Autocorrelation function adiabatic representation

It may be shown [8] that the adiabatic autocorrelation function in representation II may be written... 

Now, let us look at the autocorrelation function of the dipole moment operator within the adiabatic approach. In the representation I it is... 

By Fourier transform of the representation III of the undamped adiabatic autocorrelation function (49), one obtains the Franck-Condon progression... 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

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