Knowledge Bank

Two empirical observations about diatomic ground state potentials are made: (1) The coefficients $a1,a2$, etc. in the Dunham potential energy expansion: $$ V(\rho) = a_{0} \rho^{2} [1 + a_{1} \rho + a_{2}\rho^{2}+\cdots ], (\rho =(R-R_{e})/R_{e}) $$ are essentially constant for a large class of diatomic molecules, and (2) the coefficient $ai$ is related to $a1$ by the expression: $$ a_{i} = {(-1)}^{i-1} ia_{1}.$$ Correlations between the spectroscopic coefficients $\beta e$, $\alpha e$, $\omega e$ and $\omega eXe$ have been derived previously using the first $observation1$; however the correlations, although accurate, are isotopically inconsistent. New, useful interrelations between the vibration-rotation spectroscopic constants can be deduced by recasting the energy level expansion: $$ E(V,J) = \Sigma_{\ell , m} Y_{\ell m}\left(V+\frac{1}{2}\right)^\ell [J(J+1)]^{m} $$ in terms of mass reduced quantum numbers $\mu -1/2$ V and $a-1/2J$ and mass independent coefficients: $$ Y_{\ell m}^{*} = Y_{\ell m}\mu^{(\ell/2+m)} $$ For diatomic molecules formed from atoms from any two columns of the periodic table, an appropriate plot of the coefficients $Y\ell m\ast$ form a grid with molecules occuring at the intersections. Thus it is possible to predict the spectroscopic constants of a diatomic molecule to within a few percent even when no data are available on that molecule. The second observation is used to derive a ``universal’’ three parameter function: $$ V(\rho) = a_{0} \rho^{2} \left[ 1 + \frac{a_{1}\rho}{(1+\rho)^{2}}\right ] $$ which is shown to fit the observed energy levels of a test molecule, $I2$, over a wide range of $\rho$.