New Theoretical Method for the Accurate Calculation of Expectation Values on Functions of Internuclear Separation in 1Σ-State Diatomic Molecules

Abstract

A new theoretical method for the calculation of expectation values on ¹Σ‐state diatomic molecular vibration‐rotation states is developed. Using Dunham's theory of vibration‐rotation energies, each integral can be expressed as a rapidly converging series of terms, each of which is given to high accuracy as an analytical function of the internuclear potential parameters. Results are presented which allow the rapid calculation of expectation values on any arbitrarily specified algebraic function of intermolecular separation, complete to four orders in the ratio of the rotational to vibrational constant of the molecule. We first apply the method to review and extend the theoretical calculation of rotational energies in H₂. In these calculations, agreement with experiment to within the combined limits of accuracy is achieved, the value 60.837 cm⁻¹ resulting for Dunham's rotational constant, $Y_{01}$ . As a further example, the expectation value on the $\text{υ\hspace{0.17em}=\hspace{0.17em}0, J\hspace{0.17em}=\hspace{0.17em}1}$ state of the inverse third power of the internuclear separation in H₂ is calculated using theoretical potential constants on the one hand, and potential constants derived from spectroscopic measurements on the other. The greatest accuracy is achieved with the latter, the value ${\text{〈01\hspace{0.17em}|\hspace{0.17em}R}}^{\text{−3}}{\text{\hspace{0.17em}|\hspace{0.17em}01〉}}^{\text{−1\hspace{0.17em}/\hspace{0.17em}3}}\text{\hspace{0.17em}=\hspace{0.17em}(0.74695\hspace{0.17em}±\hspace{0.17em}0.00003) Å}$ showing good agreement with the experimental value and a value obtained from direct numerical calculation by Wolniewicz.