Subject: Re: CCL:Orbitals and Reality

From: Brian Salter-Duke <b_duke..at..octa4.net.au>
Date: Sun, 23 Jan 2005 17:49:31 +0930

On Sat, Jan 22, 2005 at 06:04:20PM -0600, Phil Hultin wrote:
> I am getting the impression that part of the problem in this discussion is
> that people mean different things when they talk about "visualizing an
> orbital".

I think it is a more fundamental difference. Here is an example I have
used for many years. In a very good text book - Quantum Mechanics in
Chemistry, 2nd Ed by M. W Hanna on page 189. there is Exercise 7-10:-

What does the fact that the ground state of N2 is doublet-Cap sigma-g
instead of doublet-Cap pi-u tell us about the ordering of the one electron
MO's of nitrogen?

The expected answer is that the sigma-g MO lies higher than the pi-u
orbital. However, it does not. Only at miminum basis set level does the
sigma-g lie higher than the pi-u. At every other MO level of theory up
to the Hartree-Fock limit the opposite is true. MO theory does not
predict the correct ordering of the two ion states in question. The
correct answer is "Nothing at all".

The fundamental conclusion to get from this is that you can not predict
theory from experiment. You predict experiment from theory. In this case
you find out that MO theory is not good enough at the Koopman's theory
level. You have to take into account both the relaxation of the orbitals 
on ionisation and the effects of electron correlation. 
> 
> There can be no argument that the mathematical formulation of orbital theory
> is a human construct, and hence that orbitals so formulated must be
> artificial models.

Indeed, but wave functions are also a human construct, but most quantum
chemists think they are more fundamental than orbitals which are
1-electron functions used to approximate the N-electron molecular wave
function. 

> These models purport to describe the spatial probability density (if that
> isn't a redundancy) for a single electron having a particular energy in the
> vicinityof a particular nucleus.  Atoms apart from hydrogen have more than
> one electron, and our mathematical model is approximate in such cases.
> Therefore, it is highly probable that even if a particular application of
> the orbital model can give a reasonable agreement with experimental energies
> in the aggregate, the individual one-electron wavefunctions will not have
> any direct resemblance to the behaviour of any one electron in reality
> (whatever that may mean).

Exactly
 
>  On the other hand, I think it is possible (albeit not probable) that our
> orbital model captures some aspect of observable electronic behaviour.
> Thus, until someone can explain to me how Villeneuve et al's tomographic
> interpretation of their spectroscopic data is fundamentally wrong, I am
> prepared to accept that they may have visualized something real, to which
> our orbital model roughly corresponds.

Yes, they probably have done this, but does the orbital model always work
and more fundamentally, are they telling us about the orbital model. The
orbital model comes from theory. Does this agree with experiment? In
many cases, yes. In many cases, no. Experiment can not predict theory.
Experiment can not tell us about orbitals. Orbitals can predict
experimental results, but they may be wrong.
 
> Note that I do not say that they have indeed visualized an orbital, because
> yes, orbitals are hypothetical objects.  And, I quite agree that you cannot
> observe a hypothetical (or perhaps imaginary) object.  So, in the strict
> sense there is no way that Villeneuve's claim to have observed an orbital
> can be correct.  Is it therefore impossible to observe the spatial
> distribution of an electron, and for this distribution to bear some
> resemblance to what our theory predicts?

It is clearly possible to observe an orbital in essentially a one-electron
system, such as an isolated H atom. I doubt it is possible to observe
the spatial distribution of an electron in N2, but one can observe, for
example, the difference between N2 and N2+. Maybe that is what they are
observing.

Yes, orbitals are hypothetical objects, but so are exact wave functions.
They are not hypothetical in the same way. The former is part of an
approximate quantum mechanical description of molecules. The latter is
valid for both approximations and exact solutions of the Schrodinger
equation. I can see that all molecules have a wave function. I understand 
that orbitals are just one way of getting to that wave function. Take the 
James-Coolodge wave function for H2 and the improvements on this once 
computers could do the grunt. There is not an orbital to be seen!
  
> I would welcome any reasoned and referenced rebuttal of my interpretation of
> Villeneuve et al's Nature paper.

I try to be reasoned.

Cheers, Brian.
 
>  
> 
> Dr. Philip G. Hultin
> 
> Associate Professor of Chemistry,
> 
> University of Manitoba
> 
> Winnipeg, MB
> 
> R3T 2N2
> 
> hultin..at..cc.umanitoba.ca
> 
> http://umanitoba.ca/chemistry/people/hultin
> 
>  
> 

-- 
            Brian Salter-Duke (Brian Duke) b_duke..at..octa4.net.au  
         Post: 626 Melbourne Rd, Spotswood, VIC, 3015, Australia
      Phone 03-92992847. http://members.iinet.net.au/~linden1/brian/