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

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.