Subject: CCL:HYBRIDIZATION NOT "REAL";V. FOCK

From: Keith Refson <[email-masked]>
Date: Tue, 12 Sep 2000 11:05:18 +0100

The question of the "reality" of hybrid orbitals seems moot since
even unhybridized orbitals are not physical observables.  

I would like to raise what seems to me to be a more pertinent question 
-- that of the usefulness of that concept.  I ran up against this recently in
teaching a course in quantum mechanics of mineral structures and
bonding, and found myself questioning whether hybridization has any
explanatory power at all.  Here's my reasoning.

"Hybridization" is not derivable from quantum mechanics.  An sp2 or
sp3 orbital is not an eigenstate of any Hamiltonian of which I am
aware. One can certainly construct linear combinations of eigenstates
in quantum mechanics, and a linear combination of eigenstates is also
an eigenstate if its components are degenerate.  But of course an s
and a p state are non-degenerate in an isolated atom, so the hybrid is
meaningless in the framework of quantum-mechanics.

What QM *does* imply is that if we have a Hamiltonian with, for
example C3 or Td symmetry then the eigenstates *will* be
symmetry-adapted functions. These can be expressed as linear
combinations of spherical harmonics which might look somewhat like 
(but are not) the sp2 and sp3 hybrid orbitals respectively.

I can not find any sound argument in the textbooks to identify in
what circumstances an sp2 hybrid might be "stable" over sp3 or vice
versa or over unhybridized orbitals.  The undergraduate physical
chemistry texts I have read simply assert without evidence that
hybridization can occur. ( Interestingly enough the small sample of
"real" quantum mechanics books on my shelf do not mention
hybridization at all!)

The semi - implicit assumption in elementary discussions of
hybridization is that the character of hybridization depends on the 
number of atoms to which bonds are being formed. 

To paraphrase the argument:

   An sp3 hybrid orbital has something of the character of
   an eigenstate of a Hamiltonian with Td symmetry.  (I can not come
   up with a better justification for its existence than this.  Anyone 
   else?).   Elements such as C and Si which form sp3 bonds tend to
   prefer tetrahedral co-ordination environments in molecules and
   solids.

Electron-counting and valence arguments won't do either, as they
can not explain why C can bond as either graphite or diamond. 

I can not see how to teach these concepts without resorting to
unsubstantiated assertion or circular reasoning.  I would really like
to give my students some idea about the bonding "preferences" of Si
and C which may be non-rigorous but can not be simply wrong.  

There is no way I am aware of to discuss energetics other than with
full MO theory -- but the fact that a full HF or DFT calculation
predicts this does not really count as an "explanation" either. I would 
appreciate any suggestions, especially those which can be treated at
an elementary level.

Keith Refson

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-- 
Dr Keith Refson,        "Paradigm is a word too often used by those who would
Dept of Earth Sciences      like to have a new idea but cannot think of one." 
Parks Road,                  -- Mervyn King, Deputy Governor, Bank of England
Oxford OX1 3PR, UK        
Keith.Refson@                       Tel: 01865 272026
             earth.ox.ac.uk         Fax: 01865 272072