Approximation of isotopic reduced partition function ratios by the WINIMAX polynomial perturbation method (WIMPER)

Diagonal H, F, and G matrix approximations to the moments of the eigenvalues are studied for a number of polyatomic molecules. For molecules containing hydrogen atoms the diagonal element approximations to each of the first six moments do not differ from the true values by more than 15%. The results further substantiate the simplified method of analysis of the isotope chemistry of hydrogen in terms of stretching and bending forces. It is shown that the coupling of bending and stretching coordinates makes a significant contribution to the motion of ’’central’’ atoms in a molecule. Corrections to the reduced partition function ratios of isotopic molecules corresponding to uncoupled internal coordinates, the diagonal element approximation, are developed through the use of the WINIMAX polynomials (WIMPER). The magnitude of the corrections are related directly to the off-diagonal F and G matrix elements. Sample calculations show that the diagonal element approximations lead to values of ln (s/s′) f for deuterium substitution within 1% of the exact values at 300°K. It is shown that the diagonal H and diagonal G–diagonal F approximations lead to values of ln(s/s′) f which differ from exact values by 7% and 13%, respectively, at 300 °K for carbon and oxygen isotope substitutions. The second order correction [(h//kT)2]2 leads to approximations of ln(s/s′) f within 3% of the exact values for carbon and oxygen isotope substitutions. Comparisons are made between the WIMPER method and the Singh–Wolfsberg perturbation method. Comparisons are also made between the WIMPER and the direct WINIMAX polynomial expansions of ln(s/s′) f. The areas of physical significance of each of the methods is discussed.