Cytoclonal Pharmaceutics Inc.

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, <S^{2}>,
thus some error
may be introduced into the calculation. This error is called spin
contamination.

As a check for the presence of spin contamination, most ab initio
programs will print out the expectation value of the total spin,
<S^{2}>.
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 <S^{2}>
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 <S^{2}> 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.

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).

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.

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.

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.*