Subject: CCL: On orbitals and "reality"

From: "Dr. N. SUKUMAR" <nagams = = = rpi.edu>
Date: Thu, 27 Jan 2005 13:52:18 EST

I have been following the discussion on "Orbitals and Reality" with
considerable interest, but wanted to hold off responding until I had a
chance to examine the Nature paper. "Elements of physical reality" formed a
major component of the early 20th century debates between Einstein and the
Copenhagen school, but I had not realized that "reality" still constitutes
an element of scientific discourse today. "Reality" is not something that
has a rigorous scientific definition (apart from the unfortunate confusion
with the real, imaginary and complex nembers of mathematics); so I prefer
not to talk about "reality" in a scientific discussion. Experimental
observations constrain the objects and language of theory; theoretical
concepts, in turn, frame the questions that experiments are designed to
answer and interpret the experimental observations. If this loop can be
closed in
an internally consistent manner, that imparts a degree of experimental
confirmation to the objects of theory. If not, it is time to either repeat
the experiments, tinker with the theoretical concepts or design new ones.
Someone mentioned unicorns. If unicorns (or dark energy or ...) help
explain experimental observations, then they will constitute legitimate
objects for scientific discourse and experimental design. That is all there
is to it! "Reality" is for philosophers (with apologies to the philosophers
here; we are scientists... why should we worry about reality? :-)

A little over a hundred years ago, theories of motion were formulated in
terms of the aether and experiments were designed to detect the motion of
bodies relative to the aether. Today our theories are formulated in terms
of spacetime and experiments are designed to detect the curvature of
spacetime. Does that make spacetime "real" and the aether "unreal"? Would
it have made a difference to our world if we had continued using the term
"aether" with new topological, rather than material, properties?

We tend to think of molecular structure as "real" -- after all, we can
"see" structure maps come out of our X-ray machines. But the X-rays are
only producing diffraction patterns and it is the software (guided by the
concepts and equations of theory) that produces the structure maps. We
often
tend to think of electrons as moving around in charge clouds and nuclei as
quasi-localized at any point in time, with definite internuclear (bond)
distances and bond angles. But there is no fundamental reason to think this
way, apart from convenience (and the thousand-fold mass difference).  The
nuclei too are delocalized in charge clouds with the protrons
indistinguishable among themselves and the neutrons likewise. But we donot
usually question whether bond lengths and bond angles are "real".

Orbitals have no meaning in VB theory or in the Hohenberg-Kohn variational
version of DFT (as distinguished from the Kohn-Sham version). These
theories cannot help design experiments to "observe orbitals" and would
interpret the observations of Villeneuve, et al, very differently. While
historically, orbitals may have derived from Bohr-Sommerfeld orbits, their
only significance in the quantum mechanics of many-electron systems derives
> from a theorem of linear algebra which states that the eigenfunctions of a
Hermitian operator (e.g. the hydrogen atom Hamiltonian) form a complete set
for the expansion of any (wave) function. Since no one uses a complete set
(which is infinite), we could as well use plane waves or Gaussians or
wavelets for our expansions... and we do! For a computational chemist doing
CI, orbitals have about as much "reality" as primitive Gaussians have for
an organic chemist doing HF computations.

Nevertheless I donot understand the reluctance to accept the interpretation
of Villeneuve, et al's observations in terms of orbitals. It is an accepted
fact that an electron undergoing ionization does not "flow" out
continuously and uniformly from throughout the atom or molecule. The hole
formed by ionization of the electron has a definite quantum of charge, a
definite quantum of spin and a characteristic shape. Let us say Villeneuve,
et al have observed the hole formed by the ionization of an electron from
N2 ... would this be more acceptable? Note that they have observed not just
the density hole, but by coherent recombination, the relative phase as
well. So what's the problem with calling this an orbital? Unfortunately,
Villeneuve, et al donot help their case by showing a diagrammatic
representation of their reconstructed wavefunction alongside a N2 2p
sigma_g orbital "from an ab initio calculation" but they omit to mention
the basis set or level of theory employed, making any quantitative
comparison moot.

The concept of an orbital has different meanings in different contexts, but
in its essence an orbital is a solution to a one-electron equation or an
"effective" one-electron equation of motion. The replacement of the
interelectronic repulsion term by an effective one-electron potential is at
the heart of the Hartree-Fock and Kohn-Sham procedures that generate a
system of one-electron equations. Within these theories, it is legitimate
to
formulate some explanations in terms of orbitals (bearing in mind the
approximations involved and thus the approximate nature of the
explanations)
and to design experiments to test these approximations (e.g. Koopmans'
theorem, Wodward-Hoffman rules).

Dr. N. Sukumar
Center for Biotechnology and Interdisciplinary Studies
Rensselaer Polytechnic Institute