Subject: SUMMARY:SOMO of the radical

From: "Adrian Radomir Jaszewski" <[email-masked]>
Date: Tue, 18 Apr 2000 17:43:25 MET

Dear CCL'ers,

Last week I asked this question on the CCL :

>From N linearly independent basis functions of the radical 
>(S=1/2) one can construct N alpha and N beta orbitals:
>N alfa = (n-1) CORE + 1 SOMO occupied + (N-n) VIRTUAL
>N beta = (n-1) CORE + 1 SOMO unoccupied + (N-n) VIRTUAL
>where n denotes the number of electrons. 
>
>Unrestricted DFT wavefunctions of the radicals are usually 
>constructed using identical (or almost identical) spatial 
>orbitals for different spins (alpha, beta) in the case of 
>CORE and VIRTUAL MO. Lowest energy (occupied) alpha-spin 
>SOMO and highest energy (unoccupied) beta-spin SOMO can be 
>also very similar to each other but sometimes they are 
>strongly asymmetrical. A knowledge of the SOMO contour can 
>be helpful in the analysis of the spin density distribution 
>as well as the values of the calculated hyperfine coupling 
>constants for the radicals.
>
>My question is:
>When SOMO occupied (alpha) and SOMO unoccupied (beta) are 
>dissimilar which of them (or what combination of these SOMOs) 
>can be used to aid spin density analysis? In some cases I 
>have observed that the spin density contours were similar 
>to the highest energy beta-spin SOMO (unoccupied!!) contour 
>rather than to the occupied alpha-spin SOMO; and what is 
>the energy (in Hartree) of that SOMO when:      
>E(lowest energy VMO)= -0.06916(alpha) -0.06008(beta)
>E(highest energy SOMO)= -0.12635(beta)  
>E(lowest energy SOMO)= -0.26301(alpha)
>E(highest energy DOMO)= -0.24908(alpha) -0.24319(beta)?
>

I summarize below the replies I obtained very quickly to my 
question. They have been very helpful. I thank very much all 
those who supplied information. 

With my best regards,

Adrian

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This is pretty normal behavior and has a fairly simple explanation.

Imagine a molecule with 3 electrons (2 alpha and 1 beta) and imagine 
orbital energy increases #1 < #2 < #3 ... This makes the occupied 
orbitals MO #1 alpha, MO #2 alpha and MO #1 beta, and the unoccupied 
orbitals MO #2 beta, MO #3 (alpha or beta) and so on. The orbitals 
that you are called "SOMO occupied (alpha)" and "SOMO unoccupied 
(beta)" are MO #2 alpha and MO #2 beta in my nomenclature.

Now consider two cases:
Case #1: MO #1 alpha looks like MO #1 beta. In this case, the spin 
density distribution looks like MO #2 alpha (this will be true even 
if MO #2 alpha and MO #2 beta do not look like each other).

Case #2: MO #1 alpha looks like MO #2 beta, and MO #2 alpha looks like
MO #1 beta. In this case, the spin density distribution looks like 
MO #1 alpha, and because of the similarity between MO #1 alpha and 
MO #2 beta, it also looks like MO #2 beta.

Bigger molecules can behave the same way. The point here is that 
UNrestricted wavefunctions allow different energy orderings for the 
alpha and beta orbitals (even when each alpha orbital looks like a 
beta orbital). This can lead to situations where the spin density 
distribution resembles an alpha orbital (or a group of orbitals) 
other than the alpha HOMO (what you call the alpha SOMO).

Hope this helps. Good luck,
-Alan

--------------
Alan Shusterman
Department of Chemistry
Reed College
Portland, OR
www.reed.edu/~alan

[email-masked] 

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It may happen that the nth alpha orbital resembles the n-1th beta
orbital and that the nth beta orbital resembles the n-1th alpha orbital.
That could be the explanation for the similarity you noticed between the
lowest beta unoccupied orbital and the total spin density.

If you are using Gaussian, you might try calculating the unrestricted
wavefunction as usual, and then read in that wavefunction as the guess
for a restricted open-shell calculation, that is, use  Guess(Read,Only).
You can do the population analysis, but you won't find orbital energies.
If you really want those orbital energies, a restricted open-shell
calculation can be done.

Curt Hoganson
University of Delaware

[email-masked]

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Your orbital energies indicate that there might be a heavy mixing of the
alpha-spin "DOMO" and "SOMO".
Is there any reason not to perform a spin-restricted DFT calculation? This makes
your analysis much easier since the spin-density comes from the SOMO alone.
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I'm happy to share my feelings about this subject with you. In my opinion
the only way to properly calculate hyperfine couplings and spin densities
is to evaluate the spin up and spin down densities, subtract them from each
other and calculate the hyperfine coupling as the expectation value of the
appropriate operator using this spin density. Using the unoccupied beta-MO
is rather unphysical and maybe quite basis set dependent. Using the alpha
MO is also unphysical because any (unitary) mixing of  alpha MOs produces
the same total wavefunction. Thus, if another MO is close in energy to the
spin up "SOMO" it will mix with it and thereby obscure the picture.
However, the physics contained in the total electron and spin density
matrices are not affected by this mixing. Having said all this, many people
have observed in open shell transition metal complexes that the unoccupied
spin down orbitals give a clearer picture of the bonding. This is because
the corresponding spin-up orbitals maybe deeply buried under or within
ligand based orbitals due to strong exchange stabilization. Noodleman
refers to this as 'inverted bonding'. In a pictorial sense it might be o.k.
to look at the unoccupied MOs but for calculation of observables I would
personally certainly not rely on anything else but the densities themselves. 
Best regards,
Frank

[email-masked]

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You appear to assume alpha means lower energy. I think most progems just 
arbritarilly assign alpha and beta spins. Hence in your case what the 
program calls the lowest unoccupied beta is actually the SOMO i.e where 
the unpaired electron (density) is.

Patrick J O'Malley

[email-masked]

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