Legendre moment method for calculating differential scattering cross sections from classical trajectories with Monte Carlo initial conditions

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Abstract
We present a method for obtaining continuous differential cross sections for molecular collisions from trajectories with initial conditions selected by Monte Carlo methods. It is a moment method and we represent the differential scattering cross section in terms of the moments of the normalized Legendre polynomials of the cosine of the scattering angle. The method is applied to five examples, for four of which we evaluate the necessary moments exactly. The fifth is a realistic example of an experimentally obtained differential cross section. For the first four cases we can obtain satisfactory convergence with as few as 400 trajectories with the proper choice of the highest order moment used in the expansion. As a working criterion we select as the highest order coefficient the highest order moment whose absolute value is larger than 0.05. Generally, the method does not converge any more rapidly than does the histogram method. It does provide a simple way of deciding the available angular resolution in the differential cross section for a given number of trajectories.