Modification of the Robert-Bonamy formalism in calculating Lorentzian half-widths and shifts

We have found an invalid assumption in the Robert-Bonamy formalism that has been widely used for calculating Lorentzian spectral line half-widths and shifts for decades. The problem results in their derivation where they assumed the cumulant expansion can be used to evaluate the Liouville matrix element 〈〈j2i2|S^|j2i2〉〉. At first sight, their assumption appears to be correct because this matrix element is diagonal in the Liouville space and as a result, it looks like that a basic requirement in applying the cumulant expansion is satisfied. However, by decomposing it into two Hilbert matrix elements associated with SI and SF(S^=SISF*), respectively, we have found that neither of these is diagonal in Hilbert space. Therefore, their assumption is not valid and their expressions for the half-widths and shifts are incorrect. We have found by choosing an average over the internal degrees of the bath molecule as the average in the cumulant expansion, one is able to apply this expansion properly and obtain the correct expressions. Numerical calculations show new half-width and shift values differ from previous ones, and the stronger the interaction between two molecules is, the larger these differences are.