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 (PPP) 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 openshell HF (ROHF) solutions. At the same time it permits the derivation of explicit expressions for the energy providing a complete description of swallowtail or Whitneyfold 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 (SCF) iterative procedures in more complex situations. KeywordsRestricted open-shell Hartree-Fock (ROHF) solutions · Doublet instability · Pariser-Parr-Pople Hamiltonian · Allyl radical π-electron model · Symmetry breaking · Potential energy surfaces · Swallowtail or Whitney-fold singularity 2 IntroductionThe Hartree-Fock (HF) approximation is undoubtedly the most often exploited version of the independent particle model (IPM) and represents nowadays a standard tool in investigations of the atomic and molecular electronic structure. Although such a description is often lacking in accuracy due to an average account of the interelectronic Coulomb interactions, it nonetheless provides a useful qualitative -and even semi-quantitative -description of various molecular properties and serves as a point of departure for most post-HF correlated methods.The relevant wave function |Φ has the form of a single anti-symmetrized product of molecular orbitals (MOs) or spin-orbitals (MSOs) |A i ,

Generalizing Frobenius inversion to quaternion matrices

“…On the other side, since the group of unit quaternions is isomorphic to the group SU(2) consisting of 2 × 2 special unitary matrices, they provide us a natural way to represent spatial rotations. 1 Because of such a surprising connection, quaternions are widely applied in areas related to rigid body mechanics and quantum mechanics [21,17,4,13,30,29,34]. Quaternions are also useful for combining relating variables into a single algebraic entity that can allow them to be manipulated collectively in a natural way, e.g., representing RGB pixel values using the i, j, and k components of a single quaternion.…”