Subject: orbitals

From: Rene Fournier - 1999-07-07 <renef.-at-.yorku.ca>
Date: Wed, 28 May 2003 16:51:44 -0400 (EDT)

Leif Laaksonen makes an important point: only observables
are "real" in QM.  So the question arises: is it possible to
define orbitals as QM observables.  It entirely depends on
how you define the word "orbital".  If one says orbitals are
one-electron functions and nothing more, then of course orbitals
can be made to be QM observables.  For example, just integrate
an exact squared wavefunction over all electron coordinates
but one to get the electron density, then take the square
root, multiply by a phase factor and you get "density orbitals".
[G Hunter, Int J Quantum Chem 9 (1975) 237].   They may not be
very useful as orbitals, but they are "real" in the sense of
QM observables.
    Orbitals are always defined as one-electron functions,
but THAT does not make them "unreal".  The electron density
is a one electron function and very real for any N-electron
system, not only for the H atom.
    One "ab initio" way to define orbitals is to start from
exact wavefunctions and calculate the overlap of a N-electron
wavefunction (neutral) and the ground and various excited
states a (N-1)-electron wavefunction (ion) (Dyson orbitals are
defined in such a way I think).  I'm sure there are definitions
that will make such orbitals unique, but does that make them
QM observables?  I don't think so, but I'd like to hear from
the CCL community on that.  There are other ways to define orbitals
> from exact wavefunctions, I'm sure: are any of them QM observables?
    As for Kohn-Sham orbitals, they may not be real individually
but the N lowest energy ones squared and summed give the true
electron density (an observable) and the negative of the KS HOMO
energy is exactly equal to the first IP (another observable) so
there are at least some bits of reality built in KS orbitals!

Here's some relevant articles:

1) Kohn, Becke, Parr, J Phys Chem 100 (1996) 12974
2) "Kohn-Sham Orbitals and Orbital Energies: Fictitious Constructs but
   Good Approximations All the Same"
   by S. Hamel, P. Duffy, M.E. Casida, and D.R. Salahub
   Journal of Electron Spectroscopy and Related Phenomena (2002?)
3) Chong, Gritsenko, Baerends, J Chem Phys 116 (2002) 1760

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