Subject: answers to "existence of HF-SCF solutions"

From: Pablo Echenique Robba <pnique +*+ unizar.es>
Date: Fri, 10 Jun 2005 08:50:57 +0200

I present here the answers to my question about the existence of HF-SCF solutions and the possibility of finding them. The answers are not all from the CCL list but they are indeed interesting.

--- My question -------------------------------------------------------

What I wanted to know is whether someone could indicate to me where may 
I find a discussion on the existence of solutions of the HF-SCF 
iterative procedure. I first enumerate briefly what I know:

1) The HF equations HAVE solutions (Lieb and Simon, 1977).

2) Any solution of the HF equations is an extremal, with no indication 
about its being a maximum, a saddle point or a minimum whatsoever.

3) The SCF method DOES NOT always converge. Of course, when it does 
converge, the wave function that results is indeed a solution of the HF 
equations. Again, with no indication about its being a maximum, a saddle 
point or a minimum.

My two questions are related to point 3:

a) When does the SCF converge?

b) How do we know, if it converged, that it did it to a minimum (and not 
to a maximum or to a saddle point)?

I would thank very much any hint to articles or books discussing this. I 
mean, further than the typical "we hope it converge because our basis 
sets and our starting point are so good".

--- Answers -----------------------------------------------------------

By now, there are indeed some results going further, from numerical 
mathematics experts; see for example Volume X of the series "Handbook
of Numerical Analysis", which is a special volume on Computational 
Chemistry (Elsevier, 2003). In the first chapter, a 268 pages long 
textbook on computational quantum chemistry in itself, discusses results 
to at least some aspects of your question.

Hope this helps,
Bernd

---
Prof. Dr. Bernd Hartke             e-mail: hartke +*+ phc.uni-kiel.de
Theoretical Chemistry              phone : +49-431-880-2753
Institute for Physical Chemistry   fax   : +49-431-880-1758
University of Kiel                 http://ravel.phc.uni-kiel.de/
Olshausenstrasse 40
24098 Kiel
GERMANY

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Dear Pablo,
We have been working on globally convergent iterative methods for
HF-SCF equations (J. Chem. Phys. 121, pp. 10863-10878 (2004)) and we
have discussed in that paper some properties of the HF equations in
that sense. Also Cancis and Le Bris published a paper on the
properties of SCF methods, in which they discuss the convergence of
the classical fixed-point iteration (Math. Modell. Numer. Anal. 34,
749 (2000)) and introduce some interesting scf methods. For an
analysis of the HF problem in a very theoretical perspective (also
much more difficult to follow) you can check the work of P. L. Lions
(COMM. MATH. PHYS. 109 (1): 33-97 1987).
We are now working on new methods and we now know several properties
of the scf energy, if you are interested we send you the paper when it
is ready.  As far as we know, we can't know whether the solution found
is a minimum, saddle point or maximum, but we know that the lower the
energy the better it is. The book suggested by Bernd probably contains
the results of Cances and Le Bris on that subject.
Best regards,
Leandro.

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Hello Pablo,
I can suggest you two references (although they are perhaps made superfluous
by the more recent reference cited by Bernd Hartke):
M. Defranceschi, C. Le Bris, "Computing a Molecule: A Mathematical
viewpoint", J. of Math. Chem. 21, 1 (1997)
M. Defranceschi, C. Le Bris, "Computing a Molecule in its Environment: A
Mathematical viewpoint", Int. J. of Quant. Chem. 71, 227 (1999)

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Hi Pablo,

I see two issues here. One is the analicity of the HF equations. And
they do have solutions. The other issue is the computational or
numerical procedure used to find those solutions.

Regarding b), we know that the converged solution is a minimum by
calculating the second derivative of the solution wrt the (MO) 
coeficients, this is the "stability" check found in some computer
programs.

Regarding a), there are many different SCF algorithms. Some first order
(use only 1st derivatives wrt to coeficients), other second order (use
1st and 2nd derivatives) and quasi-second order (use some approximation
to the 2nd derivatives). Each type has diffrent convergence properties,
and pratical/observed convergence depends on the region of the 
coeficient space which you start your optimization.

There are many reviews on these subjects, try:
author = "T. Helgaker",
title = "Optimization of Minima and saddle points",
booktitle = "Lectures Notes in Quantum Chemistry - European Summer
School in Quantum Chemistry",
editor = "B. O. Roos",
year = 1992,
volume = 1,
publisher = "Springer-Verlag",
pages = {295-324},
address = "Berlin",

or

author = "H. B. Schlegel",
title = "Optimization of equilibrium geometries and transition
structures",
journal = "Adv. Chem. Phys." ,
year =  1987,
volume = 67,
pages = {249-286},

Hope it helps,

------------------------------------------------------------------------

>Dear Guilherme,
>> 
>> Thank you very much for your precise and useful answer. I have one more 
>> question however: when you answer to my point b), you don't mention 
>> anything about checking that the minimum is not only local but also 
>> global. Are there checks for this?
>  
>
Ha, thats a completely different business! Yes, I am only saying about a 
local minimum. Wether this is the global one, there is no mathematical 
or heuristic approach to be sure. The only way to be sure we have found 
a global minimum, is to find all the other (local) minimum! heheheh G

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Hi,

Maybe the following paper could be a good starting point:

Eric Cances and Claude Le Bris, Int. J. Quantum Chem. 79 (2000) 82-90

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Thats all, hope it helps.

  Pablo Echenique

Guilherme

Regards, Lorenzo Lodi

-- 

------------------------------------
  Pablo Echenique Robba

  Departamento de Fisica Teorica
                &
  Instituto de Biocomputacion y
  Fisica de los Sistemas Complejos
  BIFI

  Universidad de Zaragoza
  50009 Zaragoza
  Spain

  Tel.:    34 976761260
  E-mail:  pnique +*+ unizar.es
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