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In [[computational chemistry]], '''spin contamination''' is the artificial mixing of different [[electron]]ic [[Spin (physics)|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-orbital]]s 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, ''Ŝ''2, but can formally be expanded in terms of pure spin states of higher [[Multiplicity (chemistry)|multiplicities]] (the contaminants).

==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<ref>{{cite book|last=Springborg|first=Michael|title=Methods of Electronic-Structure Calculations|publisher=John Wiley & Sons |isbn=978-0-471-97976-0|year=2000}}</ref>

:<math>\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).</math>

where <math>\mathcal{A}</math> 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 2''N''β spin-orbitals are forced to have the same spatial distribution:

:<math>\psi^{\alpha}_{j}(\mathbf{r}_{j}) = \psi^{\beta}_{N_{\alpha}+j}(\mathbf{r}_{N_{\alpha}+j}),\ \ \ 1\leq j\leq N_{\beta}.</math>

There is no such constraint in an UHF approach.

==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 ''Ŝ''2. The eigenvalues of ''Ŝ''2 are ''S''(''S'' + 1) where ''S'' can take the values 0 ([[Singlet state|singlet]]), 1/2 ([[Doublet state|doublet]]), 1 ([[Triplet state|triplet]]), 3/2 (quartet), and so forth.

The ROHF wave function is an eigenfunction of ''Ŝ''2: the expectation value ''Ŝ''2 for a ROHF wave function is<ref name="szabo">{{cite book|last=Szabo|first=Attila|coauthors=Ostlund, Neil S.|title=Modern Quantum Chemistry|publisher=Dover Publications|location=Mineola, New York|isbn=0-486-69186-1|year=1996}}</ref>

:<math> \langle S^{2}\rangle_{\mathrm{ROHF}} = \langle S^{2}\rangle_{\mathrm{exact}} =\left(\frac{N_{\alpha}-N_{\beta}}{2}\right)\left(\frac{N_{\alpha}-N_{\beta}}{2}+1\right).</math>

However, the UHF wave function is not: the expectation value of ''Ŝ''2 for an UHF wave function is<ref name="szabo"/>

:<math> \langle S^{2}\rangle_{\mathrm{UHF}} = \langle S^{2}\rangle_{\mathrm{exact}} + N_{\beta} - \sum_{i,j}^{\mathrm{all}}|\langle\psi_{i}^{\alpha}|\psi_{j}^{\beta}\rangle|^{2}.</math>

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 equations|Kohn–Sham]] approach to [[density functional theory]] using approximate exchange-correlation functionals.<ref>{{cite book|last=Young|first=David|title=Computational Chemistry|publisher=Wiley-Interscience|year=2001|isbn=0-471-22065-5}}</ref>

==Elimination==

Although the [[restricted open-shell Hartree–Fock|ROHF]] approach does not suffer from spin contamination, it is far less commonly available in [[quantum chemistry computer programs]]. Given this, several approaches to remove or minimize spin contamination from UHF wave functions have been proposed.

The annihilated UHF (AUHF) approach involves the annihilation of first spin contaminant of the density matrix at each step in the self-consistent solution of the Hartree–Fock equations using a state-specific [[Löwdin annihilator]].<ref>{{cite journal|last=Löwdin |first=Per-Olov|year=1955|title=Quantum Theory of Many-Particle Systems. III. Extension of the Hartree–Fock Scheme to Include Degenerate Systems and Correlation Effects|journal=Physical Review|volume=97|pages=1509–1520|doi=10.1103/PhysRev.97.1509|issue=6|bibcode=1955PhRv...97.1509L}}</ref> The resulting wave function, while not completely free of contamination, dramatically improves upon the UHF approach especially in the absence of high order contamination.<ref>{{cite journal|last=Baker|first=J|year=1988|title=Møller–Plesset perturbation theory with the AUHF wavefunction|journal=Chemical Physics Letters|volume=152|issue=2–3|pages=227–232|doi=10.1016/0009-2614(88)87359-7|bibcode = 1988CPL...152..227B }}</ref><ref>{{cite journal|last=Baker|first=J|year=1989 |title=An investigation of the annihilated unrestricted Hartree–Fock wave function and its use in second-order Møller–Plesset perturbation theory|journal=Journal of Chemical Physics|volume=91| issue =3 | pages=1789|doi=10.1063/1.457084 |bibcode = 1989JChPh..91.1789B }}</ref>

Projected UHF (PUHF) annihilates all spin contaminants from the self-consistent UHF wave function. The projected energy is evaluated as the expectation of the projected wave function.<ref>{{cite journal|last=Schlegel|first=H. Bernhard |year=1986|title=Potential energy curves using unrestricted Møller–Plesset perturbation theory with spin annihilation|journal=Journal of Chemical Physics|volume=84| issue = 8 | pages=4530–4534|doi=10.1063/1.450026|bibcode = 1986JChPh..84.4530S }}</ref>

The spin-constrained UHF (SUHF) introduces a [[Lagrange multiplier|constraint]] into the Hartree–Fock equations of the form λ(''Ŝ''2 − ''S''(''S'' + 1)), which as λ tends to infinity reproduces the ROHF solution.<ref>{{cite journal|last=Andrews|first=Jamie S.|coauthors=Jayatilaka, Dylan; Bone, Richard G. A.; Handy, Nicholas C.; Amos, Roger D.|year=1991|title=Spin contamination in single-determinant wavefunctions|journal=Chemical Physics Letters|volume=183|issue=5|pages=423–431|doi=10.1016/0009-2614(91)90405-X|bibcode = 1991CPL...183..423A }}</ref>

All of these approaches are readily applicable to unrestricted [[Møller–Plesset perturbation theory]].

==Density Functional Theory (DFT)==
Although many DFT codes simply calculate spin-contamination using the Kohn-Sham orbitals as if they
were Hartree-Fock orbitals, this is not necessarily correct.
<ref>{{cite journal | doi = 10.1063/1.2737773 | title = Evaluation of 〈Ŝ[sup 2]〉 in density functional theory | year = 2007 | last1 = Cohen | first1 = Aron J. | last2 = Tozer | first2 = David J. | last3 = Handy | first3 = Nicholas C. | journal = The Journal of Chemical Physics | volume = 126 | pages = 214104 | pmid = 17567187 | issue = 21|bibcode = 2007JChPh.126u4104C }}</ref>
<ref>{{cite journal | doi = 10.1063/1.468585 | title = Evaluation of 〈S2〉 in restricted, unrestricted Hartree–Fock, and density functional based theories | year = 1995 | last1 = Wang | first1 = Jiahu | last2 = Becke | first2 = Axel D. | last3 = Smith | first3 = Vedene H. | journal = The Journal of Chemical Physics | volume = 102 | issue = 8 | pages = 3477|bibcode = 1995JChPh.102.3477W }}</ref>
<ref>{{cite journal | doi = 10.1080/00268970110041191 | title = On the diagnostic value of (S2) in Kohn-Sham density functional theory | year = 2001 | last1 = Grafenstein | first1 = Jurgen | last2 = Cremer | first2 = Dieter | journal = Molecular Physics | volume = 99| issue = 11 | pages = 981–989|bibcode = 2001MolPh..99..981G }}</ref>
<ref>{{cite journal | doi = 10.1063/1.472497 | title = Some reasons not to use spin projected density functional theory | year = 1996 | last1 = Wittbrodt | first1 = Joanne M. | last2 = Schlegel | first2 = H. Bernhard | journal = The Journal of Chemical Physics | volume = 105 | issue = 15 | pages = 6574|bibcode = 1996JChPh.105.6574W }}</ref>

==References==
{{reflist}}

[[Category:Computational chemistry]]
[[Category:Quantum chemistry]]

Valuative criterion - Revision history

en>Monkbot: /* References */Task 3b: Fix CS1 deprecated coauthor parameter errors

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