*From*: Lingran Chen <LChen;at;mdli.com>*Subject*: Summary: References on exact solutions to Schroedinger equations?*Date*: Mon, 21 Feb 2000 23:44:51 -0800

Dear CCLers: Some days ago I asked the following question: > Dear CCLers: > > I'm looking for the references about the mathematically *exact* > solutions to Schroedinger equations for any chemical systems. > Thanks in advance. > > -Lingran I have received many replies, which are summarized below. First, let me cite the answers of my question from Ira N. Levine's book "Quantum Chemistry" (5th edition, 2000) I just bought and received today: P.78: "We solved the Schroedinger equation exactly for the particle in a box and the harmonic oscillator." P.163: "The Schroedinger equation for the one-electron atom is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schroedinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. " P.376: "The electronic Schroedinger equation for H2+ is separable, and we can get exact solutions for the eigenfunctions and eigenvalues." The answers I received are listed (in the order I received) below. Thanks a lot! -Lingran > *********************************** > Lingran Chen, Ph.D. > Senior Scientific Programmer > MDL Information Systems, Inc. > 14600 Catalina Street > San Leandro > CA 94577 > U.S.A. > > Phone: (510) 895-1313 > FAX: (510) 614-3616 > > Email: LCHEN;at;MDLI.COM > Web: http://www.mdli.com > *********************************** ----------------------------------------------------------------------------------- Subject: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 09:31:46 +0100 (CET) From: Jochen =?UNKNOWN?Q?K=FCpper?= <jochen;at;uni-duesseldorf.de> To: Lingran Chen <lchen;at;mdli.com> CC: "chemistry;at;ccl.net" <chemistry;at;ccl.net> References: 1 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 ---------- Subject: Re: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 11:40:35 +0200 (EET) From: John Kerkines <jkerk;at;arnold.chem.uoa.gr> To: Lingran Chen <lchen;at;mdli.com> Dear Lingran, That's a very interesting question you sent to the list. Can you please forward to me (or to the list) any responses? With Regards, John Kerkines ---------- Subject: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 15:50:19 +0100 From: Thomas Bligaard Pedersen <bligaard;at;fysik.dtu.dk> Organization: Physics Department, Techn. Univ. of Denmark CC: Lingran Chen <lchen;at;mdli.com>, "chemistry;at;ccl.net" <chemistry;at;ccl.net> References: 1 , 2 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 jpd;at;kemi.dtu.dk I hope this helps Thomas ---------- Subject: mathematically exact Date: Wed, 02 Feb 2000 08:02:06 -0800 From: vmohan;at;isisph.com To: LCHEN;at;MDLI.COM Hi, I would think that Quantum Monte Carlo method yields the mathematically exact solutions to Schrodinger equation. The advantage of this method is that the correlation term [exp(-rij)] can be directly used in the wavefunction. Conventional variational methods include the electron correlation indirectly. The exactness of QMC on a number of chemical systems thas been published by Jim Anderson [Penn.State]. Hope this helps, -mohan ---------- Subject: Re: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 08:14:03 -0800 From: wsteinmetz <wsteinmetz;at;POMONA.EDU> Organization: Pomona College To: Lingran Chen <lchen;at;mdli.com> References: 1 Mathematically exact solutions? The list of closed-form analytic solutions is VERY short. The list of chemical problems includes the H atom, the harmonic oscillator, the rigid rotor, possibly the Morse potential, and the ESR/NMR problem. Consult Pauling and Wilson, Introduction to Quantum Mechanics. Another good practical book with solutions is S. Flueege, Practical Quantum Mechanics, Springer-Verlag. Any good book on NMR will outiline the application of angular momentum theory to the calculation of the spectrum of a system of coupled spins. The classic was written by Pople, Schneider, and Bernstein. The solution involves the diagonalization of N x N matrices. ---------- Subject: Re: CCL: References on exact solutions to Schroedinger equation Date: Wed, 02 Feb 2000 11:54:39 -0500 (EST) From: Stefan Fau <fau;at;qtp.ufl.edu> To: lchen;at;mdli.com Hi, it may not be what you are looking for, but maybe two years ago some mathematician found an exact solution to the three body interaction problem, I think by developing it in an infinite series. I believe I read it as a short note in Scientific American. Stefan ______________________________________________________________________ Dr. Stefan Fau fau;at;qtp.ufl.edu Quantum Theory Project University of Florida Gainesville FL 32611-8435 ---------- Subject: Re: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 09:41:55 -0800 From: "Richard P. Muller" <rpm;at;wag.caltech.edu> To: Lingran Chen <lchen;at;mdli.com>, "chemistry;at;ccl.net" <chemistry;at;ccl.net> References: 1 , 2 , 3 Thomas Bligaard Pedersen wrote: > > 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 > > jpd;at;kemi.dtu.dk > > I hope this helps 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. rpm;at;wag.caltech.edu http://www.wag.caltech.edu/home/rpm ---------- Subject: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 18:52:28 +0100 From: Eberhard von Kitzing <vkitzing;at;MPImF-Heidelberg.mpg.de> To: Thomas Bligaard Pedersen <bligaard;at;fysik.dtu.dk> CC: Lingran Chen <lchen;at;mdli.com>, "chemistry;at;ccl.net" <chemistry;at;ccl.net> References: 1 , 2 At 15:50 Uhr +0100 02.02.2000, Thomas Bligaard Pedersen wrote: >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." 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: vkitzing;at;mpimf-heidelberg.mpg.de WWW: http://sunny.mpimf-heidelberg.mpg.de/people/vkitzing/ ---------- Subject: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 13:27:49 -0600 From: "Robert E. Harris" <HarrisR;at;missouri.edu> To: Chemistry <chemistry;at;ccl.net> References: 1 , 2 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 ---------- Subject: CCL:References on exact solutions to Schroedinger equations? Date: Wed, 02 Feb 2000 13:40:54 -0600 From: "Robert E. Harris" <HarrisR;at;missouri.edu> To: chemistry;at;ccl.net References: 1 , 2 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 Robert E. Harris Phone: 573-882-3274 Fax: 573-882-2754 Department of Chemistry, University of Missouri-Columbia Columbia, Missouri, USA 65211 ---------- Subject: Exact Solutions to "Chemical Systems" Date: Wed, 02 Feb 2000 15:44:11 -0500 (EST) From: Brian Williams <williams;at;bucknell.edu> To: lchen;at;mdli.com I don't know if this is exactly what you are after, but at the moment I think I have a way to define analytically solvable potentials resembling diatomic or Morse potential curves. I am at the moment attempting to see if these analytically exact solutions can be fit to experimental data for diatomics. I do not have this work written up, but would be happy to send you notes if you are interested. Brian Williams, Chemistry Bucknell University ---------- Subject: Exact solutions. Date: Thu, 03 Feb 2000 01:35:43 +0100 From: Thomas Bligaard Pedersen <bligaard;at;fysik.dtu.dk> To: lchen;at;mdli.com I was wrong in saying that only homonuclear diatomics are solved analytically. These were solved in G. Jaffé, Z. Phys. 87:535 (1934) Heteronuclear diatomic molecular ions with one electron are also solved: W.G.Barber and H.R.Hassé, Proc. Camb. Phil. Soc. 31:564 (1935) Thomas ---------- Subject: CCL:Nuclear Attraction Integrals Date: Thu, 03 Feb 2000 09:42:45 +0100 (MET) From: Christoph.van.Wuellen;at;ruhr-uni-bochum.de To: rpm;at;wag.caltech.edu (Richard P. Muller) CC: chemistry;at;ccl.net A nuclear attraction integral is just a special case of a two-electron integral: <i | 1/(r-c) | j> = (ij | kk) where k is a normalized s-Funktion centered at c with "infinitly high" exponent. Some simplifications arise naturally, so if you have a text on two-electron integrals, it also covers nuclear attr. integrals. ---------------------------+------------------------------------------------ Christoph van Wullen | Fon (University): +49 234 32 26485 Theoretical Chemistry | Fax (University): +49 234 32 14109 Ruhr-Universitaet | Fon/Fax (private): +49 234 33 22 75 D-44780 Bochum, Germany | eMail: Christoph.van.Wuellen;at;Ruhr-Uni-Bochum.de ---------------------------+------------------------------------------------ ---------- Subject: CCL:Re, References on exact solutions ... Date: Thu, 03 Feb 2000 09:49:32 +0100 (MET) From: Christoph.van.Wuellen;at;ruhr-uni-bochum.de To: chemistry;at;ccl.net I wonder if during this discussion is has been pointed out that H2+ is a three-body system. The "exact" solution discussed so far is however the solution of a one-body Schroedinger equation with the clamped nuclei hamiltonian. ---------------------------+------------------------------------------------ Christoph van Wullen | Fon (University): +49 234 32 26485 Theoretical Chemistry | Fax (University): +49 234 32 14109 Ruhr-Universitaet | Fon/Fax (private): +49 234 33 22 75 D-44780 Bochum, Germany | eMail: Christoph.van.Wuellen;at;Ruhr-Uni-Bochum.de ---------------------------+------------------------------------------------ ---------- Subject: CCL:Re, References on exact solutions ... Date: Thu, 03 Feb 2000 23:18:22 +0200 (EET) From: Tom Sundius <sundius;at;pcu.helsinki.fi> To: Christoph.van.Wuellen;at;ruhr-uni-bochum.de CC: chemistry;at;ccl.net On Thu, 3 Feb 100 Christoph.van.Wuellen;at;ruhr-uni-bochum.de wrote: > I wonder if during this discussion is has been pointed out that H2+ is > a three-body system. The "exact" solution discussed so far is however the > solution of a one-body Schroedinger equation with the clamped nuclei > hamiltonian. Of course there is no analytical solution to a three-body problem. But a complete numerical solution by separation of the variables was already given in 1927 by Oyvind Burrau in Denmark. His solution to the problem in the book by Pauling and Wilson (Introduction to Quantum Mechanics, sect. 42c). This solution was later improved by Hylleraas and Jaffe. The results were in very close agreement with experiment. Tom Sundius University of Helsinki, Department of Physics phone +358-9-191 8339 P.O.Box 9, FIN-00014 Helsinki, Finland fax +358-9-191 8680 ---------- Subject: CCL:Re, References on exact solutions ... Date: Fri, 04 Feb 2000 11:32:36 +0100 From: Ramon Crehuet <rcsqtc;at;iiqab.csic.es> Organization: C.S.I.C. To: chemistry;at;ccl.net References: 1 There is system that can be considered somehow chemical: Hooke's atom. It consists of an atom with two Coulomb interacting electrons bounded to a nucleus by an harmonic potential. This is very briefly described by Burke, Perdew and Ernzerhof in J. Chem. Phys, Vol. 109, No. 10, p. 3760 and the mathematical solution can be found in references therein. SORRY!!! Vol. 109, No. 10, p. 3760, 1998 Ramon ---------- Subject: CCL:Re, References on exact solutions ... Date: Fri, 04 Feb 2000 12:15:46 +0100 From: assfeld;at;host23.lctn.u-nancy.fr To: chemistry;at;ccl.net, rcsqtc;at;iiqab.csic.es Any FULL CI calculations are *exact* solutions, at least for the basis set considered. And there is a lot of litterature about it. Just my 2 cents... ...Xav http://www.lctn.u-nancy.fr/Chercheurs/Xavier.Assfeld