Spin Contamination

David Young
Cytoclonal Pharmaceutics Inc.

What is spin contamination

Introductory descriptions of Hartree-Fock calculations (usually using Rootaan's Self Consistent Field (SCF) method) focus on singlet systems for which all electron spins are paired. By assuming that the calculations is restricted to having two electrons per occupied orbital, the computation can be done relatively easily. This is often referred to as a spin restricted Hartree-Fock calculation or RHF.

For systems with a multiplicity other than one, it is not possible to use the RHF method as is. Often an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals, one for the alpha electrons and one for the beta electrons. Usually these two sets of orbitals use the same set of basis functions but different molecular orbital coefficients.

The advantage of unrestricted calculations is that they can be performed very efficiently. The disadvantage is that the wave function is no longer an eigenfunction of the total spin, <S²>, thus some error may be introduced into the calculation. This error is called spin contamination.

How does spin contamination affect results

Spin contamination results in having wave functions which appear to be the desired spin state, but have a bit of some other spin state mixed in. This occasionally results in slightly lowering the computed total energy due to having more variational freedom. More often the result is to slightly raise the total energy since a higher energy state is being mixed in. However, this change is an artifact of an incorrect wave function. Since this is not a systematic error, the difference in energy between states will be adversely affected. A high spin contamination can affect the geometry and population analysis and significantly affect the spin density.

As a check for the presence of spin contamination, most ab initio programs will print out the expectation value of the total spin, <S²>. If there is no spin contamination this should equal s(s+1) where s equals 1/2 times the number of unpaired electrons. One rule of thumb which was derived from experience with organic molecule calculations is that the spin contamination is negligible if the value of <S²> differs from s(s+1) by less than 10%. Although this provides a quick test, it is always advisable to double check the results against experimental evidence or more rigorous calculations.

Spin contamination is often seen in unrestricted Hartree-Fock (UHF) calculations and unrestricted Møller-Plesset (UMP2, UMP3, UMP4) calculations. It is less common to find any significant spin contamination in DFT calculations, even when unrestricted Kohn-Sham orbitals are being used.

Unrestricted calculations often incorporate a spin annihilation step which removes a large percentage of the spin contamination from the wave function at some point in the calculation. This helps minimize spin contamination but does not completely prevent it. The final value of <S²> is always the best check on the amount of spin contamination present. In Gaussian, the option "iop(5/14=2)" tells the program to use the annihilated wave function to produce the population analysis. I am not aware of any programs that use the annihilated wave function to perform the geometry optimization.

Restricted open shell calculations

It is possible to run spin-restricted open shell calculations (ROHF). The advantage of this is that there is no spin contamination. The disadvantage is that there is an additional cost in the form of CPU time required in order to correctly handle both singly occupied and doubly occupied orbitals and the interaction between them. As a result of the mathematical method used, ROHF calculations give good total energies and wave functions but the singly occupied orbital energies don't rigorously obey Koopman's theorem.

When it has been shown that the errors introduced by spin contamination are unacceptable, restricted open shell calculations are the best way to get a reliable wave function.

Within the Gaussian program, restricted open shell calculations can be performed for Hartree-Fock, density functional theory, MP2 and some semiempirical wave functions. The ROMP2 method does not yet support analytic gradients, thus the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient based method such as the Fletcher-Powell optimization must be used (note that the G94 manual implies that this may not still be functional for all cases).

Spin projection methods

Another approach is to run an unrestricted calculation then project out the spin contamination after the wave function has been obtained (PUHF, PMP2).

A spin projected result does not give the energy obtained by using a restricted open shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state rather than being optimized to describe the spin projected state.

A similar effect is obtained by using the Spin Constrained UHF method (SUHF). In this method the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice small positive values remove most of the spin contamination.

Half-electron approximation

Semiempirical programs often use the half electron approximation for radical calculations. The half electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in a consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single determinant calculation is used, there is no spin contamination.

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half electron calculation for the triplet. Koopman's theorem is not obeyed for half electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable.

Further information

Some discussion and results are in
W. J. Hehre, L. Radom, P. v.R. Schleyer, J. A. Pople "Ab Initio Molecular Orbital Theory" Wiley (1986)

An article that compares unrestricted, restricted and projected results is
M. W. Wong, L. Radom J. Phys. Chem. 99, 8582 (1995)

Some specific examples and a discussion of the half electron method are given in
T. Clark "A Handbook of Computational Chemistry" Wiley (1985)

A more mathematical treatment can be found in the paper
J. S. Andrews, D. Jayatilake, R. G. A. Bone, N. C. Handy, R. D. Amos Chem. Phys. Lett. 183, 423 (1991)

SUHF results are examined in
P. K. Nandi, T. Kar, A. B. Sannigrahi Journal of Molecular Structure (Theochem) 362, 69 (1996)

An expanded version of this article will be published in "Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems" by David Young, which will be available from John Wiley & Sons in the spring of 2001.

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