Subject: ELECTRON CORRELATION SUMMARY

From: "E. Lewars" <[email-masked]>
Date: Mon, 13 Jul 1998 17:52:11 -0400 (EDT)

Mon 1998 July 13

Thanks to all who replied to my questions about electron correlation.
Here are the answers I got.

    E. Lewars
====================
     Questions

Tues, 1998 July 7

            SOME QUESTIONS ABOUT ELECTRON CORRELATION AND THE
            HARTREE-FOCK METHOD

Hello,

The HF method (HFM), even at the basis set limit, gives an energy which is
higher than the exact expectation value of the Hamiltonian, the energy
difference being the correlation energy.  The higher value of the HF energy is
said to be due to overestimation of *potential* energy  ("the HF method always
underestimates the kinetic energies of the electrons"--Pilar, 2nd ed, p 286),
specifically electron-electron repulsion.  The HFM is also said to
underestimate the coulomb hole ("the coulomb hole is neglected almost
completely" -- Pilar, p 296/297) and to overestimate the fermi hole (Pilar 296).

  So the HFM gives an energy that is too high, because it overestimates el-el
potential E ; as far as *kinetic* energy goes, the HFM would give an energy
that is too *low*.

QUESTIONS:

1)  Does anyone question any of the above statements?

2)  If the fermi hole is  overestimated, should this decrease el-el
repulsion, since the fermi hole means a region around each el. unfriendly to
other electrons of the same spin--if any two electrons avoid one another they
repel one another less;  in which case:

3)  Shouldn't the neglect of the *coulomb hole* be the real cause of the over-
estimation of el-el repulsion?  In other words, shold not most of the el-el
repulsion in the HFM be between electrons of *opposite* spin (electrons of the
same spin avoiding one another because of the Pauli effect (i.e. because of
the fermi hole)?

4)  Overestimation of the fermi hole is simply a result of using a one-
determinant wavefunction--right?

5)  Is there a way to see *intuitively* that the HFM must overestimate
electron-electron repulsion and underestimate electron kinetic energy?

   Thanks
    E. Lewars
========================

ANSWERS:

[1]
         Jul 7  Samuel A. Abrash   (75)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 2/19 from Samuel A. Abrash                  Jul 07 '98 at 12:10 (noon)

X-Sender: [email-masked]
Subject: Re: CCL:QUESTIONS:ELECTRON CORRELATION
At 11:42 AM 7/7/98 -0400, you wrote:

>Tues, 1998 July 7
>
>            SOME QUESTIONS ABOUT ELECTRON CORRELATION AND THE
>            HARTREE-FOCK METHOD
>
>Hello,
>
>The HF method (HFM), even at the basis set limit, gives an energy which is
>higher than the exact expectation value of the Hamiltonian, the energy
>difference being the correlation energy.  The higher value of the HF
energy is
>said to be due to overestimation of *potential* energy  ("the HF method
always
>underestimates the kinetic energies of the electrons"--Pilar, 2nd ed, p 286),
>specifically electron-electron repulsion.  The HFM is also said to
>underestimate the coulomb hole ("the coulomb hole is neglected almost
>completely" -- Pilar, p 296/297) and to overestimate the fermi hole (Pilar
296).
>
>  So the HFM gives an energy that is too high, because it overestimates el-el
>potential E ; as far as *kinetic* energy goes, the HFM would give an energy
>that is too *low*.
>5)  Is there a way to see *intuitively* that the HFM must overestimate
>electron-electron repulsion and underestimate electron kinetic energy?
>
Intuitively, the Hartree-Fock method has the electron interacting with the
average position of the other electron in its two electron orbital.
However, when an electron in a two electron orbital is at a given point in
space, the wavefunction of the other electron is substantially less
isotropic than the wavefunction used in the Hartree fock calculation, and
is peaked as far as possible from the position of the electron under
consideration.  Thus the hartree fock equation has the electron of interest
interacting with the other electron at close distances far too much of the
time.  Since the average radius of interaction of the two electrons in the
orbital is smaller than the true value, the electron-electron repulston
energy is too high.
>    E. Lewars
>========================
Samuel A. Abrash
Associate Professor
Department of Chemistry
University of Richmond
Richmond, VA  23174
(804) 289-8248
Fax: (804)289-8482
[email-masked]

"I believe in the open mind, but not so open your brain falls out."
====================
[2]
          Jul 7  Alan Shusterman    (24)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 3/19 from Alan Shusterman                        Jul 07 '98 at 9:45 am

Date: 07 Jul 98 09:45:57 PDT
Subject: Re: CCL:QUESTIONS:ELECTRON CORRELATION
To: [email-masked]

I think your 3rd point, repulsion is overestimated between electrons of
opposite spin, is correct. HF assigns an electron pair to the same orbital and
this must create most of the correlation error.

Alan

----------------
Alan Shusterman
Department of Chemistry
Reed College
Portland, OR
www.reed.edu/~alan
=============
[3]
          Jul 7  Vitaly Rassolov    (62)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 4/19 from Vitaly Rassolov                    Jul 7 '98 at 12:02 (noon)

X-Sender: [email-masked]
Reply-To: Vitaly Rassolov <[email-masked]>
Subject: Re: CCL:QUESTIONS:ELECTRON CORRELATION

Most of the features of electron correlation can be "understood" if one
remembers that HF model is equivalent to describing the motion of every
electron in the averaged field of the others.  For instance, if this field
is not averaged, it structure is more complicated and electrons have to
do a more complicated movement, thus increasing their kinetic energy (it
does take kinetic energy to avoid other electrons!).  The same goes for
the electron repulsion energy - it should go down.  Both of these
statements are universal i.e. they apply to all "reasonable" (i.e. no
"funny" potentials with non-isolated singularites, etc) systems.

The "overestimation" of Fermi hole is en exception.  It is either not
true, or true only for some specific cases.  We know that in dense
electron gas the Fermi hole is underestimated, just like a Coulomb one.
line 1 [h for help]In fact, they are underestimated in a very similar manner, which is
somewhat surprising.  One can also see that a triplet state of
two-electron system (like He) also has its Fermi hole underestimated by
the HF model (the electron repulsion energy should be smaller for the
correlated wave function, and there is no Coulomb hole in the system, so
the whole effect is in the Fermi hole).

To summarize:

1. Yes (with respect to Fermi hole).

2. Yes.

3. Yes.

4.  The whole question is probably wrong.

5.  The averaged field explains it all.

I hope it helps,

Vitaly Rassolov.

Vitaly Rassolov                      [email-masked]
Chemistry Department                 tel. (847) 491-3423
Northwestern University              fax  (847) 491-7713
==========
[4]

  NU 5   Jul 7  Thomas A Adler     (70)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 5/19 from Thomas A Adler                         Jul 7 '98 at 10:29 am
Organization: Albany Research Center, DOE
To: "E. Lewars" <[email-masked]>
Date: Tue, 7 Jul 1998 10:29:48 PST
Subject: Re: CCL:QUESTIONS:ELECTRON CORRELATION
Message-ID: <[email-masked]>

Dear Dr. Lewars,

The best explanation that I have seen for the correlation problem is 
in Bethe and Salpeter, Quantum Mechanics of One- and Two-Electron 
Systems.  The correlation problem is due to the form of the wave 
function.  The Hartree-Fock method uses only single electron 
functions, while an exact solution to the Hamiltonian must contain 
two electron terms.

One electron function:

U = u(r_1)v(r_2)

where r_1 is the distance from the nucleus to electron 1 and r_2 is 
the distance from the nucleus to electron 2.

Two electron function:

W = u(r_1)v(r_2)w(r_12)

where r_12 is the distance from electron one two electron 2.  w(r_12) 
is missing in the HFM and is the reason the electron-electron 
potential energy is too high and the kinetic energy is too low.

The overestimate in the coulomb term is greater than the 
underestimate in the exchange term.  Therefore the net 
electron-electron potential is overestimated.  The underestimate in 
the kinetic energy is nearly the same as the net overestimate in the 
electron-elctron potential.  The electron-nuclear potential is nearly 
correct.  (In other words, I do not question the statements from 
Pilars.)

> Tues, 1998 July 7
> 
>             SOME QUESTIONS ABOUT ELECTRON CORRELATION AND THE
>             HARTREE-FOCK METHOD
> 

Thomas A. Adler
Albany Research Center, Department of Energy
1450 Queen Avenue, SW
Albany OR 97321-2198

E-mail:  [email-masked]
(541) 967-5853
=========
[5]

 N  7   Jul 7  [email-masked] (41)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 7/19 from [email-masked]                Jul 7 '98 at 2:22 pm

Subject: Re: CCL:QUESTIONS:ELECTRON CORRELATION
To: [email-masked] (E. Lewars)
Date: Tue, 7 Jul 1998 14:22:15 -0400 (EDT)
X-Mailer: ELM [version 2.4 PL23]
Mime-Version: 1.0
Content-Transfer-Encoding: 8bit

> 
> 5)  Is there a way to see *intuitively* that the HFM must overestimate
> electron-electron repulsion and underestimate electron kinetic energy?

I'm not familiar with the literature in this area, but as I read your
question I thought about it to see if it made sense, so I'd thought I'd
try responding directly to this last point.

What is missing from the HFM is the correlation, which corresponds
roughly to allowing the electrons to avoid one another.  I think this is
also what you refer to as the coulomb hole.  In that case, it seems
obvious that the coulomb repulsion integral between two mean-field
orbitals will be necessarily larger than the average of the instantaneous
repulsion between two moving electrons.  The other effects would seem to
be compensating for that.  The variational calculation still tries to
minimize the energy as best it can, and so it takes more advantage of the
Pauli repulsion than it would otherwise, and it uses larger orbitals
than it really needs which lowers the kinetic energy.

-John.
==========

[6]

   Jul 7  J. Sichel          (37)   Re: CCL:QUESTIONS:ELECTRON CORRELATCommand: Read MessageMessage 8/19 from J. Sichel                               Jul 7 '98 at 3:22 pm

X-Authentication-Warning: bosoleil.ci.umoncton.ca: sichelj owned process doing -
bs
Date: Tue, 7 Jul 1998 15:22:17 -0300 (ADT)
X-Sender: [email-masked]
Reply-To: "J. Sichel" <[email-masked]>

On Tue, 7 Jul 1998, E. Lewars wrote:

>             SOME QUESTIONS ABOUT ELECTRON CORRELATION AND THE
>             HARTREE-FOCK METHOD
>
> 5)  Is there a way to see *intuitively* that the HFM must overestimate
> electron-electron repulsion and underestimate electron kinetic energy?

The exact and HF wave function both obey the virial theorem (Levine
Quantum Chem 4/e p.441) =20
That is, <V> =3D -2 <T> and therefore <E> =3D <T> + <V> =3D - <T>.=20

So since the variational theorem guarantees that E(HF) > E(exact), we must
have T(HF) < T(exact) and V(HF) > V(exact).

John Sichel
Universit=E9 de Moncton, NB, Canada
=============