Subject: SUMMARY DFT ORBITALS, WF

From: elewars <[email-masked]>
Date: Fri, 25 Aug 2000 12:55:45 -0400

Friday, 2000 Aug 25

Hello,

here is the summary of replies to my question about orbitals and
wavefunctions in DFT. I got 12 replies:
#1 D. Mathieu, #2 and #3 E. Scerri, #4 E. Proynov, #5 H. van Dam, #6 A.
Bittner, #7 J. Spanget-Larsen, #8 U. Richter, #9 A. Shusterman, #10 K.
Nguyen, #11 J. Smith, #12 R. Jensen. I hope no one has been overlooked.
Thanks very much to all who responded.

I don't think question 3 can really be answered definitively yet. I'll
leave this topic with the thought that an electron density is an
observable (it can be measured), while a wavefunction is a recipe, a
prescription, a Vorschrift--use it in a certain way and you get the
right answers.

THE QUESTION
Sat 2000 Aug 19

Hello,

We know that in pure DFT (in contrast to wavefunction theory) there are
are no orbitals and there is no wavefunction; the Kohn-Sham orbitals of
current practical DFT methods were introduced to make the calculation of

the electron density tractable.

QUESTIONS
(1) An important feature of ab initio (and semiempirical) calculations
is that one can visualize the orbitals, particularly the HOMO and LUMO
(frontier orbitals) and make chemical deductions. In a _pure_ DFT
calculation a question about the shape and energy of these orbitals
would be meaningless, right?

(2) The Fukui function (R. G. Parr and W. Yang, J Am Chem Soc, 1984,
106, 4049) seems to be always discussed in the context of DFT; yet this
function is "a chemical reactivity index in the sense of the
frontier-electron theory of reactivity..." (R. G. Parr and W. Yang,
"Density-Functional Theory of Atoms and Molecules", Oxford, New York,
1989, p. 101). But in pure DFT theory there are no frontier orbitals.
Isn't this strange?

(3)
If, as has been suggested (Parr and Yang, "Density-Funtional Theory...",

p. 53) wave mechanics can be reformulated without the wavefunction
concept, what are we to make of the 80-year-old debate in physics and
philosophy about the meaning of the wavefunction (wavefunction collapse,

Schroedinger's cat, the many-worlds theory of QM, etc, etc, etc; see
e.g. "Einstein, Bohr and the Quantum Dilemma", A. Whitaker, Cambridge
University Press, Cambridge, 1996, and _many_ other books and
papers...). Was this debate a meaningless exercise that would never have

occurred if QM had been originally formulated in terms of electron (or
more generally, particle) density rather than the wavefunction?

Thanks,

E. Lewars
====
THE REPLIES

#1  Didier Mathieu, [email-masked]

> In a _pure_ DFT
> calculation a question about the shape and energy of these orbitals
> would be meaningless, right?

While the concept of orbitals is not necessary, it might be possible to
define
orbitals from the density obtained from "pure DFT"and its responses to
changes of
the Hamiltonian. It might then turn out that such orbitals, wile no
physical
observables, can be given some physical meaning a posteriori.

> 1989, p. 101). But in pure DFT theory there are no frontier orbitals.
> Isn't this strange?

Why ? Fukui functions are defined independently of the notion of
orbitals. Thus, I
see them as an natural alternative - whithin DFT - to frontier orbitals
and
perturbation theory in order to interpret chemical reactivity.

> papers...). Was this debate a meaningless exercise that would never
have
> occurred if QM had been originally formulated in terms of electron (or

> more generally, particle) density rather than the wavefunction?

In that case, I wonder how these physical concepts would have emerged.
Clearly, the one-particule density is not suitable as it does not
contains all the
information about the system.
However, QM has also been formulated in terms of the density matrix
since the
beginning.
For me, probably for historical reasons, it is easier to grasp many QM
ideas in
terms of the wavefunction, also because some of the basic notions
involve
"interference effects", (sorry for being so conditioned by the
wavefunction
picture) and the phase of the wavefunction might be more suitable than
the
coherences of the density matrix to study them.
On the other hand, in some cases, the influence of the environment on
the quantum
system considered is more naturally introduced in the density matrix
formulation:
for instance when adding the phenomenological relaxation rates T1 and T2
to the
evolution equation in NMR. This sounds more natural than modifying the
Shrodinger
equation through adding a non-hermitian term to the hamiltonian.

Cheers,

Didier.

===
#2  Eric Scerri,  [email-masked]

But surely the shape of the electron density which would correspond
to these orbitals could still be visualized if necessary, just as
electron density can be in the ab initio wavefunction approaches.

The two approaches are alternative ways of looking at the micro
world.  Whether one chooses to visualize, or not, is always an option
in any theory.  But perhaps you ar asking a more specific question
like what do HOMO and LUMO electron density plots look like?

Please clarify your question (1)

eric scerri

-------------------------------------------------
Eric Scerri PhD,
Visiting Professor,
Department of Chemistry & Biochemistry,
UCLA,
Los Angeles,  CA 90095
USA
---
I replied:
Hello,

I'll summarize the answers, maybe with some comments. But concerning
your answer:
in wavefunction theory we have a series of atomic or molecular orbitals,
psi_1,
psi_2,..., with increasing energy. I think that in a pure density
functional theory
we just have an electron density function, rho(r); not a series of
electron
densities. So there is no electron density rho_1 corresponding to a
certain
orbital , and another density rho_2 corresponding to another orbital,
etc. Just one density function. But maybe I have overlooked
something--so I am
asking the chemical community for their thoughts. And it just occurrs to
me that
one might wonder how this fits in with photoelectron spectroscopy, which
indicates
that molecules have groups of electrons of different energies.

thanks

EL
====
#3  Eric Scerri

I dont know enough about DFT to know the answer to this further
point.  I would guess that it is possible to recover some kind of
shell tructure from DFT.  Please let me know what you discover from
others.  I am also still puzzling over your No 3.

You ask interesting philosophical questions.  Please consider joining
the Philchem list.

  the address can be found on the web pages for the International
Society for the Philosophy of Chemistry after the section on the
recent meeting in Poznan.

====
#4 Emil Proynov,  [email-masked]

Dear Dr Elewards,

I would suggest not to get so upset by those who claim that the
Kohn-Sham (KS) determinantal
wavefunction does not have a physical meaning. The SCF potential, as
defined
in the KS DFT scheme, determines a self-consistent field such that the
KS determinant
minimizes the electron kinetic energy and yields the exact electron
density. It
differs from the Hartree-Fock (HF) self-consistent field in what the
latter is associated
with the HF determinant (minimizes the total energy but does not yield
the
exact electron density). To any meaningful self-consistent field one can

assign SCF one electron orbital picture with a portion of
physical/chemical meaning in it.
To my opinion there is nothing mysterious in the KS SCF field, just to
keep in mind that
it does not originate directly from the Schrodinger equation
but from the Euler equation of the DFT variational task. It is true that
in the latter
one does not deal directly wavefunctions, but all the quantum mechanics
with its
postulates, theorems and paradoxes is hidden there in the
exchange-correlation potential.

You can find different opinions on this in the literature indeed. The
matter is perhaps
more complicated that I am trying to make it with this comment. Just to
note that
an increasing amount of theoretical studies use and refer explicitly to
the KS
SCF orbitals demonstrating their practical utility (see for example: I.
Verdernikova et.al.
Int. J. Quant. Chem. vol.77, 161 (2000)).

Best wishes,

Emil Proynov

+-----------------------------+
| Emil Proynov                |
| [email-masked] |
+-----------------------------+

=====

#5  Hub van Dam,  [email-masked]

(Question 1)
People try that sort of thing, yes. But I am not sure if this really
works in
cases where there is strong static correlation. In other words where you
have a
wavefunction (again) with multi configurations with large coefficients.

(Q2)
Yes, this seems very strange.

(Q3)
The 74 year old debate, autumn next year the Schroedinger equation will
have been
around for 75 years (I haven't heard any plans yet of how people want to

commemorate this anniversary). I would like to think that quantum
mechanics only
obtained a firm foundation with Schroedingers wave-equations. Anyway, I
don't
think this was a meaningless exercise. Although the wavefunction in
itself has no
direct physical interpretation, we still need it to get the probability
density
right. I.e. the probability to find a particle at a certain point in
space behaves
like there is some superposition of waves involved. Trying to formulate
quantum
mechanics in terms of the electron density only, and at the same time
including
these superposition effects correctly is a far from trivial matter (I
think it is
save to say that this problem is unsolved). Therefore I think that
although the
wavefunction has not direct physical interpretation it is far from
"meaningless"
concept.

Kind regards, Huub

--

========================================================================

Huub van Dam                               E-mail: [email-masked]

CCLRC Daresbury Laboratory                  phone: +44-1925-603362
Daresbury, Warrington                         fax: +44-1925-603634
Cheshire, UK
WA4 4AD

======
#6  Eric Bittner,  [email-masked]

No, There is the issue of measurement, interference, causality, etc..
which
you can not simply sweep under the rug by invoking a density.
The debate over the ontology of quantum mechanics continues and is
perhaps
extremely relevant in the advent of quantum computers.  Perhaps if your
only
interested in using quantum theory as an algorithm for making
predictions
about physical reality,  the debate is pointless.  However, if you
really
and truly want to understand the inner workings of nature, it is not.

IF you do choose to only work with the density, rho(x), you are not
representing the full physical picture.
QM is completely formulated in terms of
the density, which evolves under the continuity equation, AND the
quantum
Hamilton-Jacobi equation,

\frac{\partial S}{\partial t} - H(p,q) -
\frac{\hbar^2}{2m}\frac{1}{\sqrt{\rho}}\nabla^2 \sqrt{rho} = 0

where H(p,q) is the Hamiltonian with the canonical substitution p->
\nabla
S/m and the last term is the quantum potential, Q.

If you seek stationary states, i.e. d\rho/dt = 0, then the QHJ  reduces
to
the time-independent Schrodinger equation.

These equations of motion
are easily derived from the Schrodinger eqn. by writing
 psi = R exp(i S/hbar)
and defining the density as rho = |psi|^2.

On the computational side of this:
Bob Wyatt and I have recently developed methods to solve these equations
of
motion using ONLY the trajectories they imply...no wave function, no
basis
functions...only quantum paths.
A pre-print version of one of our papers is included as an attachment
(pdf
format).

-Cheers,
Eric R. Bittner
U. Houston

--
Prof. Eric R. Bittner
Department of Chemistry
Univ. of Houston
==========

#7  Jens Spanget-Larsen,  [email-masked]

(Q1)
Or, in other words: "What Do the Kohn-Sham Orbitals and Eigenvalues
Mean?".
Stowasser and Hoffmann address this question in their recent paper -

Ralf Stowasser & Roald Hoffmann: J. Am. Chem. Soc. 121, 3414-3420
(1999).

Yours, Jens >--<

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
JENS SPANGET-LARSEN         Phone:  +45 4674 2000  (RUC)
Department of Chemistry             +45 4674 2710  (direct)
Roskilde University (RUC)   Fax:    +45 4674 3011
P.O.Box 260                 E-Mail: [email-masked]
DK-4000 Roskilde, Denmark
=========

#8  Uwe Richter,  [email-masked]

Hi,

I'm very interested in answers to this questions. Please could you
summarize
or send me the responses directly? There are some people who even
discuss
virtual KS orbitals.

Thanks,
Uwe
===========

#9  Alan Shusterman,  [email-masked]

These are great questions. I'd be grateful if you would pass along any
answers you receive. Thanks,

-Alan
[email-masked]

====
Alan Shusterman
Department of Chemistry
Reed College
Portland, OR
=========

#10  Kiet Nguyen,  [email-masked]

Although there is considerable controversy in the literature concerning
the meaning of
Kohn-Sham orbitals,  I think they are far from "meaningless".  This
topic has been carefully
analyzed and reviewed.[1-5]   Parr and Yang stated that "... one should
expect no simple
physical meaning for the Kohn-Sham orbital energies."[3]  However,
using the Janak theorem
which "provides a meaning for the eigenvalues of the Kohn-Sham
equation", they have connected
the HOMO and LUMO energies to the ionization potential and electron
affinity,
respectively.[3]

[1]   E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997)
5383.
[2]   R. G. Parr and W. Yang, Density Functional Theory of Atoms and
Molecules (University
Press,  Oxford, 1989).
[3]   J. P. Perdew, in: Density Functional Methods in Physics, edited by
R. M. Dreizler and
J. da Providencia New York, 1985), p. 265
[4]   E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997)
5383.
[5] R. Stowasser and R.  Hoffman, J. Am. Chem. Soc. 121 (1999),  3414.

Kiet A. Nguyen
AFRL/MLPJ
Laser Hardened Materials Branch
Wright-Patterson AFB, OH  45433
Phone (937) 255-6671, Ext 3178
FAX   (937) 255-1128

==========

#11  Jack Smith,  [email-masked]

(We know that in pure DFT...no wavefunction)
At least not explicitly.  One-electron orbitals are not required for
wavefunction theory either.  I would say that the two approaches
converge
when you reduce it to N-electron density matrices (probably even at the
2-electron density matrix level).

(The KS orbitals...electron density tractable)
Particularly to make the kinetic energy part tractable.  Orbitals also
made
wavefunction theory more tractable by introducing the
pseudo-one-particle
SCF equations, which further allowed for the formulation of the Roothan
equations (for a basis set expansion approach).

> My feeling is that Natural Orbitals from a CI calculation
(eigenvectors of
> 1-particle density matrix) should correspond to KS orbitals with
> fractional occupations.  Is there an analog to Natural Geminals (from
the
> 2-particle density matrix) in DFT?

(In a pure DFT calc., a question about the shape and energy of these
orbitals would be meaningless, right?)
Orbitals, yes; orbital densities, no.  The density differences between
the
neutral system and its various ionic states are in principle observable
quantities (assuming you can get xray diffraction data for the ions).
Under
Koopmans' approximation these density differences should be the same as
the
orbital-derived densities.  And the orbital energies should correspond
to
the ionization (electron attachment) energies.  A similar correspondence
can
be defined for orbital density differences among different excited
states.
One thing that's missing from a pure density picture is the concept of
overlap and orthogonality, which depend on the sign of the underlying
one-electron orbitals (wavefunctions).  Overlap is a key component of
frontier orbital theory.  Should it be?  Is there a pure density analog?

(The Fukui function.. But in a pure DFT theory these are no orbitals.
Isn't this strange?)
But there ARE frontier orbital DENSITIES.  The +/- Fukui functions
should
correspond to the orbital densities mentioned above for whole electron
changes (N +/-n).  However, in an "open" sysem (in a Grand Canonical
sense)
where fractional electrons can come and go, the Fukui function can also
be
defined for infinitessimal changes in electron number - a continuous
functional of the number of electrons.  Thus, derivatives in the density
wrt
electron number can be defined even at N electrons.  The corresponding
derivative in energy would be the chemical potential, and the second
derivative would be the hardness -- with no need to define a HOMO or
LUMO
and use crude finite difference approximations (along with Koopmans'
approximation).

> (3)
> If, as has been suggested (Parr and Yang, "Density-Funtional
Theory...",
> p. 53) wave mechanics can be reformulated without the wavefunction
> concept, what are we to make of the 80-year-old debate in physics and
> philosophy about the meaning of the wavefunction (wavefunction
collapse,
> Schroedinger's cat, the many-worlds theory of QM, etc, etc, etc; see
> e.g. "Einstein, Bohr and the Quantum Dilemma", A. Whitaker, Cambridge
> University Press, Cambridge, 1996, and _many_ other books and
> papers...). Was this debate a meaningless exercise that would never
have
> occurred if QM had been originally formulated in terms of electron (or

> more generally, particle) density rather than the wavefunction?
>
It just means that all the debate is focused on the form of energy
functional itself, the exchange and correlation parts in particular,
rather
than the wavefunction.  One still has to explain the particle-wave
duality
of the electron beam experiments.  The outcome is still probabilistic
(non-deterministic for a given electron) whether you speak in terms of
electron density or a wavefunction.  The concepts of exchange and
correlation are still very much "quantum" phenomena, whether you
encapsulate
in a quantum-constrained probabilistic wavefunction and a classical
Hamiltonian (energy) operator, or in a classical-like probabilistic
electron
density and a quantum-constrained energy functional.

DFT-KS and HF only converge when HF exchange is used, no correlation
term is
included, AND the Grand Canonical (GC) form of the density operator is
used.
DFT-KS and MC-SCF converge as the MC-SCF approach a full CI and the
correlation functional approaches the "exact" expression, but again only
if
the GC form of the density operator is used.  An interesting middle
ground
would be a Natural Orbital iteration on a small CI, or a VB wavefunction

using non-orthogonal orbitals, or GC UHF using localized fractionally
occupied nearly-orthogonal orbitals.  (As you might tell, I'm a an
advocate
of the latter - which should have a correspondence with DFT-KS, given
the
right functional).

My 2 cents' worth.

- Jack

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
 Jack A. Smith             ||
 Union Carbide             || Phone:  (304) 747-5797
 Catalyst Skill Center     || FAX:    (304) 747-4672
 P.O. Box 8361             ||
 S. Charleston, WV  25303  || [email-masked]
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=====

#12  Roy Jensen,  [email-masked]

Have you had any response to your questions. I would be very
interested to hear the results. Please post a summary to the CCL.

Roy Jensen
===================
================
=========