%0 Journal Article %1 springerlink:10.1007/s10910-011-9926-0 %A Bhattacharjee, Shayak %A Ray, D. %A Bhattacharjee, J. %D 2012 %I Springer Netherlands %J Journal of Mathematical Chemistry %K analysis asymptotic mechanics physics potential semiclassical series %N 4 %P 819-832 %R 10.1007/s10910-011-9926-0 %T Bohr Sommerfeld quantisation and molecular potentials %U http://dx.doi.org/10.1007/s10910-011-9926-0 %V 50 %X We combine, within the Bohr Sommerfeld quantization rule, a systematic perturbation with asymptotic analysis of the action integral for potentials which support a finite number of bound states with E < 0 to obtain an interpolation formula for the energy eigenvalues. We find interpolation formulae for the Morse potential as well as potentials of the form $$V=V_0 łeft łeft( ax \right)^2k-łeft( ax\right)^k \right$$. For k = 6 i.e. the well known Lennard Jones potential this yields results within 1 per cent of the highly accurate numerical values. For the Morse potential this procedure yields the exact answer. We find that the result for the Morse potential which approaches zero exponentially is the $$k\rightarrowınfty$$ limit of the Lennard Jones class of potentials.

@article{springerlink:10.1007/s10910-011-9926-0, abstract = {We combine, within the Bohr Sommerfeld quantization rule, a systematic perturbation with asymptotic analysis of the action integral for potentials which support a finite number of bound states with E < 0 to obtain an interpolation formula for the energy eigenvalues. We find interpolation formulae for the Morse potential as well as potentials of the form $${V=V_0 \left[ {\left( {\frac{a}{x}} \right)^{2k}-\left( {\frac{a}{x}}\right)^{k}} \right]}$$. For k = 6 i.e. the well known Lennard Jones potential this yields results within 1 per cent of the highly accurate numerical values. For the Morse potential this procedure yields the exact answer. We find that the result for the Morse potential which approaches zero exponentially is the $${k\rightarrow\infty}$$ limit of the Lennard Jones class of potentials.}, added-at = {2012-03-15T02:19:11.000+0100}, affiliation = {Department of Physics, Indian Institute of Technology Kanpur, Kanpur, 208016 India}, author = {Bhattacharjee, Shayak and Ray, D. and Bhattacharjee, J.}, biburl = {https://www.bibsonomy.org/bibtex/2d204066e3f405c2f0c8701760b872d0a/drmatusek}, doi = {10.1007/s10910-011-9926-0}, interhash = {a9d7fec0e467cfe8962ee4eea4fe2329}, intrahash = {d204066e3f405c2f0c8701760b872d0a}, issn = {0259-9791}, issue = {4}, journal = {Journal of Mathematical Chemistry}, keyword = {Chemistry and Materials Science}, keywords = {analysis asymptotic mechanics physics potential semiclassical series}, month = {April}, number = 4, pages = {819-832}, publisher = {Springer Netherlands}, timestamp = {2012-10-28T18:24:09.000+0100}, title = {Bohr Sommerfeld quantisation and molecular potentials}, url = {http://dx.doi.org/10.1007/s10910-011-9926-0}, volume = 50, year = 2012 }