Subject: Summary Hartree-Fock, Coulomb, Fermi

From: elewars <[email-masked]>
Date: Thu, 20 Jun 2002 22:11:59 -0400

2002 June 20

Many thanks to all who replied to my query about Coulomb and Fermi
effects in HF calculations.
These replies have been very helpful.

E. Lewars
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Question:

2002 June 16

Hello,

The HF method gives "absolute" energies that are too high, because it
overestimates electron-electron repulsion. However, there are two kinds
of electron-electron repulsion: classical Coulomb (-->Coulomb hole),
arising from electric charge, and quantum mechanical or Fermi repulsion
(--> Fermi hole), which exists between electrons of like spin. The HF
approximate treatment of electron correlation  arises from the use of
just one determinant, and from the "smeared electron cloud" integration
used to get the J and K integrals.

Questions: (1) Is it now well-recognized which effect, Coulomb or Fermi,

causes the main problem with HF energies?  (2) Is there general
agreement on which approximation is worse, single-determinant or average

electron cloud (or maybe the two can't be disentangled)?

Thanks

E. Lewars
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Replies:

#1
  rothstein <[email-masked]>

Dear Dr. Lewars:

You ask an interesting question; I am keen to read the replies from
people
who are more knowledgeable on this issue than I am.

My hunch is that the repulsion between electrons of different spin will
dominate. The determinant has the Fermi correlation "built in" as it
will
vanish when electrons of the same spin occupy the same spatial
positions.
However, I don't know of any numerical studies on this.

If I understand your second question, in principle one could obtain a
smeared electron cloud from a multideterminantal wavefunction, not just
> from
a single determinant. As this is a better approximation to the true
reduced
electron distribution, it seems to me the single-determinant will be the

worse approximation.

Yours sincerely,

Stuart M. Rothstein,
Professor of Chemistry and Physics,
Director, Brock Institute for Scientific Computation (BISC),
Department of Chemistry
Brock University,
St. Catharines, Ontario,
L2S 3A1 CANADA

e_mail: [email-masked]
http://chemiris.labs.brocku.ca/~chemweb/faculty/rothstein
http://www.brocku.ca/bisc
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#2
 Ulrike Salzner <[email-masked]>
 Organization:
 Bilkent University

Dear Dr. Lewars,
the Fermi part of the correlation energy (exchange) is treated exactly
within the HF approximation. The Coulomb part is missing completely.
This is one of the reasons why DFT/hybrid methods borrow exchange from
HF.
Whether the appoximate treatment of the Coulomb correlation or the
single determinant wave function is the main problem depends on the
system. In general the missing Coulpmb correlation is the one causing
trouble because many closed shell systems are well represented by a
single determinant. If you have a system like singlet carbene or ozone
where two or several electronic configurations have almost the same
energy, HF fails badly. Also for calculating EAs or IPs where one
species is closed shell and the other is not, HF gives very bad results.

Regards,
Uli Salzner
==================
#3
  [email-masked]

Dear dr. Lewars,

The HF approximation (single determinant wave function) completely
accounts
for the Fermi correlation, i.e. the correlation of electrons of the same

spin. What is neglected in HF theiry is the Coulomb correlation and its
neglect is the main cause of the problems with SCF energies.

Robert Ponec
Institute of Chemical Process Fundamentals
Czech Academy of Sciences
Rozvovova 135
165 02 Prague 6, Suchdol 2
e-mail:[email-masked]
phone: 420 2 20390271
fax: 420 2 20920661
=================
#4
  Christoph van Wüllen <[email-masked]>

If I were to add my §0.02 to the topic, here they are:

1) The existence of the Fermi "hole" has nothing to do with spin,
quantum
or other stuff. It simply comes from the fact that a particle cannot be
at
two different locations at the same time. So, if you see an electron at
a
given place r1, then the  probability of seeing an electron at a
different place r2, integrated over r2, is N-1, where N is the number
of electrons.

2) Quantum effects (namely the wavefunction antisymmetry) require that
the
probability of finding another electron at r1 goes to zero if all spins
have
the same spin. This determines not the Fermi hole, but its value at
r1=r2 (i. e. the depth of the hole).

3) There is only one kind of electron repulsion (in nonrelativistic
theory) --
two electrons repel each other with 1/r12. The "exchange" essentially is
a
self-interaction correction: the electrostatic Coulomb term contains
the interaction of each electron with itself (this is the "smeared
cloud" approximation). However, since one electron cannot be at two
places at the same time (see above), this approximation contains the
unphysical interaction of each electron with itself. (This would also
be the case for bosons).
Hartree-Fock just usually does a little bit more than just removal of
self-interaction (because of the depth of the fermi hole).

4.) To see what is the large effect and what is a minor correction,
consider a two-electron singlet system. Here we have no difference
between Hartree and Hartree-Fock, and the "Exchange" just removes the
self-interaction.

5.) In Hartree-Fock, two electrons of like spin avoid each other at
very short distances, but two electrons of unlike spin can still bump
into each other. So the dynamical correlation should mainly come from
neglect of the "Coulomb hole".

6.) Hartree Fock is not the same as "smeared electron cloud" (see
above) -- The smeared electron cloud problems are there, however, in
most DFT treatments. Here the self-interaction is not fully
eliminated, causing wrong long-range tails of the
exchange-correlation potential.

+---------------------------------+----------------------------------+
| Prof. Christoph van Wüllen      | Tele-Phone (+49) (0)30 314 27870 |
| Technische Universität Sekr. C3 | Tele-Fax   (+49) (0)30 314 23727 |
| Straße des 17. Juni 135         | eMail                            |
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+---------------------------------+----------------------------------+
==========================
#5
  Frank Jensen <[email-masked]>

        Ad. 1) The unlike-spin correlation is significantly larger
than the like-spin correlation, and thus the Coulomb hole is the
main deficiency in HF.
        Ad. 2) The single-determinant/average electron cloud is
very closely related. I guess that a small MCSCF for e.g. a biradical
will fix the single-determinant error, but not the average electron
cloud error, but for "well-behaved" (i.e. non-MC) systems I don't think
you can disentangle the errors.

        Frank
===================
#6
   Patrick Bultinck <[email-masked]>

You were not wrong. The term Fermi correlation (a term I prefer over
Fermi
repulsion) refers to the correlation due to the antisymmetry of the
single
slater determinant.

At 08:05 17/06/2002 -0400, you wrote:
>2002 June 17
>
>Re "Fermi repulsion": that should have been Pauli repulsion, of course
>(related to the Pauli exclusion principle).
>
>E. Lewars

[Since the Fermi hole is an avoided region around an electron _A_ for
electrons of the same spin as _A_, I thought a better name for the
*cause* of the hole might be Pauli repulsion. --EL]
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