Spin contamination

In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. 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. Approximate wave functions with a high degree of spin contamination are undesirable. 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). In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. 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. Approximate wave functions with a high degree of spin contamination are undesirable. 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). Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals. 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). In general, an N-electron Hartree–Fock wave function composed of Nα α-spin orbitals and Nβ β-spin orbitals can be written as where A {displaystyle {mathcal {A}}} is the antisymmetrization operator. This wave function is an eigenfunction of the total spin projection operator, Ŝz, with eigenvalue (Nα − Nβ)/2 (assuming Nα ≥ Nβ). 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. 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