Marechal-Witkowski equivalence

Now, we will show that the SD (71) that is the Fourier transform of the ACF (68) via Eq. (69), is equivalent to that obtained by Marechal and Witkowski in their pioneering work [18], For this purpose, look at the ACF (68). It is possible to write the translation operator appearing in it, which has to be averaged on the Boltzmann density operator, as a product of two translation operators. According to Eq. (F.4), it is possible to write... 

In Eq. (80), one may recognize the Franck-Condon progression appearing in the model of Marechal and Witkowski [18] dealing with weak H-bonds. Of course, since Eq. (69) and Eq. (79) are two different, but equivalent, expressions of the ACF, the two expressions of the line shape given by Eqs. (80) and (71), which are, respectively, the Fourier transforms of Eq. (69) and Eq. (79) are also equivalent. The advantage of the expression (80) with respect to that (71), is it allows a pictorial representation of the line shape. This is performed in Fig. 5 for absolute zero temperature that implies c —> 8o,m and for which the SD (80)... 

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