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 Nα α-spin orbitals and Nβ 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, Ŝz, with eigenvalue (NαNβ)/2 (assuming NαNβ). 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.

Read more about this topic:  Spin Contamination

Famous quotes containing the words wave and/or functions:

    The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.
    Aldous Huxley (1894–1963)

    In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.
    Urie Bronfenbrenner (b. 1917)