Irreducible correlation functions of the S matrix in the coordinate representation: application in calculating Lorentzian half-widths and shifts.
By introducing the coordinate representation, the derivation of the perturbation expansion of the Liouville S matrix is formulated in terms of classically behaved autocorrelation functions. Because these functions are characterized by a pair of irreducible tensors, their number is limited to a few. They represent how the overlaps of the potential components change with a time displacement, and under normal conditions, their magnitudes decrease by several orders of magnitude when the displacement reaches several picoseconds. The correlation functions contain all dynamical information of the collision processes necessary in calculating half-widths and shifts and can be easily derived with high accuracy. Their well-behaved profiles, especially the rapid decrease of the magnitude, enables one to transform easily the dynamical information contained in them from the time domain to the frequency domain. More specifically, because these correlation functions are well time limited, their continuous Fourier transforms should be band limited. Then, the latter can be accurately replaced by discrete Fourier transforms and calculated with a standard fast Fourier transform method. Besides, one can easily calculate their Cauchy principal integrations and derive all functions necessary in calculating half-widths and shifts. A great advantage resulting from introducing the coordinate representation and choosing the correlation functions as the starting point is that one is able to calculate the half-widths and shifts with high accuracy, no matter how complicated the potential models are and no matter what kind of trajectories are chosen. In any case, the convergence of the calculated results is always guaranteed. As a result, with this new method, one can remove some uncertainties incorporated in the current width and shift studies. As a test, we present calculated Raman Q linewidths for the N2-N2 pair based on several trajectories, including the more accurate "exact" ones. Finally, by using this new method as a benchmark, we have carried out convergence checks for calculated values based on usual methods and have found that some results in the literature are not converged.
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NASA/Goddard Institute for Space Studies and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10025, USA. qma@giss.nasa.gov
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