Modifications of the Robert-Bonamy Formalism and Further Refinement Challenges

We have made two modifications of the Robert-Bonamy (RB) formalism that has been widely used for calculating Lorentzian spectral line half-widths and shifts for decades. The first one comes from a correction of their derivation where they as- sumed 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 S I and S F (?=S IúS*F ) respectively, we have found 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 aver- age 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. The second correction is the expression for S 1 (i.e., the first term in the expansion of the S matrix) that is essential in calculating vibration-rotation pressure-broadened shifts that is not correctly given in the RB formalism. In this case, the problem resulted when they considered effects of the vibrational dephasing on S 1; they made the incorrect assumption that the trajectories of interest are vibrationally independent. As a result, the current expression is not applicable in calculating shifts for molecular lines involving vibrational transitions. Based on a vibration-dependent trajectory model, which is physically sound, we derive the correct expression for S 1 . In comparison with the original expression, the new formula contains extra terms which represent the contributions from vibration- dependent trajectories. We find for some molecular systems of interest, calculated shifts based on the new formula differ signif- icantly from those calculated using the existing formalism. Beside these corrections, we point out that there are several other assumptions introduced in the RB formalism. Some lack theoretical justifications and others may limit the accuracy of the results. One must address these problems in order to make further refinements of the RB formalism.