The distribution of density matrices over potential-energy surfaces: Application to the calculation of the far-wing line shapes for CO2

Abstract

Within the formalism developed previously for the calculation of the far-wing line shape for molecular systems, most of the computer resources were used to diagonalize anisotropic potential-energy matrices whose sizes are determined by the number of states included. As this number is increased, one expects the results to converge. However, for some systems of atmospheric interest, e.g., CO2, the convergence is so slow that one is unable to obtain converged results within reasonable computer 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 unnecessary and one can include as many states as desired. The main computational task is transformed from a diagonalization procedure to the carrying out of multidimensional integrations over the continuous orientational variables. In practice, the integrals are approximated by multidimensional summations over discrete values, the number of which is determined by the resolution required so that the approximated integrals are close to their true values. By choosing reasonable resolutions based on the smooth functional behavior of the integrands, one is able to evaluate the required integrations within reasonable computer time. Furthermore, by introducing weighting functions which are the distribution of the density matrices over potential-energy surfaces, one can reduce the multidimensional integrations to two-dimensional ones. The calculation of the weighting functions can also be carried out with reasonable CPU time and furthermore needs only to be done once for a given molecular system at a specified temperature. Using these as input data, the remaining calculations of the line shapes and corresponding absorption for given potential parameters become straightforward. The formalism is applied in the present paper for linear molecular systems and sample calculations for CO2–CO2 and CO2–N2 are presented. To our knowledge, these are the first, first-principle calculations for the far-wing line shape of CO2 except for the much simpler CO2–rare gas systems.