canonical MO χ₁₉₅^α at |χ₁₉₅^α(r)| = 0.009 in C₆₀ calculated at the CAM-B3LYP/6-31+G* level of theory using an -symmetrized geometry (see Table 2). This MO has and symmetry.

Figure 5

Contours of the Pipek–Mezey-localized MO χ₃₄^β of solution A in the yz-plane. The inequivalence between the interactions of the (CN)⁻ π orbitals in the y- and z-directions with the Fe-based d_yz orbital accounts for the T_1g ⊕T_2g symmetry breaking of this MO.

Figure 6

Simplistic depiction of the MOs in hydrogen fluoride at zero field.

Figure 7

(a) Energy landscapes of the frontier m_s = +1/2 MOs in hydrogen fluoride at various magnetic field strengths and angles. (b) Cross sections through these landscapes at parallel (top) and perpendicular (bottom) field orientations. Annotated on these cross sections are the MO symmetries and isosurfaces plotted at |χ(r)| = 0.100. The color at each point r on an isosurface indicates the value of arg χ(r) at that point according to the accompanying color wheel. The numerical value next to each isosurface gives the value of the orbital electronic dipole moment ⟨χ|μ̂_z|χ⟩ for the associated MO. In , the one-dimensional irreducible representation Γ₁ has character function χ^Γ₁[Ĉ_∞(ϕ)] = exp(iϕ), and the corresponding complex-conjugate one-dimensional irreducible representation Γ̅₁ has character function χ^Γ̅₁[Ĉ_∞(ϕ)] = exp(−iϕ), where Ĉ_∞(ϕ) denotes an anticlockwise rotation through an angle ϕ as viewed down the z-axis.