From:	Victor Geskin <Victor..at..averell.umh.ac.be>
Date:	Wed, 4 Jun 2003 11:31:51 +0200 (DFT)
Subject:	CCL: Orbitals: a practical approach of a theoretical chemist

Dear CCLers,

I was absent from office, only by now I have read all the discussion - a
very stimulating one, and I would like to dwell on several points that
have not received due attention, unless I overlooked something. A
practical approach of a theoretical chemist.

An almost trivial introduction.

Scientific papers contain words, not only tables with numbers and graphs.
And these words not only describe which number in a table is greater than
another one, but tend to explain why. That is, as natural scientists and
human beings we need images and qualitative models. We strive at
discussing our results, not just cutting-pasting the numbers provided by
our computers. If the computers were able to provide any number with
necessary precision at reasonable computational cost, it would be of great
importance for industry and the death of natural science; of theoretical
chemistry in particular, as far as chemical research is concerned.
Hopefully, this is not expected soon.

The words in theoretical chemistry papers are supposed to be not just
anything plausible we invent to make our numbers look less dull. (It could
be like that before QM.) We have got an underlying physical theory, which
is extremely well cast and of unquestionable applicability in our domain:
QM, of course. Our words are rooted in its formalism and can be used only
provided they do not contradict this formalism; the latter is taken for
granted.

Now closer to orbitals to illustrate this approach.

A simple-minded question (one of those I used to ask myself as a
high-school student, before getting acquainted with QM): are electrons
just immobile clouds shaped as orbitals or do electrons move within the
orbitals? Now I think that the correct answer is: no matter what electrons
are, they do move, that is their velocity is non-zero, because the
expectation value of the kinetic energy operator is non-zero.

Are orbitals something like flats in which electron pairs (or poor lonely
electrons) live? That could be an acceptable image for a Hartree product
wave function, but not for a Slater determinant: in the latter it is
blurred which electron is where. Therefore, MOs should not be viewed as
flats in a molecule-hostel.

It is widely accepted (in DFT, though not strictly proven, as far as I
know) that any electron density in our 3D space can be obtained from one
Slater determinant, squared (assuming all the numbers real), by
integrating over the coordinates af all the electrons but one. Since the
orbitals (the one-electron functions of which the determinant is built)
are orthogonal by construction, the total electron density is the sum of
orbitals squared (multiplied by the normalization factor, of course) - no
cross terms. Therefore, MOs are good for decomposing the total
electron density; I do not see a real problem in the fact that in this
sense they are defined up to a phase factor.

A corollary: the necessity of multideterminant CI wavefunctions is but an
illusion; having obtained a precise electron density, one can reconstruct
it from one determinant - built of Kohn-Sham, not Hartree-Fock orbitals. I
suppose, they do it in the group of Baerends.

However, these orbitals are not unique, even supposing that the detrminant
is unique: one more reason why they are not flats for electrons.

I would say that the electrons do not "live in" orbitals, but in many
cases they "leave from" them. I mean the following. For ionization, as
long as Koopmans works, it is reasonable to suppose that the electron
density decreases in the spatial region of a given orbital. For
excitation, when a CI or TD-DFT calculation shows which excited
determinants constitute the excited state, it means in the region of which
orbitals the electron density decreases and where it increases. In this
sense, ionization and excitation take place between (among) canonincal
orbitals.

I would like also to mention experimental spin-density determinations in
organic radicals performed some ten years ago in Grenoble, in the
laboratory of Jacques Schweitzer, by neutron scattering (neutron has spin
but no charge: most useful for spin density probing). If some MO based
calculations (I suppose these have been done) show that the total spin
density can be traced back to one MO predominantly - then we can even
admit that, in a sense, the unpaired electron even lives in a given
orbital.  This bearing in mind that we cannot say which electron is
unpaired: they are all antisymmetrized in a determinant.

In particular for Kohn-Sham orbitals, there was a tradition to justify
their meaning by Janak's theorem, stating that the partial derivative of
the total energy wrt a given orbital's occupation number is equal to the
eigenenergy of this orbital. A most confusing theorem based on an
assumption that an electron can be ionized in infinitesimal parts! (Not
the same as fractional occupation numbers of e.g. natural orbitals!) I
think, better to forget about this theorem (it must already be forgotten
since noone has mentioned it in this discussion).

In conclusion, I consciously avoided any direct reply to whether the
orbitals are real or not. I tried to exemplify that in many situations
they can be used in a quantum-mechanically well-defined sense and help us
building models that do not contradict QM.

If you are not yet too tired of this discussion, I would be most grateful
for any remarks and criticism.
 ___________________________________________________________________

      Dr Victor GESKIN         e-mail:   Victor..at..averell.umh.ac.be

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