Sum rules and the symmetry of the memory function in spectral line shape theories

Abstract

Starting from the well-known expression between the absorption coefficient and the spectral density F( ω), the Fourier transform of the dipole correlation function, we use a projection operator to average out the degrees of freedom of the bath variables. As a result, F( ω) can be written in terms of a memory function and effects due to initial correlations, all of which are confined to the line space of the absorber molecule. The initial correlations influence the form of the spectral density and are necessary in order to satisfy the principle of detailed balance. There is a relationship between the existence of sum rules and the symmetry of the memory function matrix and the latter depends on the projection operator introduced. In the present case, the matrix of the memory function is asymmetric and only one-sided sum rules are valid. The above conclusions remain true within the binary collision approximation that is applicable to low density gases. Finally, we discuss previously published two-sided sum rules that have been derived using a different projection operator and consequently a different memory function.