Subject: HF Limit Summary

From: David Young <[email-masked]>
Date: Tue, 26 Aug 1997 07:36:44 -0500 (CDT)

Hi,

	Thank you, to all that replied about the HF limit.  The references 
that were given pointed me to answers to all but one of my questions.
Here is a summary of the replies received.

	The reference that I have still not found is the rigorous proof of
why single determinant calculations in the infinite basis set limit converge 
to an incorrect energy (the error is called the correlation energy). 
I did track down a 1951 paper mentioning that this proof had not been 
achieved at that time.  It is

T. Kato Trans. Amer. Math Soc. 70, 212 (1951)

                                Dave Young
                                [email-masked]

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The original question:

	I've searched a number of sources and not come up with anything,
so I'm hoping someone can send me a reference or point me in the right
direction.

	What I am looking for is a proof of the existence of the
Hartree-Fock limit.  Is it possible that this is an observed result for
which no derivation has ever been done?

                                Dave Young
                                [email-masked]

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From: david heisterberg <[email-masked]>

Kato proved that the hamiltonian is bounded below, at least, I think, for
molecular cases, so that establishes that there is a lower limit to the
energy.  Is that what you're asking?

Dave Heisterberg

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From: [email-masked]

Dave,

Is it the Variation Theorem that you are seeking?  If so, see:
Ira N. Levine, "Quantum Chemistry", Fourth Edition, Prentice Hall, 1991,
Chapter 8.

Bob Stolow
Department of Chemistry
Tufts University
Medford, MA  02155

e-mail:  [email-masked]

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From: Georg Schreckenbach <[email-masked]>

Hi Dave,

maybe I don't understand your question right, but isn't the HF limit
defined as the exact -- in contrast to approximate, by means of basis
functions -- solution of the HF equations? In that case, I don't think
a proof would be required, or would it ? (i.e., you are only left with
the problem that solutions must exist at all, but this is a fairly esotheric
question, from the point of view of quantum chemistry or solid state physics)
   Further, it follows from vector algebra that any function can be
represented by a complete set of functions for this space. Thus, you
must only make sure that your basis set converges to a complete set.
This should be given, for instance, for simple Gaussians or Slaters
situated on just one center (plane waves are another example of such
a set).

I would like to see your summary -- whether the above is worth two cents
of it or mothing ...

Regards, Georg

--
==============================================================
Dr. Georg Schreckenbach           Tel:     (USA)-505-667 7605
Theoretical Chemistry T-12        FAX:     (USA)-505-665 3909
M.S. B268, Los Alamos National      E-mail:  [email-masked]
Laboratory, Los Alamos, New Mexico, 87545, USA
Internet:    http://www.t12.lanl.gov/~schrecke/
==============================================================

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From: David Woon <[email-masked]>

The HF limit is a direct implication of the variation principle, i.e.,
the HF wavefunction is a variational wavefunction, possessing a rigorous
upper bound.

Dave Woon

David E. Woon                    |     [email-masked]
Molecular Research Institute     |
845 Page Mill Road               |     (650)424-9924 (voice)
Palo Alto, CA 94304              |     (650)424-9501 (fax)

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From: "Nathan A. Baker" <[email-masked]>

Isn't the proof of the Hartree-Fock limit simply the proof of the 
validity for a expansion of a function in terms of a complete set?  
Pinsky describes this in terms of Parseval's theorem in his book "Partial 
Differential Equations and Boundary-Value Problems with Applications, 2nd 
ed." on page 20.

In other words, we postulate that some trial function, a linear 
combination of n basis functions, is the solution to the Schrodinger 
equation.  If the n basis functions form a complete set (usually 
infinite), then some linear combination of these (as n goes to infinity) 
is the solution to the Schrodinger equation.

The difference between the true solution and the linear combination is 
called the mean squared convergence, which goes to zero in the limit of a 
complete set.

This probably could have been stated better, but I think I conveyed the 
general gist.

-----------------------------------------------
Nathan Baker * [email-masked]

HELP! MY TYPEWRITER IS BROKEN!
	-- E. E. CUMMINGS
-----------------------------------------------

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From: Patrick Cassam-Chenai <[email-masked]>

Dear Dave,

Check the review paper by M. Defranceschi and C. Le Bris, J. Math. Chem.
21, p.1-30 (1997) on this and related topics.
You'll find the refs. of the proof you're after. It is not
indeed as simple as Nathan Baker has written.

  Patrick Cassam-Chenai
  Laboratoire d'Etude Theorique des Milieux Extremes,
  Ecole Normale Superieure,
  24 rue Lhomond, 75231 PARIS Cedex 05, France
  Tel. (33)/(0) 1 44 32 34 70
  Fax. (33)/(0) 1 44 32 39 92

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From: "Dr. Heinz Schiffer" <[email-masked]>

Hi David,
if I understand you correctly, then there are 2 questions. The 1st is
the one about the existence of solutions of the Hartree-Fock equations
(do they have any solution at all !). The 2nd is about convergence
of basis set expansions. The 1st questions might be answered with yes
(but not conclusively, as far I understand ), see e.g.

	Elliott H. Lieb, Barry Simon
	The Hartree-Fock Theory for Coulomb Systems
	Commun. math. Phys. 53 (1977) 185-194

	Elliott H. Lieb, Barry Simon
	On solutions to the Hartree-Fock problem for atoms
	and molecules
	J. Chem. Phys. 61(2) (1974) 735-736

	G. Fonte, R. Mignani, G. Schiffrer
	Solution of the Hartree-Fock Equations
	Coomun. math. Phys. 33 (1973) 293-304

The 2nd question is very difficult to answer. People like Bob Hill,
John D. Morgan, and Werner Kutzelnigg had a hard time thinking of
it, here a few references :

	Werner Kutzelnigg
	Theory of the Expansion of Wave Functions in a Gaussian Basis
	Int. J. Quantum Chem. 51 (1994) 447-463

	John D. Morgan III
	The analytic structure of atomic and molecular wavefunctions
	and its impact on the rate of convergence of variational
	calculations
	in : M. Defranceschi, J. Delhalle (eds.), Numerical 
	Determination of the Electronic Structure of Atoms,
	Diatomic and Polyatomic Molecules, Kluwer, 1989, pp. 49-84

	Robert Nyden Hill
	Rates of convergence and error estimation formulas for the 
	Rayleigh-Ritz variational method
	J. Chem. Phys. 83(3) (1985) 1173-1196

Hope, this helps
Ciao
Heinz
-- 
Dr. Heinz Schiffer              Phone   ++49-69-305-2330                        
Hoechst CR&T                    Fax     ++49-69-305-81162                       
Scientific Computing, G864      Email   [email-masked]           
65926 Frankfurt am Main                 [email-masked]

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From: "Dr. Bruno Manunza" <[email-masked]>

Dear Dave, if you are looking for a mathematical proof of the variational 
theorem on which the Hartree-Fock theory is based, you may have a look to:
H. Margenau, G.M. Murphy. The Mathematics of Physics and Chemistry. Van 
Nostrand. Toronto. 1956. The theorem is discussed in the chapter 11.18 
(Quantum Mechanics-Variational Method).

Bye
Bruno

Dr Bruno Manunza
DISAABA (Dept. of Agricultural Environm. Sci)
University of Sassari
V.le Italia 39
07100 Sassari, ITALY
phone: 39 79 229215
fax:   39 79 229276
e-mail: [email-masked]
e-mail: [email-masked]
e-mail: [email-masked]
web: http://antas.agraria.uniss.it

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From: Richard Bone <[email-masked]>

I believe that the answer given to you by Bruno Manunza is closer to what you
need. 

Some of the other references broadcast to the list are pertinant to perturb-
ation theory methods.

Anyway, if I understand you right, you are looking for the theoretical proof
that there is a "Hartree Fock Limit": i.e., that, in the limit of an infinite
basis set, the lowest eigenvalue solution of the H-F equations converges from
above.  This is a feature of variational methods and was proved rigorously by
MacDonald though I think that you might need to look up some discussion of the
Rayleigh-Ritz "variational principle" in mathematics text books. 

See:

J. K. L. MacDonald, Phys. Rev., 43, 830, (1933)

Hope this is on the right track.

Yours,

Richard Bone

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