Spin contamination and noncollinearity in general complex Hartree-Fock wave functions

TL;DR

This paper derives an expression for the extless{} S^2 extgreater{} expectation value in complex Hartree-Fock wave functions, analyzing spin contamination, noncollinearity, and perpendicularity contributions with applications to H extsubscript{2}O extsuperscript{+}.

Contribution

It introduces a comprehensive decomposition of spin expectation values in general complex Hartree-Fock wave functions, including new insights into noncollinearity and perpendicularity effects.

Findings

Decomposition of extless{} S^2 extgreater{} into four contributions.

Comparison of contributions for H extsubscript{2}O extsuperscript{+} system.

Optimization of the collinearity axis discussed.

Abstract

An expression for the square of the spin operator expectation value, \textless{} S^2\textgreater{}, is obtained for a general complex Hartree-Fock (GCHF) wave function and decomposed into four contributions: The main one whose expression is formally identical to the restricted (open-shell) Hartree-Fock expression. A spin contamination one formally analogous to that found for spin unrestricted Hartree-Fock wave functions. A noncollinearity contribution related to the fact that the wave function is not an eigenfunction of the spin-S\_z operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H\_2O^+ system. The optimization of the collinearity axis is also considered.