Re: CCL:CCL: (1) Morse potential and (2) hindered rotations

 Dear Alexei,
 I have done quite a bit of work with partition functions,
 and so I find myself able to answer your questions.
 (1) The Morse potential does indeed admit analytical solutions
 for rotationless motion.
 If you read Morse's original paper
 P.M. Morse, Phys. Rev. v.34, p.57 (1929)
 the eigenfunctions and eigenvalues are given. The eigenvalues are
 all that you need to get the partition function. Unfortunately,
 you will have to sum up the energy levels using a spreadsheet
 or small program, as the partition function is not analytically
 summable (Another example of this is the quantum rigid
 rotator, which has analytical eigenvalues but not a closed-
 form partition function).
 To summarize: the Morse potential is
 U = D{1-exp[-a(r-re)]}^2
 where D= dissociation energy, re = equilibrium bond distance
 and a is a parameter used to fit the curvature to experiment.
 From Steinfeld's "Molecules and Radiation", 2nd ed. (1985),
 I read that to get the harmonic frequency exactly in agreement
 with experiment, one chooses a to be
 a = omega * sqrt [ pi*c*mu/(hbar * D)]
 with a, omega and D all in cm^-1 and r, re in cm.
 omega is the harmonic vibrational
 frequency, mu is the reduced mass in units
 appropriate to those used for hbar, and
 c is the speed of light in cm/sec.
 You may want to check that by evaluating the second derivative
 of the Morse potential at R = Re.
 The energy eigenvalues, again in cm^-1, are
 E = omega * (n + 0.5) - kappa * (n + 0.5)^2
 kappa = hbar* a^2 / 4*pi*c*mu
 Note that the energy eigenvalues are of the form
 E = E (harmonic oscillator) + correction, which is
 kind of cool. Spectroscopists have traditionally
 expanded vibrational energy levels in term series
 and this formula resembles the first 2 terms in such
 a series (sans rotations).
 (2) I believe you're thinking of a formula that Truhlar
 developed - although there is no truly exact solution
 this one comes close, and has good properties:
 D.G. Truhlar, J.Comp.Chem v12, p266 (1991)
 I hope this helps.
 Best, Robert Topper
 Robert Q. Topper                        email:   topper (- at -)
 Asst. Professor of Chemistry            phone:   (212) 353-4341
 The Cooper Union                        fax:     (212) 353-4378
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 New York, NY 10003                               and you're there!
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 > 	Dear CCL'ers,
 > 	Could anybody hint me (or give some reference) concerning the
 > energy levels for uni-dimensional motion in the Morse potential? (I read
 > that this problem has been solved exactly, but may be I confused something
 > and this is not the case). What I need in exactly is the partition
 > function for the vibrational motion just in this Morse potential.
 > 	Secondly, I also heard (but not found anywhere) that there is an
 > exact expression of the partition function for the hindered internal
 > rotations. Does anybody know, where one could find it?
 > 	Best regards,
 > 	Alexei