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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''<sup>2</sup>, but can formally be expanded in terms of pure spin states of higher [[Multiplicity (chemistry)|multiplicities]] (the contaminants).
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| ==Open-shell wave functions==
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| 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''<sub>α</sub> α-spin orbitals and ''N''<sub>β</sub> 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>
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| :<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>
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| where <math>\mathcal{A}</math> is the [[antisymmetrization operator]]. This wave function is an eigenfunction of the total spin projection operator, ''Ŝ''<sub>z</sub>, with eigenvalue (''N''<sub>α</sub> − ''N''<sub>β</sub>)/2 (assuming ''N''<sub>α</sub> ≥ ''N''<sub>β</sub>). For a ROHF wave function, the first 2''N''<sub>β</sub> spin-orbitals are forced to have the same spatial distribution:
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| :<math>\psi^{\alpha}_{j}(\mathbf{r}_{j}) = \psi^{\beta}_{N_{\alpha}+j}(\mathbf{r}_{N_{\alpha}+j}),\ \ \ 1\leq j\leq N_{\beta}.</math>
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| <!--where in the ROHF case the, first ''M'' = ''N''<sub>β</sub> electrons are constrained to share common spatial orbitals and the renaming ''N'' - 2''M'' are unpaired, with ''N'' = ''N''<sub>α</sub> + ''N''<sub>β</sub>.-->There is no such constraint in an UHF approach.
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| ==Contamination==
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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 ''Ŝ''<sup>2</sup>. The eigenvalues of ''Ŝ''<sup>2</sup> 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.
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| The ROHF wave function is an eigenfunction of ''Ŝ''<sup>2</sup>: the expectation value ''Ŝ''<sup>2</sup> 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>
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| :<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>
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| However, the UHF wave function is not: the expectation value of ''Ŝ''<sup>2</sup> for an UHF wave function is<ref name="szabo"/>
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| :<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>
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| 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
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| is often no contamination if the number of α and β electrons is the same. A small basis set could also constrain the
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| wavefunction sufficiently to prevent spin contamination.
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| 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>
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| ==Elimination==
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| 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.
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| 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> | |
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| 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>
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| The spin-constrained UHF (SUHF) introduces a [[Lagrange multiplier|constraint]] into the Hartree–Fock equations of the form λ(''Ŝ''<sup>2</sup> − ''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>
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| All of these approaches are readily applicable to unrestricted [[Møller–Plesset perturbation theory]].
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| ==Density Functional Theory (DFT)==
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| Although many DFT codes simply calculate spin-contamination using the Kohn-Sham orbitals as if they
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| were Hartree-Fock orbitals, this is not necessarily correct.
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| <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>
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| <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>
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| <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>
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| <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>
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| ==References==
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| {{reflist}}
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| [[Category:Computational chemistry]]
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| [[Category:Quantum chemistry]]
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Wish to know how to turn out to be credit cards hacker? No, I'm not discussing becoming a member of Task Blitzkrieg and doing something unlawful that could property you in the slammer.
I've 22 cards on my credit file now. To give you a concept of the result: my credit score is leaner than it usually is (around 720 now, generally it's nearer to 760), but that's generally because I'm floating a big balance on a 0% cards. (Why pay it back now when I could earn 3% APY in a high-yield bank checking account?), therefore my utilization % isn't optimal. I also have plenty of applications during the past year, because I went insane on apps soon after my mortgage went through this past year. EASILY didn't have the high stability, I think I'd be around 740+, so that it hasn't hurt as well badly.
In my view, there is no real advantage to having more than four to six 6 credit cards. You only need to rotate their usage monthly which means you keep them active. Just use them to cover stuff you'd normally spend with cash-a handbag of groceries or haircut. The main thing is to find that couple of cards that matches your needs-points, miles, no costs, etc But stop and let them get old with you.
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