Rotational energy transfer in gaseous mixtures has been considered within the framework of the infinite order sudden (IOS) approximation. A new derivation of the IOS from the coupled states Lippmann–Schwinger equation is given. This approach shows the relation between the IOS and CS T matrices and also shows in a rather transparent fashion Secrest’s result that the IOS method does not truncate closed channels but rather employs a closure relation to sum over all rotor states. The general CS effective cross section formula for relaxation processes is used, along with the IOS approximation to the CS T matrix, to derive the general IOS effective cross section. It is then observed that this cross section can be factored into a finite sum of ’’spectroscopic coefficients’’ Fn(j′aj′b‖jajb ‖L) and ’’dynamical coefficients’’ QL(k). The Fn(j′aJ′b‖jajb ‖L) can be calculated once and tabulated since they do not depend at all on the particular system considered. The QL(k) can be shown to equal the integral inelastic cross section for the transition j=0 to j=L, so that if these cross sections are evaluated, either theoretically or experimentally, other types of cross sections can be computed without any further dynamical calculations. In principle, the factorization permits one to calculate other types of cross sections if any one type of cross section has been obtained by some procedure. The functional form can also be used to compact data. This formalism has been applied to calculate pressure broadening for the systems HD–He, HCl–He, CO–He, HCN–He, HCl–Ar, and CO2–Ar. In order to test the IOS approximation, comparisons have been made to the CS results, which are known to be accurate for all these systems, as well as to several exact close coupling, semiclassical, and experimental values for some of the systems. The IOS approximation is found to be very accurate whenever the rotor spacings are small compared to the kinetic energy, provided closed channels do not play too great a role. For the systems CO–He, HCN–He, and CO2–Ar, these conditions are well satisfied and the IOS is found to yield results accurate to within 10%–15%.
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).
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