Spin contamination

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^[1]

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

where ${\displaystyle {\mathcal {A}}}$ 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:

${\displaystyle \psi _{j}^{\alpha }(\mathbf {r} _{j})=\psi _{N_{\alpha }+j}^{\beta }(\mathbf {r} _{N_{\alpha }+j}),\ \ \ 1\leq j\leq N_{\beta }.}$

There is no such constraint in an UHF approach.^[2]

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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 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. The Ŝ² eigenvalues of the most common spin multiplicities are listed below.

Spin quantum numbers, spin multiplicities and Ŝ² eigenvalues

Spin quantum number S	Spin multiplicity 2S+1	Ŝ2 eigenvalue S(S+1)
0	1 (singlet)	0.00
1/2	2 (doublet)	0.75
1	3 (triplet)	2.00
3/2	4 (quartet)	3.75
2	5 (quintet)	6.00
5/2	6 (sextet)	8.75
3	7 (septet)	12.00
7/2	8 (octet)	15.75
4	9 (nonet)	20.00

The Ŝ² operator can be decomposed as:^[3]

${\displaystyle {\hat {S}}^{2}={\hat {S}}_{z}+{\hat {S}}_{z}^{2}+{\hat {S}}_{-}{\hat {S}}_{+}}$

Slater determinants have well-defined spin projections:

${\displaystyle \langle {\hat {S}}_{z}\rangle =m_{s}={\frac {N_{\alpha }-N_{\beta }}{2}}\quad }$ ${\displaystyle \quad \langle {\hat {S}}_{z}^{2}\rangle =m_{s}^{2}=\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}}$

${\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}}$ can be expressed in terms of individual electron operators: ${\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}=\sum _{i}{\hat {s}}_{-}(i)\sum _{j}{\hat {s}}_{+}(j)}$.

For an Unrestricted Hartree-Fock (UHF) wavefunction, the expectation value Ŝ² is^[4]

${\displaystyle \langle \Phi _{\mathrm {UHF} }|{\hat {S}}^{2}|\Phi _{\mathrm {UHF} }\rangle ={\frac {N_{\alpha }-N_{\beta }}{2}}+\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}+N_{\beta }-\sum _{i}^{N_{\alpha }^{\mathrm {occupied} }}\sum _{j}^{N_{\beta }^{\mathrm {occupied} }}|\langle \psi _{i}^{\alpha }|\psi _{j}^{\beta }\rangle |^{2}}$

where the first two terms follow directly from the decomposition of Ŝ², the third term corresponds to the diagonal contribution of ${\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}}$ and the last term involves the overlap between the ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ spin-orbitals with the negative sign arising from the antisymmetry of the Hartree-Fock wavefunction.^[3]

For Slater determinants constructed from restricted spin-orbitals, the spatial parts of corresponding ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ orbitals are the same, thus the ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ overlaps evaluate to 1 for ${\displaystyle i=j}$ and 0 otherwise. This simplifies to:

${\displaystyle {\begin{aligned}\langle \Phi _{\mathrm {restricted} }|{\hat {S}}^{2}|\Phi _{\mathrm {restricted} }\rangle &={\frac {N_{\alpha }-N_{\beta }}{2}}+\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}+N_{\beta }-\sum _{i}^{N_{\alpha }^{\mathrm {occupied} }}\sum _{j}^{N_{\beta }^{\mathrm {occupied} }}\delta _{ij}\\&={\frac {N_{\alpha }-N_{\beta }}{2}}+\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}+N_{\beta }-N_{\beta }^{\mathrm {paired} }\\&={\frac {N_{\alpha }-N_{\beta }}{2}}+\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}+N_{\beta }^{\mathrm {unpaired} }\end{aligned}}}$

In Restricted open-shell Hartree-Fock (ROHF), all unpaired electrons have the same spin, yielding:^[3]

${\displaystyle \langle \Phi _{\mathrm {ROHF} }|{\hat {S}}^{2}|\Phi _{\mathrm {ROHF} }\rangle =|m_{s}|+m_{s}^{2}=S(S+1)}$

making ROHF wavefunctions eigenfunctions of Ŝ².

For multi-configurational wavefunctions expressed as ${\displaystyle |\Psi \rangle =\sum _{I}c_{I}|\Phi _{I}\rangle }$ with ${\displaystyle \Phi _{I}}$ being Slater determinants, ⟨Ŝ²⟩ is :

${\displaystyle \langle \Psi |{\hat {S}}^{2}|\Psi \rangle =\sum _{I,J}c_{I}^{*}c_{J}\langle \Phi _{I}|{\hat {S}}^{2}|\Phi _{J}\rangle }$.

The diagonal terms ${\displaystyle \langle \Phi _{I}|{\hat {S}}^{2}|\Phi _{I}\rangle }$ are calculated as above, while cross-terms ${\displaystyle \langle \Phi _{J}|{\hat {S}}^{2}|\Phi _{I}\rangle }$ (where I≠J) require computing ${\displaystyle \langle \Phi _{J}|{\hat {S}}_{-}{\hat {S}}_{+}|\Phi _{I}\rangle }$ using individual electron operators. ${\displaystyle \langle \Phi _{J}|{\hat {S}}_{z}|\Phi _{I}\rangle }$ and ${\displaystyle \langle \Phi _{J}|{\hat {S}}_{z}^{2}|\Phi _{I}\rangle }$ vanish when I≠J.

The sum of the last two terms in the UHF equation 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. Therefore, the deviation of the UHF expectation value of Ŝ² from the exact Ŝ² eigenvalue as would be expected from the spin multiplicity (see the table above) is usually taken as a measure of the severity of the spin contamination. Naturally, there is no contamination if all electrons are the same spin. Also, there is often (but not always, as in open-shell singlets) 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 even some that employ a restricted wave function) and, to a much lesser extent, in the unrestricted Kohn–Sham approach to density functional theory using approximate exchange-correlation functionals.^[5]