Subject: Chem Topic: Spin Contamination

From: [email-masked]
Date: Fri, 4 Apr 1997 07:51:43 -0600

Hello all,

        I have written the following short essay for my users and am
posting it here for your enjoyment and comments.  Please let me know
if I missed any important points.

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                                Dave Young
                                [email-masked]

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                       		Spin Contamination

                                   David Young

                        Division of University Computing
                                 144 Parker Hall
                                Auburn University
                                Auburn, AL 36849

WHAT IS SPIN CONTAMINATION 

	Introductory descriptions of Hartree-Fock calculations (usually 
using Rootaan's 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 restricted 
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, <S**2>, 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 orbitals which appear to be
the desired spin state, but have a bit of some other spin state mixed in.
This results in slightly lowering the computed total energy.  However,
this lowering 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, <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.

	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(6/15=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).  This gives a correction to the energy, but does not
improve other properties.

	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.

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)

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