The generating function approach to the formulation of the effective rotational Hamiltonian: A simple closed form model describing strong centrifugal distortion in water-type nonrigid molecules

Abstract

We deal with the problem of the divergence of the standard power series expansion of the effective rotational Hamiltonian for high rotational levels. The paper describes a very simple model (G-operator) which takes into account the dominant part of the anomalous centrifugal distortion in H 2O-type nonrigid quasilinear molecules in a closed form. This model is derived by taking account of the most important part of the bending-rotational interaction in the zero-order approximation. The main features of the model discussed above are the following. The high-order coefficients of the Taylor series expansion of the model are in accordance with experimental centrifugal distortion parameters for H 2O. To a realistic approximation, the new model may be considered as a generating function for Watson's expansion of the rotational Hamiltonian. The “asymptotic behavior” of the calculated energies as K a → J is in qualitative agreement with available experimental data up to K a ≤ 20, contrary to the traditional approach. Direct calculations using the G-operator give much more satisfactory E JK a K c calc than those with a truncated standard power series expansion of H rot. For the H 2O ground state in the interval of quantum numbers J < 20 and 0 < K a < ( 3 4 )J we have typical discrepancies Δ E = | E obs - E cal| ≤ 0.5 cm −1 and relative errors ( ΔE E ) ≤ 0.00008 (without fitting). The analytical formulas for convergence radii of the standard rotational Hamiltonian derived with the use of the G-operator are in accordance with numerical tests and with previous “empirical” estimates. The conclusion is made that the generating function approach is applicable for quantum number values which are above the convergence radius of the standard expansion.