3.7: The Morse Oscillator

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The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. In this model, the potential \(E_j(R)\) is expressed in terms of the bond dissociation energy \(D_e\) and a parameter a related to the second derivative k of \(E_j(R)\) at \(R_e k = \frac{d^2E_j}{dR^2} = 2a^2D_e\) as follows:

\[ E_j(R) - E_j(R_e) = D_e \left[ 1 - e^{-a(R-R_e)} \right]^2. \nonumber \]

The Morse oscillator energy levels are given by

\[ E^0_{j,v} = E_j(R_e) + \hbar \dfrac{\sqrt{k}}{\mu}\left( v+\dfrac{1}{2} \right) - \dfrac{\hbar^2}{4}\left( \dfrac{k}{\mu D_e} \right) \left( v + \dfrac{1}{2} \right)^2 \nonumber \]

the corresponding eigenfunctions are also known analytically in terms of hypergeometric functions (see, for example, Handbook of Mathematical Functions , M. Abramowitz and I. A. Stegun, Dover, Inc. New York, N. Y. (1964)). Clearly, the Morse solutions display anharmonicity as reflected in the negative term proportional to \((v+\dfrac{1}{2})^2.\)