Spin contamination ($⟨{\hat{S}}^2⟩-S(S+1){\hslash }^2$)

A wavefunction intended for a specified total-spin state contains components of other total spins. For an unrestricted determinant with one alpha and one beta orbital, $⟨S^2⟩=1-|⟨{\varphi }_{\alpha }|{\varphi }_{\beta }⟩{|}^2$ in units of ${\hslash }^2$.