Knowledge Bank

The transformation of the potential function expressed in normal coordinates from the principal isotopic species of carbon dioxide to other isotopic species is shown. Required in the intermediary normal to valence coordinate transformation was the use of the Eckart condition as a constraint along with the use of the computer algebra system, MACSYMA. This tool was used in both determining the coordinate transformation and its expansion in Taylor series form. After expansion to seventh order in valence coordinates, the potential function coefficients were calculated to seventh order in normal coordinates via the method presented by Hoy $etal.1$ The transformed potential function surfaces were then used to calculate the eigenstate energies for all relevant isotopic species of $CO2$ by the direct numerical diagonalization $method.2$ Some kinetic energy terms had to be generalized for the asymmetric species. The dipole moment function was also transformed from the principal isotopic normal coordinate form to the valence coordinate form. This in turn was again transformed to the appropriate form for the various isotopic species including the asymmetric ones; the transition intensities were then calculated. Comparison with measured transitions of the oxygen-18 enriched asymmetric isotope (628), which are forbidden in the principal isotope, gives some indication of the errors inherent in the technique.