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

:$\backslash Psi^\{\backslash mathrm\{HF\}\}(\backslash mathbf\{r\}\_\{1\}\backslash sigma(1)\backslash cdots\backslash mathbf\{r\}\_\{N\}\backslash sigma(N))\; =\; \backslash mathcal\{A\}\backslash left(\backslash psi\_\{1\}^\{\backslash alpha\}(\backslash mathbf\{r\}\_\{1\}\backslash alpha\_\{1\})\backslash cdots\backslash psi\_\{N\_\{\backslash alpha\}\}^\{\backslash alpha\}(\backslash mathbf\{r\}\_\{N\_\{\backslash alpha\}\}\backslash alpha\_\{N\_\{\backslash alpha\}\})$

\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 $\backslash mathcal\{A\}$ 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:

:$\backslash psi^\{\backslash alpha\}\_\{j\}(\backslash mathbf\{r\}\_\{j\})\; =\; \backslash psi^\{\backslash beta\}\_\{N\_\{\backslash alpha\}+j\}(\backslash mathbf\{r\}\_\{N\_\{\backslash alpha\}+j\}),\backslash \; \backslash \; \backslash \; 1\backslash leq\; j\backslash leq\; N\_\{\backslash beta\}.$

There is no such constraint in an UHF approach.{{cite journal | last1 = Glaesemann | first1 = Kurt R. | last2 = Schmidt | first2 = Michael W. | title = On the Ordering of Orbital Energies in High-Spin ROHF† | journal = The Journal of Physical Chemistry A | volume = 114 | issue =33 | pages = 8772–8777 | year = 2010 | pmid = 20443582 | doi = 10.1021/jp101758y| bibcode = 2010JPCA..114.8772G }}

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.

class="wikitable" |+ 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:Schlegel, H. Bernhard. "Spin contamination." Encyclopedia of Computational Chemistry 4 p.2665 (1998).

:$\backslash hat\{S\}^2\; =\; \backslash hat\{S\}\_z\; +\; \backslash hat\{S\}^2\_z\; +\; \backslash hat\{S\}\_-\; \backslash hat\{S\}\_+$

Slater determinants have well-defined spin projections:

$\backslash langle\backslash hat\{S\}\_z\backslash rangle\; =\; m\_s\; =\; \backslash frac\{N\_\backslash alpha\; -\; N\_\backslash beta\}\{2\}\; \backslash quad$ $\backslash quad\; \backslash langle\backslash hat\{S\}\_z^2\backslash rangle\; =\; m\_s^2\; =\; \backslash left(\backslash frac\{N\_\backslash alpha\; -\; N\_\backslash beta\}\{2\}\backslash right)^2$

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

For an Unrestricted Hartree-Fock (UHF) wavefunction, the expectation value Ŝ² is{{cite book|last=Szabo|first=Attila|author2=Ostlund, Neil S. |title=Modern Quantum Chemistry|publisher=Dover Publications|location=Mineola, New York|isbn=978-0-486-69186-2|year=1996}}

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

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

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

This simplifies to:

:$$

\begin{align}

\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{align}

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

:$\backslash langle\; \backslash Phi\_\{\backslash mathrm\{ROHF\}\}\; |\; \backslash hat\{S\}^2\; |\; \backslash Phi\_\{\backslash mathrm\{ROHF\}\}\; \backslash rangle\; =\; |m\_s|\; +\; m^2\_s\; =\; S(S+1)$

making ROHF wavefunctions eigenfunctions of Ŝ².

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

$\backslash langle\; \backslash Psi\; |\; \backslash hat\{S\}^2\; |\; \backslash Psi\; \backslash rangle\; =\; \backslash sum\_\{I,J\}\; c^*\_I\; c\_J\; \backslash langle\; \backslash Phi\_I\; |\; \backslash hat\{S\}^2\; |\; \backslash Phi\_J\; \backslash rangle$.

The diagonal terms $\backslash langle\; \backslash Phi\_I\; |\; \backslash hat\{S\}^2\; |\; \backslash Phi\_I\; \backslash rangle$ are calculated as above, while cross-terms $\backslash langle\; \backslash Phi\_J\; |\; \backslash hat\{S\}^2\; |\; \backslash Phi\_I\; \backslash rangle$ (where I≠J) require computing $\backslash langle\; \backslash Phi\_J\; |\; \backslash hat\{S\}\_-\; \backslash hat\{S\}\_+\; |\; \backslash Phi\_I\; \backslash rangle$ using individual electron operators. $\backslash langle\; \backslash Phi\_J\; |\; \backslash hat\{S\}\_z\; |\; \backslash Phi\_I\; \backslash rangle$ and $\backslash langle\; \backslash Phi\_J\; |\; \backslash hat\{S\}^2\_z\; |\; \backslash Phi\_I\; \backslash rangle$ 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.{{cite book|last=Young|first=David|title=Computational Chemistry|publisher=Wiley-Interscience|year=2001|isbn=978-0-471-22065-7}}

Although the ROHF approach does not suffer from spin contamination, it is 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.{{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}} The resulting wave function, while not completely free of contamination, dramatically improves upon the UHF approach especially in the absence of high order contamination.{{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 }}{{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=1789–1795|doi=10.1063/1.457084 |bibcode = 1989JChPh..91.1789B }}

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.{{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 }}

The spin-constrained UHF (SUHF) introduces a constraint into the Hartree–Fock equations of the form λ(Ŝ² − S(S + 1)), which as λ tends to infinity reproduces the ROHF solution.{{cite journal|last=Andrews|first=Jamie S.|author2=Jayatilaka, Dylan |author3=Bone, Richard G. A. |author4=Handy, Nicholas C. |author5= 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 }}

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

Although many density functional theory (DFT) codes simply calculate spin-contamination using the Kohn-Sham orbitals as if they were Hartree-Fock orbitals, this is not necessarily correct.

{{cite journal | doi = 10.1063/1.2737773 | title = Evaluation of〈Ŝ²〉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 }}{{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 }}{{cite journal | doi = 10.1080/00268970110041191 | title = On the diagnostic value of〈Ŝ²〉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 | s2cid = 101554092 }}{{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 }} The unrestricted Kohn-Sham (UKS) wavefunction of a spin-pure electronic state may have a Ŝ² expectation value that is inconsistent with a spin-pure state, if it is calculated using the UHF formula. Therefore, the benefits of using a restricted open-shell treatment over an unrestricted one is smaller in a DFT context than in HF. In particular, the exact spin density of the S_z=S spin-component of an open-shell molecule generally adopts negative values in small parts of the real space due to spin polarization, but a restricted open-shell Kohn-Sham (ROKS) treatment necessarily gives non-negative spin density everywhere, which proves that, in general, only UKS can possibly give the exact spin density of a system.{{cite journal | doi=10.1002/qua.560560414 | title=Spin-unrestricted character of Kohn-Sham orbitals for open-shell systems | date=1995 | last1=Pople | first1=John A. | last2=Gill | first2=Peter M. W. | last3=Handy | first3=Nicholas C. | journal=International Journal of Quantum Chemistry | volume=56 | issue=4 | pages=303–305 }} In practical calculations, the energy error of an unrestricted DFT calculation due to spin contamination can sometimes be remedied by a spin projection approach, particularly if the spin contamination is large (e.g. when there is antiferromagnetic coupling in the system).{{cite journal | doi=10.1016/S0009-2614(00)00166-4 | title=Ab initio computations of effective exchange integrals for H–H, H–He–H and Mn2O2 complex: Comparison of broken-symmetry approaches | date=2000 | last1=Soda | first1=T. | last2=Kitagawa | first2=Y. | last3=Onishi | first3=T. | last4=Takano | first4=Y. | last5=Shigeta | first5=Y. | last6=Nagao | first6=H. | last7=Yoshioka | first7=Y. | last8=Yamaguchi | first8=K. | journal=Chemical Physics Letters | volume=319 | issue=3–4 | pages=223–230 | bibcode=2000CPL...319..223S }} However, for moderate spin contamination, spin projection may actually be counterproductive, and create unphysical cusps (i.e. derivative discontinuities) on potential energy curves.

On the other hand, in TDDFT calculations, one frequently observes severe spin contamination in the excited states, even if the ground state is described by ROKS (i.e. even when the ground state itself is not spin contaminated). This is because the single excitations out of an open-shell reference state (usually the ground state) do not span a spin-complete manifold, i.e. it is impossible to unitarily transform the set of all single excitations of an open-shell reference state, such that all of the transformed states are eigenstates of the Ŝ² operator.{{cite journal | doi=10.1063/1.3463799 | title=Spin-adapted open-shell random phase approximation and time-dependent density functional theory. I. Theory | date=2010 | last1=Li | first1=Zhendong | last2=Liu | first2=Wenjian | journal=The Journal of Chemical Physics | volume=133 | issue=6 | page=064106 | pmid=20707560 | bibcode=2010JChPh.133f4106L }} The excited state 〈Ŝ²〉 values of a TDDFT calculation of an open-shell state can have a maximum error of 2 when the ground state is described by ROKS, and sometimes slightly larger than 2 when the ground state is described by UKS.{{cite journal | doi=10.1063/1.3573374 | title=Spin-adapted open-shell time-dependent density functional theory. II. Theory and pilot application | date=2011 | last1=Li | first1=Zhendong | last2=Liu | first2=Wenjian | last3=Zhang | first3=Yong | last4=Suo | first4=Bingbing | journal=The Journal of Chemical Physics | volume=134 | issue=13 | page=134101 | pmid=21476737 | bibcode=2011JChPh.134m4101L }} To eliminate the excited state spin contamination in TDDFT, spin-adapted TDDFT methods{{cite journal | doi=10.1063/5.0025428 | title=Analytic energy gradients of spin-adapted open-shell time-dependent density functional theory | date=2020 | last1=Wang | first1=Zikuan | last2=Li | first2=Zhendong | last3=Zhang | first3=Yong | last4=Liu | first4=Wenjian | journal=The Journal of Chemical Physics | volume=153 | issue=16 | page=164109 | pmid=33138406 | bibcode=2020JChPh.153p4109W | s2cid=226241125 }} must be used, which either explicitly or implicitly include some double excitations and thereby go beyond the adiabatic approximation of the TDDFT response kernel. Spin-adapted TDDFT methods not only improve the prediction of absorption spectra,{{cite journal | doi=10.1063/1.3660688 | title=Spin-adapted open-shell time-dependent density functional theory. III. An even better and simpler formulation | date=2011 | last1=Li | first1=Zhendong | last2=Liu | first2=Wenjian | journal=The Journal of Chemical Physics | volume=135 | issue=19 | page=194106 | pmid=22112065 | bibcode=2011JChPh.135s4106L }}{{cite journal | doi=10.1021/acs.jctc.5b01158 | title=Critical Assessment of TD-DFT for Excited States of Open-Shell Systems: I. Doublet–Doublet Transitions | date=2016 | last1=Li | first1=Zhendong | last2=Liu | first2=Wenjian | journal=Journal of Chemical Theory and Computation | volume=12 | issue=1 | pages=238–260 | pmid=26672389 }} but also improve the prediction of the fluorescence spectra{{Cite journal|arxiv=2307.07886 |last1=Wang |first1=Xingwen |last2=Wu |first2=Chenyu |last3=Wang |first3=Zikuan |last4=Liu |first4=Wenjian |title=When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin |journal=Frontiers in Chemistry |date=2023 |volume=11 | pmc=10667454|doi=10.3389/fchem.2023.1259016 |doi-access=free |pmid=38025061 |bibcode=2023FrCh...1159016W }} and excited state relaxation pathways of open-shell molecules.

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Category:Computational chemistry

Category:Quantum chemistry