P. Senn ETH-Zentrum, CHN H32,8092 Zurich, Switzerland
Schrodinger equations having analytic solutions represent special cases or simply starting points in many quantum mechanical orohlems of nractical imnortance. A number of potentials are known which reproduce potentials of diatomic molecules more or less realisticallv and which also eive rise to analytic expressions for the energy levels and for the wave functions (1-4). Beyond the simple harmonic oscillator, only the Morse oscillator appears to be used frequently, even though its quantum mechanical treatment is not straightforward (5).The motive for using the Morse oscillator rests on the fact that the energy levels are in a mathematical form analogous to a truncated power series which spectroscopists use to express the vibronic energy levels of diatomic molecules (6)
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TheMorse os'cillator model has also been applied topolyatomic molecules (7).For many of the normal modes in polyatomic molecules the Morse oscillator model is not appropriate due to its asymmetry. This would he the case, for example, for the bending modes of linear molecules. It would therefore be useful to have a model with an analytic expression for the energy levels in the same mathematical form as shown in eq (I), hut based on a symmetric potential. I t turns out that such a model has been known for quite some time. I t is the modified Poschl-Teller (PT) potential as described, among others, by Fliigge (8)
The treatment shown here differs from the one by Fliigge. In eq (2), we make the following substitutions *"(u) = (1 - u2)8~ " ( 4
(4)
where 2p2 = -,~c.l(hu)~. We obtain the following A PTpotential hole with A = 55. This potential hole holds seven bound states whose normalized wave functions are shown.
A power series expansion of x,
substituted in eq (5), gives rise to the following recursion relation among the expansion coefficients
Through the transformation in eq (3) the range from -m to + m on the x axis is mapped onto the interval -1 < u < 1. Wedemand that il.,(u)besquare integrahleover this interval and vanish a t its limits. I t can be shown that il." tends to infinity a t u = f 1 unless the power seriesin eq (6) terminates with a finite number of terms (9). This is the case if, for a maximum k, the right-hand side of (7) vanishes. By having applied the "infinity condition" we ohtain the following quantization of the energy
where in eq (6) and eq (7) k = 0 , 2 , . . ., u for even solutions and k = 1, 3, . . ., v for odd solutions. In eq ( 5 ) ,we have the differential equation of the ultraspherical polynomials which are also referred to as Gegenbauer polynomials. Using the notation of Ahramowitz and Stegun (10) we obtain the following +,(x) = Nu [eosh(a~)]~o C,(b'(tanh(olr))
where 26 = 40
(9)
+ 1and the normalization factor is (11)
Volume 63
Number 1 January 1986
75
ThePTpotential hole hasa finite number of bound stares. Reauirinr that in eu (8) 3 be ousitive. weubtain thefollowine constraint on u
The figure shows a P T potential hole with A = 55, u,hich h ~ l d s s c ~ bound ~ : n states. Tfthe bottomo1'the well is takenas the point of reference for the energy levels, we obtain the following
explaining the appearance of such spectra (12).Morse potentials have been used as diagonal bond potential functions for the re~resentationof ootential enerw surfaces of ~ o l v a tomic moiecules in conjumtion with wxat is known a s the local mode model (13).The modified P T potential could be used in such schemes. For this, various types of matrix elements will be needed. With the expression of the wave functions in terms of orthogonal polynomials the derivation of analytical ex~ressionsfor most or all of the matrix elements of interest should be possible. Anharmonic potential energy surfaces are often needed for a realistic description of molecular dynamics. In some cases the dominant channels of decay are accessible only due to anharmonic terms in the potential energy surface (14). Llterature Clted
If A is large, i.e., for deep potential wells, the second term on the right-hand side of the above equation dominates for the lowest few states. In this case the PT potential hole appears as nearly harmonic. The concept of normal modes together with the underlying assumption of the harmonic approximation to potential energy surfaces has been firmly engrained in the theory of molecular vibrations. As advances in conventional experimental techniaues and the emereence of new tvnes of snectroscopies have made possible thk observationh'f vibraLionallv highly excited species, it became more and more evident that the normal mode picture in many cases was incapable of
78
Journal of Chemical Education
Steele, D.: Lippincott, E. R.: Vanderaliee, J. T. Reus. M d . Phys. 1962,34.239 Singh, R. B.; Rai, D. K,Indian J. &re And. Phvs. 1966.1.102. Varsndaa. A. J. C. J . Chom. Sac, Farod. ~ m r n11 . 1980.76.129. Mohammad, S. N. Physiro 1979.96C. 410. Berrsndo. M.:Palma.A. J.Phv8.A: Moth. Gsn. 1980.13.778. Hemberg, G. ~ p & a and Structure. I. ~ ~ of& Diatomic ~ s Molecules"; Van Nostrand: New York, 1950. Wallace, R.Chsm. Phya. Left. 1974 37,115. Fliigge. S. "Practical Quantum Mechanics": Springer: Berlin. 1971, (a translation of ' ~ R e ~ h ~ n r n ~ t hrlpr n do,,.n'mrh.nri.'~, ~n ........ ,. Sen", P.. unpubli~hedwork. Abramawite. M;Stegun, I. A. "Handbook of Mathematleal Functions. With Formulas, Graphs, and Mathematics1 Tsbles".9th ed.: Dover: New York, 1970. Gradrhteyn, I. S.; Ryzhik, I. W. "Table of Integrals, Series and Products", 4th ed.; Academic:NeluYork, 196S;sect7.313. (Multiply out8 fsetorofll +x! toobtainthe dosired formula.! Child, M. S.: Halonen,L.Adu. Chem.Phys. 1984.37.1. Nold. D. W.;Kmzykowski, M.L.;Marcus.R.A. J. Chsm.Edue., 1980.57.624, Mataea. N.: Tanekzu. K. "Molaeular Intereetions and Electronic Soectrs": Dekker:
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