Perturbation-Theoretic Approach to Potential-Energy Curves of Diatomic Molecules

Abstract

A perturbation theory is developed whereby the diatomic molecular potential energy $\mathrm{W(R)}$ as a function of the internuclear distance $R$ is expressed, for $R$ near $R_e$ , as a power series in the parameter ${\text{λ\hspace{0.17em}=\hspace{0.17em}1\hspace{0.17em}−\hspace{0.17em}(R}}_e\mathrm{\hspace{0.17em}/\hspace{0.17em}R)}$ , $${\mathrm{W(\lambda )\hspace{0.17em}=\hspace{0.17em}w}}_0\mathrm{\hspace{0.17em}+\hspace{0.17em}}{\displaystyle \sum _{\text{n\hspace{0.17em}=\hspace{0.17em}1}}^{\infty }}{\mathrm{(w}}_n{\mathrm{\hspace{0.17em}-\hspace{0.17em}w}}_{\text{n−1}}{\mathrm{)\lambda }}^n.$$ Truncations of this series have the form of finite power series in $R^{\text{−1}}$ . The quantities $w_n$ are obtained simply as perturbation energies for a purely kinetic‐energy perturbation at $R_e$ , by setting up the problem in confocal elliptic coordinates, in which the kinetic‐energy part of the Hamiltonian is $R^{\text{−2}}$ times an R‐independent operator and the potential‐energy part is $R^{\text{−1}}$ times an R‐independent operator. Expressions for the successive vibrational force constants $k_e{\mathrm{,\; l}}_e{\mathrm{,\; m}}_e\mathrm{,\; ···}$ , are given, and it is shown how it happens, through cancellation of effects in the molecule near $R_e$ against effects in the separated atoms, that truncation of the power series in $\lambda$ at the ${\lambda }^2$ level is often a good approximation, as has been shown empirically.