Symmetry-breaking in the independent particle model: nature of the singular behavior of Hartree–Fock potentials

Abstract

The nature of the singular behavior of Hartree–Fock (HF) potential energy surfaces (PESs) that arises in the presence of a spin-preserving instability of the relevant restricted HF solutions is illustrated by a simple π-electron model of the allyl radical as described by the Pariser–Parr–Pople semi-empirical Hamiltonian. The simplicity of this three-electron model system stems from a low dimension of the appropriate variational space which enables an independent direct analytical approach illustrating the appropriateness of doublet stability conditions for restricted open-shell HF (ROHF) solutions. At the same time it permits the derivation of explicit expressions for the energy providing a complete description of swallowtail or Whitney-fold catastrophe singularities on the corresponding PES that arise with the onset of a doublet instability. In particular, this simple model enables the computation of the part of the PES that is associated with unstable ROHF solutions and which would be difficult if not impossible to generate in full generality via standard self-consistent field iterative procedures in more complex situations.