Knowledge Bank

“The problem of the determination of potential energy functions from the observed vibration-rotation data is considered. With the exception of Dunham’s, the functions that have been proposed are generally inadequate for the obvious reason that they do not contain enough disposable parameters. Moreover, if a particular $\pi$-parameter function is used, it may not be possible to reproduce a given set of $\pi$ observed constants. The reasons for this limitation will be illustrated by a discussion of the Poschl-Teller potential function. General types of potential functions having the correct behaviour at $r=\infty$ may be constructed in many ways. For example, the potential V may be conveniently defined by means of an equation such as \begin{equation}\frac{{d}({V}^{1/2})}{{d}\zeta}={f}({V}^{1/2})\end{equation} (1) or its integrated form \begin{equation}\zeta=\varphi({V}^{1/2})\end{equation} (2) where $\zeta =(r-re)/re$. The function f($V1/2$) may be taken as a power series in $V1/2$ \begin{equation}{f}({V}^{1/2})={a}{0}+{a}{1}{V}^{1/2}+{a}_{2}{V}+....\end{equation} Many of the familiar potential energy functions emerge now as simple particular cases. If only the constant term in Eq. (3) is retained, Eq. (1) gives the harmonic oscillator. A linear expression for f($V1/3$) leads to Morse’s function. The quadratic expression leads to the potential functions of Kratzer (for $\Delta =a12-4a0a2=0$). Rosen-Morse (). Manning-Rosen ($\Delta >O,a2>O$). The expansion (3) has better convergence properties than Dunham’s form of expansion. Taking a finite number of terms gives functions which converge as $r\to \infty$, provided f($V1/2$) has a zero (at $V=D$). Definition of V by means of equation (2) has considerably interest, as the expression for the energy levels is practically independent of the even part of the function $\phi (V1/2)$, the most important factor being the width of the potential well. The expressions for the vibration-rotation constants take rather simple forms in terms of the parameters appearing in Eqs. (1) and (2), since these represent convenient changes of variable for the integration of Schr^{\prime\prime}{o}Udinger’s equation. Some of the implications of the present work as regards the practical determination of V from the experimental data are discussed.”