Spatial and Spin Symmetry Breaking in Semidefinite-Programming-Based Hartree-Fock Theory

Area of Science:

• Quantum chemistry
• Computational physics

Background:

• The Hartree-Fock (HF) problem is a cornerstone of electronic structure theory.
• Previous work recast HF as a semidefinite optimization over rank-constrained two-body reduced density matrices (RDMs).
• This formulation transferred non-convexity to the rank constraint.

Purpose of the Study:

• To explore an equivalent optimization over positive semidefinite one-electron RDMs (1-RDMs).
• To investigate the symmetry-breaking properties of this RDM-based approach in molecular systems.
• To compare the RDM-based method with generalized Hartree-Fock theory.

Main Methods:

• Reformulation of the Hartree-Fock problem as an optimization over 1-RDMs.
• Inclusion of ensemble N-representability and spin-state conditions as constraints.
• Application to molecular systems to study spatial and spin symmetry breaking.
• Relaxation of symmetry constraints to compare with generalized Hartree-Fock theory.

Main Results:

• The 1-RDM optimization retains the non-convexity of the HF energy functional.
• Imposing spin symmetry (Ŝ² and Ŝ³) but relaxing spatial symmetry often reveals symmetry-broken solutions.
• A smooth, spatially symmetry-broken potential energy curve was found for the Be-H₂ insertion pathway.
• Relaxing spin symmetry constraints showed equivalence to real-valued generalized Hartree-Fock theory.

Conclusions:

• The RDM-based approach provides a robust framework for exploring electronic structure beyond standard methods.
• This method facilitates the discovery of complex symmetry-broken states in molecules.
• The RDM formulation offers a valuable alternative for advanced quantum chemical calculations.