Spin Contamination - Contamination

Contamination

The total spin-squared operator commutes 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 can take the values 0 (singlet), 1/2 (doublet), 1 (triplet), 3/2 (quartet), and so forth.

The ROHF wave function is an eigenfunction of Ŝ2: the expectation value Ŝ2 for a ROHF wave function is

However, the UHF wave function is not: the expectation value of Ŝ2 for an UHF wave function is

The sum of the last two terms is a measure of the extent of spin contamination in the unrestricted Hartree–Fock approach and is always non-negative – the wave function is usually contaminated to some extent by higher order spin eigenstates unless a ROHF approach is taken. Naturally, there is no contamination if all electrons are the same spin. Also, there is often no contamination if the number of α and β electrons is the same. A small basis set could also constrain the wavefunction sufficiently to prevent spin contamination.

Such contamination is a manifestation of the different treatment of α and β electrons that would otherwise occupy the same molecular orbital. It is also present in Møller–Plesset perturbation theory calculations that employ an unrestricted wave function as a reference state and, to a much lesser extent, in the unrestricted Kohn–Sham approach to density functional theory using approximate exchange-correlation functionals.