answers to "existence of HF-SCF solutions"

 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
 -----------------------------------------------------------------------
 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.
 -----------------------------------------------------------------------
 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)
 -----------------------------------------------------------------------
 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 
 -----------------------------------------------------------------------------------------------------
 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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 Centre de Saint-Jerome      13397 - Marseille Cedex 20, France
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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
 ------------------------------------