Spin contamination

In computational chemistry, spin contamination is the artificial mixing of different electronic 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-orbitals are allowed to differ. Approximate wave functions with a high degree of spin contaminated wave functions 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 multiplicities (the contaminants).


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 comprised 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]

:Psi^{mathrm{HF(mathbf{r}_{1}sigma(1)cdotsmathbf{r}_{N}sigma(N))=mathcal{A}left(psi^{alpha}_{1}(mathbf{r}_{1})alpha(1)cdotspsi^{alpha}_{N_{alpha(mathbf{r}_{N_{alpha)alpha(N_{alpha})psi^{eta}_{N_{alpha}+1}(mathbf{r}_{N_{alpha}+1})eta(N_{alpha}+1)cdotspsi^{eta}_{N_{alpha}+N_{eta(mathbf{r}_{N_{alpha}+N_{eta)eta(N_{alpha}+N_{eta}) ight)

where 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 2"N"β spin-orbitals are forced to have the same spatial distribution:

:psi^{alpha}_{j}(mathbf{r}_{j}) = psi^{eta}_{N_{alpha}+j}(mathbf{r}_{N_{alpha}+j}), 1leq jleq N_{eta}.

There is no such constraint in a UHF approach.


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), 1/2 (doublet), 1 (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 iscite 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]

: langle S^{2} angle_{mathrm{ROHF = langle S^{2} angle_{mathrm{exact =left(frac{N_{alpha}-N_{eta{2} ight)left(frac{N_{alpha}-N_{eta{2}+1 ight).

However, the UHF wave function is not: the expectation value of "Ŝ"2 for an UHF wave function is

: langle S^{2} angle_{mathrm{UHF = langle S^{2} angle_{mathrm{exact + N_{eta} - sum_{i,j}^{mathrm{all|langlepsi_{i}^{alpha}|psi_{j}^{eta} angle|^{2}.

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 always contaminated to some extent by higher order spin eigenstates unless a ROHF approach is taken.

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 approach to density functional theory using approximate exchange-correlation functionals. [cite book|last=Young|first=David|title=Computational Chemistry|publisher=Wiley-Interscience|date=2001|isbn=0-471-22065-5]


The ROHF approach does not suffer from spin contamination however 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. [cite journal|last=Löwdin |first=P.-O|date=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] 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|date=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] [cite journal|last=Baker|first=J|date=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|pages=1789|doi=10.1063/1.457084 ]

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 |date=1986|title=Potential energy curves using unrestricted Møller–Plesset perturbation theory with spin annihilation|journal=Journal of Chemical Physics|volume=84|pages=4530-4534|doi=10.1063/1.450026]

The spin-constrained UHF (SUHF) introduces a constraint in to the Hartree-Fock equations of the form λ("Ŝ"2-"S"("S"+1)), which as λ tends to infinity reproduces the ROHF solution. [cite journal|last=Andrews|first=Jamie S.|coauthors=Jayatilaka, Dylan; Bone, Richard G. A.; Handy, Nicholas C.; Amos, Roger D.|date=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]

All of these approach are readily applicable to unrestricted Møller-Plesset perturbation theory.


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