Subject: CCL:References on exact solutions to Schroedinger equations?

From: "Jochen Küpper" <[email-masked]>
Date: Wed, 2 Feb 2000 09:31:46 +0100 (CET)

Lingran Chen writes:
 > I'm looking for the references about the mathematically *exact*
 > solutions to Schroedinger equations for any chemical systems.

As far as I understand there are none :-( You could look at
textbooks-solutions of H-atoms, though ).
Jochen
--
Heinrich-Heine-Universität
Institut für Physikalische Chemie I
Jochen Küpper
Universitätsstr. 1, Geb. 26.43.02.29
40225 Düsseldorf, Germany
phone ++49-211-8113681, fax ++49-211-8115195
http://www.jochen-kuepper.de

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From: Thomas Bligaard Pedersen <[email-masked]>
Date: Wed, 02 Feb 2000 15:50:19 +0100

>  > solutions to Schroedinger equations for any chemical systems.

The only molecules, for which there exists exact analytical solutions, are the
molecular ions of homonuclear diatomics with only one electron: H2(+), He2(3+),
...

"Where as the equation for the H atom is separable in sperical polar coordinates,
the equation for the molecule-ion is separable in ellipsoidal coordinates in which
the two nuclei are the foci of the ellipses."
quote P.W.Atkins, Molecular Quantum Chemistry 2nd ed. p 250-251
An actual derivation is not made in this book, but I have seen one in lecture
notes by Jens Peder Dahl. These might be difficult to obtain, but he is very
helpfull, and can probably direct you to the original references if you contact
him on

[email-masked]

I hope this helps

Thomas

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From: "Richard P. Muller" <[email-masked]>
Date: Wed, 02 Feb 2000 09:41:55 -0800

> helpfull, and can probably direct you to the original references if you contact

If you're interested in close-to-exact solutions, I would suggest
looking at the papers of Hylleraas or Pekeris for the He atom (I can
send you software that reproduces Pekeris' work, if you'd like).

I also highly recommend Flugge's _Practical_Quantum_Mechanics_, which
has a wide variety of problems solved analytically. Bethe and Jackiw'w
_Intermediate_Quantum_Mechanics_ is also quite good for this.
-- 
Richard P. Muller, Ph.D. 
[email-masked] 
http://www.wag.caltech.edu/home/rpm

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From: Eberhard von Kitzing <[email-masked]>
Date: Wed, 2 Feb 2000 18:52:28 +0100

>the two nuclei are the foci of the ellipses."

You may find something in

Hund Z Phys 36, 657 (1926)
Mulliken J Chem Phys 3, 375 (1935)
Coulson Trans Farad Soc 33, 1479 (1937)

==========================================================

Eberhard von Kitzing

Abteilung Zellphysiologie
Max-Planck-Institut fuer Medizinische Forschung
Jahnstr. 29
D 69120 Heidelberg        Tel: +49 6221 486 467
Germany                   FAX: +49 6221 486 459

email: [email-masked]
WWW: http://sunny.mpimf-heidelberg.mpg.de/people/vkitzing/

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From: "Robert E. Harris" <[email-masked]>
Date: Wed, 2 Feb 2000 13:27:49 -0600

The H2 molecule positive ion solution is discussed on pp. 201-203 of
Eyring, Walter, and Kimball's book, "Quantum Chemistry".  They refer to
work;  E. Teller, Z. Physik, 61, 458 (1930), O. Burrau, Kgl. Danske
Videnskab. Selskab., 7, 1 (1927), E. Hylleras, Z. Physik, 71, 739 (1931),
and G. Jaffe, Z. Physik, 87, 535 (1934).  It is also discussed on pp.
134-136 of Pitzer's "Quantum Chemistry".  See also pp. 1-40 of Slater's
"Quantum Theory of Molecules and Solids, Volume 1, Electronic Structure of
Molecules".  Slater says quite complete numerical information about the
solution is given by D. R. Bates, K. Ledsham, and A. L. Stewart, Phil.
Trans. Roy. Soc. London, 246, 215 (1953).
One should note that the solution for the ground state is not closed form
in terms of elementary transcendental functions.

REH

Robert E. Harris  Phone: 573-882-3274  Fax:  573-882-2754
Department of Chemistry, University of Missouri-Columbia
Columbia, Missouri, USA 65211

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From: "Robert E. Harris" <[email-masked]>
Date: Wed, 2 Feb 2000 13:40:54 -0600

I neglected Pauling and Wilson's "Introduction to Quantum Mechanics", pp.
327-340 in the Dover reprint.
The exact solutions for H2+ ion are of course numerical solutions; these
are exact in the sense that the exponential solution for the H atom is
exact.  The exponential function is well-know and frequently tabulated,
while the solutions to the H2 + ion are not familiar.  So, some would say
the H atom is solved "exactly" while the H2 + ion isn't, but really, is a
mutt more of a dog than an Afgan ound jsut because mutts are more familiar?

REH