Lindstedt Poincare technique applied to molecular potentials

Abstract

The Lindstedt–Poincare technique has traditionally been used to deal with oscillators with power-law potentials. We show how this method can be extended to deal with molecular potentials for which the frequency goes to zero as the energy approaches zero. The extension requires the use of an asymptotic analysis which is combined with perturbation theory. For the Morse potential, we get an exact answer while for the Lennard Jones class of potentials ${{\rm V}={\rm V}_0 \left[ {\left( {\frac{{a}}{{\rm x}}}\right)^{2{\rm n}}-\left({\frac{{\rm a}}{{\rm x}}}\right)^{{\rm n}}}\right]}$ , the answer is generally approximate with some values of n giving exact results. For the widely studied case, n=6, our approximation gives better than 1% accuracy at the lowest order of calculation. We show that as ${{\rm n} \rightarrow \infty}$ , the result tends to that for the Morse potential. We also point out that the time period obtained by us can be used to obtain the quantum mechanical energy levels of these potentials within the Bohr-Sommerfeld scheme.