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Spin Contamination (Read 5292 times)
Gerrit-Jan Linker
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Spin Contamination
22.12.09 at 10:29:39
 
Spin Contamination
 
Hartree Fock calculations on singlet systems (Total spin S=0; multiplicity S(S+1)=1) are usually done using spin restricted Hartree Fock (RHF). in RHF each occupied MO contains 2 electrons.
Open shell systems can be calculated using unrestricted HF (UHF) in which there is a separate set of orbitals for alpha and beta electrons. The resulting wavefunction is however no longer an eigenfunction of <S2>. The deviation depends on the non orthogonal character of both sets of orbitals.  
Other configurations with the same Ms value can mix in. For a singlet this mixture can contain the closed shell singlet, the open shell singlet and the Ms=0 component of the triplet. In systems that are not well described by a single determinant (multi-configurational systems) the value for <S2> is found to be higher. This is called spin contamination and it gives information on the multi-configurational character of the system.
 
A measure for the spin contamination is the deviation of the computed expectation value <S2> and s(s+1) where s is half the number of unpaired electrons. In organic molecules spin contamination is negligible if the difference is less than 10%.
 
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 can be state higher or lower in energy. Spin contamination is not a systhematic error and can lead to a too high or too low energy.
 
As a workaround for spin cotamination ROHF can be used instead of UHF.
 
Source:
Spin Contamination  
http://www.ccl.net/cca/documents/dyoung/topics-orig/spin_cont.html
ROHF
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1285938955/0#0
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« Last Edit: 27.04.13 at 16:51:02 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Gerrit-Jan Linker
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Posts: 75
]Spin Contamination
Reply #1 - 02.05.13 at 13:48:37
 
Spin Contamination
 
The determinant |ab| is not an eigenfunction of S^2 but the following linear combinations are:
 
|ab|+|ba|= (ab+ba)(α-α)      (Ms=0 ; Singlet)
|ab|-|ba|= (ab-ba)(α+α)       (Ms=0 ; Triplet)
 
Calculating the expectation value of S^2 for these 2 determinantal wavefunctions:
 
<|ab|+|ba| S^2 |ab|+|ba|>=  
<(ab+ba)(α-α) |S^2| (ab+ba)(α-α)> =
<(ab+ba) | (ab+ba)><(α-α) |S^2| (α-α)> =  
<(α-α) |S^2| (α-α)> =
<(α-α) |0 (α-α)> = 0 = S(S+1) => S=0
 
<|ab|-|ba| S^2 |= |ab|-|ba|> =
<(ab-ba)(α+α) |S^2| (ab-ba)(α+α)> =
<(ab-ba) | (ab-ba)> <(α+α) |S^2| (α+α)> =
<(α+α) |S^2| (α+α)> =
<(α+α) |2 (α+α)> = 2 = S(S+1) => S=1
 
In short:
<(α-α) |S^2| (α-α)> = 0
<(α+α) |S^2| (α+α)> = 2
 
Now we can write the single determinant |ab| as being half singlet and half triplet:
|ab| = |S> + |T> =
(|ab|+|ba|) + (|ab|-|ba|) =
|ab|+|ba| + |ab|-|ba| = |ab|
Edited:
The factor should be 1/sqrt(2)

The orbitals a and b can be seen as average orbitals for the singlet and the triplet.
 
When we write the single determinant as half singlet and half triplet, S^2 could also be half the singlet value and half triplet value. Then this is not a proper expectation value (would it be an average value?) and the value is 1.
 
The expectation value of S^2 for a broken symmetry solution <BS|S^2|BS> where alpha and beta spins are described by other orbital spaces, can have different values as listed above. Orbitals from the alpha orbital space and those from the beta orbital space are not necessary orthogonal. This is the source for spin contamination.
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« Last Edit: 17.03.14 at 08:44:49 by Gerrit-Jan Linker »  

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Spin Contamination
Reply #2 - 02.05.13 at 17:39:59
 
Spin Contamination
 
Unrestricted states singlet |S>, double |D> and triplet |T> can be expanded using exact singlet |1>, doublet |2>, triplet |3>, ... states:
 
|S> = c_s1|1> + c_s3|3> + c_s5|5> + ...
|D> = c_d2|2> + c_d4|4> + c_d6|6> + ...
|T> = c_t3|3> + c_t5|5> + c_t7|7> + ...
 
If the leading term in the expansions is dominant then the unrestricted determinant is a reasonable to a good approximation.
 
The expectation value for S^2 for an unrestricted determinant is always too large because the contaminants have a larger value of S.
 
<S^2>UHF = <S^2>exact +N_beta + ∑ij|Sij|2
where:
<S^2>UHF = expectation value of S^2 for a UHF determinant
<S^2>exact = exact value of S  
N_beta = number of beta electrons (N_alpha>N_beta)
ij = sums running over N alpha orbs i and N beta orbs j
Sij = overlap of alpha orbital i and beta orbital j.
 
Source: Szabo and Oslund.
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« Last Edit: 02.05.13 at 17:45:13 by Gerrit-Jan Linker »  

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