This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial parts of α and β spin-orbitals are permitted to differ.
In particular, they are not eigenfunctions of the total spin-squared operator, Ŝ2, but can formally be expanded in terms of pure spin states of higher multiplicities (the contaminants).
For an open-shell system, the mean-field approach of Hartree–Fock theory gives rise to different equations for the α and β orbitals.
Consequently, there are two approaches that can be taken – either to force double occupation of the lowest orbitals by constraining the α and β spatial distributions to be the same (restricted open-shell Hartree–Fock, ROHF) or permit complete variational freedom (unrestricted Hartree–Fock UHF).
For a ROHF wave function, the first 2Nβ spin-orbitals are forced to have the same spatial distribution: There is no such constraint in an UHF approach.
[2] The total spin-squared operator commutates with the nonrelativistic molecular Hamiltonian so it is desirable that any approximate wave function is an eigenfunction of Ŝ2.
The eigenvalues of Ŝ2 are S(S + 1), where S is the spin quantum number of the system and can take the values 0 (singlet), 1/2 (doublet), 1 (triplet), 3/2 (quartet), and so forth.
[4] Although the ROHF approach does not suffer from spin contamination, it is less commonly available in quantum chemistry computer programs.