Kratzer-Fues Potential

The Kratzer-Fues potential is a reasonable model for a diatomic molecule rotating and vibrating in 3 dimensions, or even a ion-pair complex (e.g., an single ion pair from an ionic solvent in the gas phase). It has the form $\( V_{\text{Kratzer-Fues}}(r) = \frac{a}{r^2} - \frac{b}{r} \qquad \qquad a>0 \text{ and } b>0 \)\( and so the Schrödinger equation has the form: \)\( \left(-\frac{\hbar^2}{2m}\nabla^2 +\frac{a}{r^2} - \frac{b}{r} \right) \psi(\mathbf{r}) = E\psi(\mathbf{r}) \)$

This is a spherically-symmetric Schrödinger equation in 3 dimensions. This means that the solution is the product of a radial part and a spherical harmonic, $\( \psi_{nlm_l}(r,\theta,\phi) = R_{nl}(r) Y_l^{m_l} (\theta,\phi) \)\( and the radial wavefunction, \)R_{nl}(r)\( satisfies the radial Schrödinger equation for a 3-dimensional system, \)\( \left(-\frac{\hbar^2}{2m} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{\hbar^2 l(l+1)}{2mr^2} + \frac{a}{r^2} - \frac{b}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r) \)\( or, simplifying slightly, \)\( \left(-\frac{\hbar^2}{2m} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{\hbar^2 l(l+1) + 2ma}{2mr^2} - \frac{b}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r) \)$

This is a spherically-symmetric Schrödinger equation in 3 dimensions. This means that the solution is the product of a radial part and a spherical harmonic, $\( \psi_{nlm_l}(r,\theta,\phi) = R_{nl}(r) Y_l^{m_l} (\theta,\phi) \)\( and the radial wavefunction, \)R_{nl}(r)\( satisfies the radial Schrödinger equation for a 3-dimensional system, \)\( \left(-\frac{\hbar^2}{2m} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{\hbar^2 l(l+1)}{2mr^2} + \frac{a}{r^2} - \frac{b}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r) \)\( or, simplifying slightly, \)\( \left(-\frac{\hbar^2}{2m} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{\hbar^2 l(l+1) + 2ma}{2mr^2} - \frac{b}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r) \)$

Notice that the radial Hamiltonian for the Kratzer-Fues potential looks a lot like the Hydrogenic atom, which has the Hamiltonian, $\( \left(-\frac{1}{2} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{l(l+1)}{2r^2} - \frac{Z}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r) \)\( Returning to the Kratzer-Fues case, if one uses atomic units multiplies both sides of the equation by \)m\(, then \)\( \left(-\frac{1}{2} \left( \frac{d^2}{dr^2}+ \frac{2}{r} \frac{d}{dr}\right) + \frac{l(l+1) + 2ma}{2r^2} - \frac{bm}{r} \right) R_{n,l}(r) = \left(mE_{n,l}\right)R_{n,l}(r) \)\( If you then defined a value, \)l’\(, such that \)\( l'(l'+1) = l(l+1) + 2ma \)\( the solutions would be Hydrogenic, \)\( E_{\text{Kratzer-Fues},n} = -\frac{mb^2}{2n^2} \)\( and \)\( R_{n,l}(r) \propto \left( \frac{2mbr}{n} \right)^{l'} L_{n-1-l'}^{2l'+1}\left(\frac{2mbr}{n}\right) e^{-\frac{mbr}{n}} \)\( This treatment is quite naive, insofar as there is no guarantee that \)l’\( is an integer. In practice, this solution is not quite right except for the exceptional cases where \)l’\( is an integer less than \)n$. But this is the idea of the solution. The exact solution can be found various places, see, for example. The detailed solution can be found at in

S. Flügge Practical Quantum Mechanics (Berlin:Springer, 1957)