  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.
