Application of the Kubo–Anderson band shape equation to vibrational relaxation studies in the frequency domain and to an improved determination of spectral second moments from experimental data

The well-known stochastic equation of Kubo–Anderson, widely adopted in the studies of time vibrational correlation functions, is put in a single-parametric form (defining a parameter α that describes the modulation regime of the dephasing) and Fourier transformed. It is shown how the corresponding analytical expression in the frequency domain can be compared with any experimental band shape, allowing the computation of the dynamical parameters. In particular, the more extensively addressed problems, in literature, in the field of vibrational relaxation studies, have been afforded and a contribution to their solution has been given; these are: the baseline positioning, the uncertainties of the dynamical variables involved in the stochastic model, and the frequency second moment computation. The algebraic development has been implemented in the KUBOFREQ© computer program; it has been applied in the fitting of two sets of experimental data: ν4 and ν5 mode of liquid CH3NO2 at various temperatures. The comparison of the dynamical variables computed with KUBOFREQ© and those previously obtained following the conventional time domain approach, shows a substantial agreement between the two methods: the former, however, gives more accurate values, because the baseline positioning and the second moment computation are based upon stringent criteria, allowing to correctly express the physical uncertainties of the variables. In the case of the ν5 mode, the uncertainty on α is about 5%, denoting that the Kubo–Anderson model is appropriate for the description of the vibrational relaxation of this mode; the uncertainties connected to the second moment are of the order of 10%–12%. The ν4 mode gives a band shape that still ‘‘wears’’ the theoretical profile, but in a worse way with respect to the ν5 profile: the α uncertainty is, in fact, of the order of 10%, while the second moment uncertainties are around 30%–40%. Finally, the analytical equation in the frequency domain may be regarded as a new representation of any band profile, directly dependent only on one parameter.