On the factorization and fitting of molecular scattering information

Abstract

The factorization of cross sections of various kinds resulting from the infinite order sudden approximation is considered in detail. Unlike the earlier study of Goldflam, Green, and Kouri, we base the present analysis on the factored IOS T-matrix rather than on the S-matrix. This enables us to obtain somewhat simpler expressions. For example, we show that the factored IOS approximation to the Arthurs–Dalgarno T-matrix involves products of dynamical coefficients TLl and Percival–Seaton coefficients fL(jl‖j0l0‖J). It is shown that an optical theorem exists for the TlL dynamical coefficients of the T-matrix. The differential scattering amplitudes are shown to factor into dynamical coefficients qL(χ) times spectroscopic factors that are independent of the dynamics (potential). Then a generalized form of the Parker–Pack result for Σj(dσ/d?)(j0→j) is derived. It is also shown that the IOS approximation for (dσ/d?)(j0→j) factors into sums of spectroscopic coefficients times the differential cross sections out of j0=0. The IOS integral cross sections factor into spectroscopic coefficients times the integral cross sections out of j0=0. The factored IOS general phenomenological cross sections are rederived using the T-matrix approach and are shown to equal sums of Percival–Seaton coefficients times the inelastic integral cross section out of initial rotor state j0=0. This suggests that experimental measurements of line shapes and/or NMR spin–lattice relaxation can be used to directly give inelastic state-to-state degeneracy averaged integral cross sections whenever the IOS is a good approximation. Factored IOS expressions for viscosity and diffusion are derived and shown to potentially yield additional information beyond that contained in line shapes. They are however expected to be dominated by the elastic scattering integral cross section. Factored IOS expressions are also shown to hold for thermal rates and averages and the same spectroscopic coefficients apply. By measuring the line shapes over a range of temperatures, deconvolution methods can be used to obtain the definite energy pressure broadening cross section. This can then yield the inelastic integral cross sections. Computations are given illustrating the use of the factored IOS expressions as fitting functions and for predictions of integral cross sections for the systems CO+He and HCl+He, and of thermal rates for the systems CO+H, HCN+He, N2H++He, and CO, CS, and OCS with H2 (treated as a structureless atom).