*From*: Pablo Echenique Robba <pnique +*+ unizar.es>*Subject*: answers to "existence of HF-SCF solutions"*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 -------------------------------------------------------

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

My two questions are related to point 3: a) When does the SCF converge?

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

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

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,

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 morequestion however: when you answer to my point b), you don't mentionanything about checking that the minimum is not only local but alsoglobal. Are there checks for this?

----------------------------------------------------------------------------------------------------- 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 -- *********** Dr. Nicolas Ferre' *********** Laboratoire de Chimie Theorique et de Modelisation Moleculaire UMR 6517 - CNRS Universites Aix-Marseille Case 521 Centre de Saint-Jerome 13397 - Marseille Cedex 20, France Tel : (+33)4.91.28.27.33 Fax : (+33)4.91.28.87.58 http://www.up.univ-mrs.fr/wsrep/ctheo/lctmm.html *************************************************************** --------------------------------------------------------------------------------------------------- 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 ------------------------------------