On combined statistical and j z-conserving coupled states approximations

Publisher Website
Google Scholar
Abstract
In this paper we consider the collision of an atom and diatomic molecule within the framework of the jz‐conserving coupled states approximation. Particular attention is focused on the dependence of jz CCS cross sections upon the rotor magnetic quantum number. It is found computationally that for the He+H2 and He+CO2 systems, the jz CCS partial cross sections σJλ0j0→j are approximately independent of the particular value of λ0, so that the full cross section can be estimated quite well by σj0→j?[π/k2j0(2j0+1) ΣJ(2J+1)(2 min(j0,j,J)+1) ‖TJλ0(j0‖j) ‖2, where ‖λ0(j0, j, J) ‖? min (j0, j, J). This is equivalent to a statistical approximation so far as the magnetic states of the rotor is concerned. By utilizing this fact, one need carry out jz CCS calculations only for λ0=0 and then employ the above expression for σj0→j to obtain all possible integral cross sections. This leads to an approximation scheme requiring the same computational effort as Rabitz’s effective potential method. However, the accuracy of the present approximation is expected to be better than the E.P. for the strongly interacting He+CO2 system. Furthermore, if only transitions involving larger j0, j are required, one may choose λ0=J for J< min (j0, j) and λ0=min (j0, j) for J? min (j0, j). This results in even fewer equations to solve than the effective potential method. It enables one to systematically truncate the number of states coupled in order to calculate a given j0→j cross section since j′<λ0 do not enter the equations. Thus, the truncated statistical jz approximation is the first fully quantal approach to inelastic molecular collisions which permits one to (a) select a transition ji→jf in which one is interested and (b) within the jzCCS approximation, rigorously eliminate all states j′< min (ji, jf, J) for each J. Yet another level of approximtation can be obtained by combining the results of the statistical jz and truncated statistical jz approximations. The averaged results are found to be much more accurate than either approximation alone and in fact are within a very few percent of the full jzCCS results for the He+H2 and He+CO2 systems. The computational effort is simply the sum of that for the statistical jz and truncated statistical jz approximations. This is still very much less than a full jzCCS calculation and only slightly more work than the E.P. method. The results for the j0, j transitions with j0, j<λ0 will be the same as the statistical jz method. Numerical results are presented for the He+H2 and He+CO2 systems to illustrate these new statistical jz approximations.