[Paper Review] Exploitation of complex Abelian point groups in quantum-chemical calculations

Marios-Petros Kitsaras, Stella Stopkowicz|arXiv (Cornell University)|Feb 12, 2026

Magnetism in coordination complexes1 citations

TL;DR

The paper extends symmetry exploitation to complex Abelian point groups with complex characters in quantum-chemical calculations, enabling efficiency gains in finite-magnetic-field HF and post-HF methods, demonstrated on four small hydrocarbons.

ABSTRACT

Quantum-chemical calculations often make use of point-group theory to exploit molecular symmetry, resulting in a reduction of the computational cost and in insights into the electronic structure. This exploitation is often limited to subgroups of $D_{2h}$ which are Abelian with real characters. Here, we extend the symmetry exploitation to Abelian point groups with complex characters. Such point groups are often encountered in calculations that involve finite magnetic fields, though their occurrence is not limited to these cases alone. We present the evaluation of integrals over symmetry-adapted orbitals using the double-coset decomposition, as well as the use of these symmetries in the contractions needed within post Hartree Fock calculations in the context of block tensors. Efficiency gains are discussed for four simple hydrocarbons that exhibit a complex Abelian point group in the presence of a magnetic field.

Motivation & Objective

• Motivate the use of molecular symmetry to reduce computational cost and gain electronic-structure insight.
• Generalize symmetry exploitation from real Abelian point groups to complex Abelian point groups with complex characters.
• Develop and apply the double-coset decomposition framework for complex Abelian groups in integral evaluations.
• Implement complex Abelian point group symmetry handling in the cfour and qcumbre software packages.
• Demonstrate computational savings in finite-field HF and post-HF calculations through benchmark calculations on representative systems.

Proposed method

• Revisit and adapt the double-coset decomposition (DCD) for complex Abelian point groups to eliminate redundant contributions in integral evaluations.
• Define symmetry-adapted orbitals (SAOs) via projection operators that account for complex characters, including necessary complex conjugation of representations.
• Use stabilizers of atomic centers to construct left and right subgroups for efficient DCD factorization in one- and two-electron integrals.
• Incorporate complex-valued irreducible representations into second-quantization formalism to prescribe selection rules for amplitudes and density matrices.
• Employ symmetry-blocked tensor contractions to reduce computational cost in HF and post-HF methods, including CC and EOM-CC approaches.
• Implement the approach in cfour and qcumbre to enable exploitation of complex Abelian PGs (e.g., Cn, Cnh, Sn) for finite magnetic field calculations.

Experimental results

Research questions

• RQ1 How can double-coset decomposition be formulated and implemented for complex Abelian point groups with complex characters?
• RQ2 What are the theoretical and practical changes required for symmetry exploitation in SCF and post-HF methods when using complex Abelian PGs?
• RQ3 What is the impact of complex Abelian PG symmetry on integral evaluation, tensor contractions, and overall computational cost in quantum-chemical calculations under magnetic fields?
• RQ4 How do the implemented complex PG symmetries perform in benchmark calculations compared to conventional symmetry-free approaches?
• RQ5 Which complex Abelian PGs (and subgroups) are practically useful within standard quantum-chemical workflows under finite fields?

Key findings

• The authors provide working equations for evaluating one- and two-electron integrals using DCD in complex Abelian PGs.
• Symmetry handling in second quantization correctly assigns creation/annihilation operators to IRREPs and their complex conjugates, enabling proper selection rules.
• Block-tensor techniques for symmetry-adapted amplitudes reduce memory and computational requirements by exploiting the complex-group structure, with contractions performed block-by-block.
• The implementation supports complex Abelian PGs of order up to 8 (e.g., Cn, Cnh with n=3–4, and Sn with n=4,6,8) in cfour and qcumbre for finite-field calculations.
• Preliminary benchmarks on four small hydrocarbons show substantial computational savings when exploiting complex Abelian PGs compared to symmetry-unexploited calculations.
• The study demonstrates ff-CCSD + London orbitals as a representative workflow where complex PG symmetry yields practical performance gains.