I have the following equation:

$$-\frac{{{\hbar }^{2}}}{2I{}_{r}}\frac{{{d}^{2}}\psi }{d{{\theta }^{2}}}+V\left( \theta \right)\psi =E\psi, $$


$$V\left( \theta \right)=a+\sum\limits_{n=1}^{N}{{{b}_{n}}\cos \left( n\theta \right)}+\sum\limits_{m=1}^{M}{{{c}_{m}}\sin \left( m\theta \right)}$$

where the notations are straightforward ($\theta \in \left[ 0,2\pi \right)$). I would like to get $K$ eigenvalues of the spectrum. How to integrate it (with some derivations)? Is there any non-commercial program which is able to provide them? I used MSTOR by Truhlar's group, but the potential does not contain the sine-terms.

  • 2
    $\begingroup$ If I remember correctly, Gordy and Cook (Microwave Molecular Spectra) discuss the solution in the case where the potential consists of one sine term. They expand the solution as a Fourier series with appropriate symmetry properties which then results in the evaluation of an infinite determinant. Truncating this determinant and diagonalizing it results in the eigenvalues. I believe this should also be applicable to this case. There is also a review by Lin and Swalen (Rev. Mod. Phys. 31, 84 (1959)) where they discuss the solutions. $\endgroup$ – Paul May 27 '17 at 12:04
  • 1
    $\begingroup$ I found this article which shows how to solve your problem. Solution of the Schrödinger Equation for One-Dimensional Anharmonic Potentials: An Undergraduate Computational Experiment. J. Chem. Educ., 2011, 88 (7), pp 929–931 DOI: 10.1021/ed1000137. The abstract says 'A method of solving the Schrödinger equation using a basis set expansion is described and used to calculate energy levels and wavefunctions of the hindered rotation of ethane and the ring puckering of cyclopentene.' $\endgroup$ – porphyrin May 27 '17 at 22:48

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