CCL:G: Summary: Spin Density vs SOMO surfaces

 Sent to CCL by: Pascal Gosselin [Pascal.Gosselin(!)univ-lemans.fr]
 Dear CCLers,
I realized that I never posted a summary of the answers I received to the following question. 
 Here is a fix for this negligence:
 Thanks to everyone who replied to my original question about the
  differences between the Spin Density and SOMO surfaces:
Alan Shusterman, David Shobe, Noel O'Boyle, Doug Fox, Christophe Morell. 
  The answers are given below.
  Regards,
  Pascal Gosselin.
  --------------------
  Original question:
  --------------------
 I wanted to visualize the localization of the unpaired electron
 in radical-cations whose structures were optimized at the UB3LYP/6-31G*
 level in Gaussian98.
For that purpose, I asked Gaussview to compute both Spin Density and SOMO surfaces. 
 In most cases, both pictures were very similar. However, in some cases,
 I was surprised to observe two very different pictures, i.e.
the surfaces were located over completely different regions of the radical-cation.
In some cases, alphaSomo, alphaSomo-1 and betaSomo were found very close in energy. I tried to obtain a "mean view" of the system by adding the alphaSomo and alphaSomo-1 
 then subtracting betaSomo (with the cubman utility within g98 package).
 The resulting picture remained different of the Spin Density surface.
 Has somebody an explanation?
  --------------------
  Answers:
  Alan Shusterman:
 --------------------
You might want to search thru the CCL archives. I believe that this question has been asked before and rather complete answers were given.
The short answer is this: unrestricted methods (UHF, UB3LYP, etc.) assign a unique orbital to every electron (rigorously, every unrestricted orbital is a SOMO). The spin density surface is based on the difference in total alpha and total beta electron density. Since spin density depends on *all* of the electrons, there is no reason to expect the spin density surface to resemble a particular MO.
However, for many radicals, all of the occupied beta MO#i have a corresponding occupied alpha MO#i of similar shape. These orbitals are essentially double-occupied and do not contribute to the spin density. In this case, one alpha MO (usually alpha HOMO, or "SOMO" according to GaussView) contributes to the spin density. 
 --------------------
 David Shobe
 --------------------
Because it's an unrestricted calculation *every* orbital is a SOMO! (= singly occupied molecular orbital). What you're calling the SOMO is actually the highest occupied alpha molecular orbital. I would choose the spin density as more representative of where the ion has "radical character." 
 --------------------
 Noel O'Boyle
 --------------------
Perhaps you should square the SOMOs (using cubman) before adding and subtracting 
 them (since the density is the square of the MO).
 --------------------
 Doug Fox
 --------------------
    I can only guess but note that the orbitals in a spin unrestricted
 calculation are ordered by occupation and then by energy. Thus it is
 possible that the real SOMO, i.e. one which has no beta counterpart,
 can be somewhere other than alphasomo or alphasomo-1.  Check the top
 few orbitals to try and match up alpha/beta SOMOs for the top few
 orbitals.
 --------------------
 My reply to Doug Fox
 --------------------
You are right: I had already found what you call the "real" SOMO in several examples. It was for instance, the alphaHOMO-2 and the beta counterpart (same shape) was the first unoccupied beta orbital (betaLUMO). But, that's what disturbs me ! Shouldn't the "real" SOMO be the highest HOMO according to the Aufbau principle?
Or is that just an artifact due to the Unrestricted method used (UB3LYP)? As a matter of fact, ROHF results show a SOMO very similar to the Spin Density surface (ie -> no problem) but, I never succeeded in making the ROB3LYP calculations to converge, 
 despite numerous tries with different initial guesses...
On the other hand, if we admit that the SOMO is, ie alphaHOMO-2, could that fact 
 reflect a particular stabilization of the radical cation?
 --------------------
 Doug Fox 2nd reply
 --------------------
    The Aufbau principle is an ideal but not a guarantee.  The energy
 of orbitals depends a good deal on what element is involved.  For
 example the lone pair electrons on F often get in the way of defining
 an active space because the bonding orbitals fall above them in
 energy.  ROHF artifically forces the SOMO to be the highest alpha
 orbital by pairing the other orbitals.
 --------------------
 Christophe Morell (in French, not posted on the mailing list)
 --------------------
 Etes-vous sûr que vous visualisez la SOMO? En fait je
 me suis rendu compte recemment que la SOMO peut être
 quelques fois enterrée, c'est à dire que parfois ce
 n'est pas la plus haute en énergie. Je vous conseille
 donc de visualiser  les couples d'orbitales alpha et
 béta proches de la HOMO afin de vérifier quelle est
 exactement la SOMO
 --------------------
My reply to Christophe Morell (in French, not posted on the mailing list) 
 --------------------
 Merci pour vos conseils.
Effectivement, j'ai trouvé dans plusieurs cations-radicaux étudiés au niveau UB3LYP, une alphaHOMO "enterrée" dont l'allure rappelle fortement la surface de densité de spin. De plus, cette (alphaHOMO - n) ressemble à la première betaLUMO ce qui correspond bien à ce qu'on attend d'une SOMO. Dans ces exemples, le diagramme d'énergie des OM alpha et béta met en évidence le caractère "isolé" (non-apparié à première vue) de cette SOMO par rapport aux autres OM pour lesquelles on trouve pour chaque alpha une béta d'énergie très voisine. Ce qui me gêne, c'est de considérer comme SOMO une orbitale qui ne soit pas la plus haute en énergie. N'est-ce pas contraire au principe Aufbau? Par ailleurs, considérant une SOMO d'énergie inférieure à la HOMO, cela pourrait-il indiquer une stabilisation particulière du cation- radical correspondant par rapport à un autre R+. dont la SOMO serait bien la HOMO? 
 --------------------
 Christophe Morell 2nd reply (in French, not posted on the mailing list)
 --------------------
 vos questions sont délicates, je n'ai malheureusement
 pas de réponses définitives en la matière.
 Néanmoins, ce qui est certain c'est qu'il faut être
 prudent avec les radicaux. Car en général les radicaux
 sont des réacitfs mous, c'est à dire dont les
 orbitales frontières sont mals séparées. Cela implique
 que plusieurs configurations électroniques sont à
 envisager (d'un point de vue chimie quantique
 plusieurs déterminants).
 Si vous regarder le livre de Nguyen Trong Ahn
 orbitales frontières manuel pratique, à la page 95,
 les différentes configurations possible pour un
 radical sont proposées. Si la configuration
 fondamentale, la SOMO est la HOMO, dans d'autres
 configuration la SOMO est "enterrée". Le poids de
 chacune des configuration est pris en compte dans un
 calcul CI interaction de configuration. Je serais pret
 à prendre le pari que le calcul en MP2 de votre
 radical implique un forte contamination de spin.
 Pour ce qui concerne votre conclusion concernant la
 réactivité j'aurais tendance à y souscrire d'un point
 de vue purement intuitif. Pour comparer la réactivité
 de vos radicaux, une solution est de calculer la
 dureté de vos radicaux.
See p.95 in N. Trong Anh, Orbitales frontières : manuel pratique, InterEditions / CNRS Editions, 1995