THE DISTRIBUTION OF DENSITY-MATRICES OVER POTENTIAL-ENERGY SURFACES -APPLICATION TO THE CALCULATION OF THE FAR-WING LINE-SHAPES FOR CO2

Within the formalism developed previously for the calculation of the f ar-wing line shape for molecular systems, most of the computer resourc es were used to diagonalize anisotropic potential-energy matrices whos e sizes are determined by the number of states included. As this numbe r is increased, one expects the results to converge. However, for some systems of atmospheric interest, e.g., CO2, the convergence is so slo w that one is unable to obtain converged results within reasonable com puter limitations. In the present paper, a new formalism is presented in which the eigenfunctions of the orientations of the system, not the states themselves, are chosen as the complete set of basis functions in Hilbert space. In this case, the diagonalization procedure is unnec essary and one can include as many states as desired. The main computa tional task is transformed-from a diagonalization procedure to the car rying out of multidimensional integrations over the continuous orienta tional variables. In practice, the integrals are approximated by multi dimensional summations over discrete values, the number of which is de termined by the resolution required so that the approximated integrals are close to their true values. By choosing reasonable resolutions ba sed on the smooth functional behavior of the integrands, one is able t o evaluate the required integrations within reasonable computer time. Furthermore, by introducing weighting functions which are the distribu tion of the density matrices over potential-energy surfaces, one can r educe the multidimensional integrations to two-dimensional ones. The c alculation of the weighting functions can also be carried out with rea sonable CPU time and furthermore needs only to be done once for a give n molecular system at a specified temperature. Using these as input da ta, the remaining calculations of the line shapes and corresponding ab sorption for given potential parameters become straightforward. The fo rmalism is applied in the present paper for linear molecular systems a nd sample calculations for CO2-CO2 and CO2-N-2 are presented. To our k nowledge, these are the first, first-principle calculations for the fa r-wing line shape of CO2 except for the much simpler CO2-rare gas syst ems. (C) 1998 American Institute of Physics. [S0021-9606(98)01809-1].