Spin Contamination - Open-shell Wave Functions

Open-shell Wave Functions

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 N_α α-spin orbitals and N_β β-spin orbitals can be written as

$\Psi^{\mathrm{HF}}(\mathbf{r}_{1}\sigma(1)\cdots\mathbf{r}_{N}\sigma(N)) = \mathcal{A}\left(\psi_{1}^{\alpha}(\mathbf{r}_{1}\alpha_{1})\cdots\psi_{N_{\alpha}}^{\alpha}(\mathbf{r}_{N_{\alpha}}\alpha_{N_{\alpha}}) \psi_{N_{\alpha}+1}^{\beta}(\mathbf{r}_{N_{\alpha}+1}\beta_{N_{\alpha}+1})\cdots\psi_{N}^{\beta}(\mathbf{r}_{N}\beta_{N})\right).$

where is the antisymmetrization operator. This wave function is an eigenfunction of the total spin projection operator, Ŝ_z, with eigenvalue (N_α − N_β)/2 (assuming N_α ≥ N_β). For a ROHF wave function, the first 2N_β spin-orbitals are forced to have the same spatial distribution:

There is no such constraint in an UHF approach.

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