Correlation functions vibrational coordinate

We shall now briefly review the Kubo-Oxtoby theory of vibrational line-shape. The starting point for most theories of vibrational dephasing is the stochastic theory of lineshape first developed by Kubo [131]. This theory gives a simple expression for the broadened isotropic Raman line shape (/(< )) in terms of the Fourier transform of the normal coordinate time correlation function by... 

For deriving an expression for the frequency-time correlation function the formulation of Oxtoby is followed. If V is the anharmonic oscillator-medium interaction, then by expanding V in the vibrational coordinate Q using Taylor s series we obtain... 

The experimental dephasing time is defined by the decay of the correlation function of the vibrational coordinate q ... 

To calculate this correlation function, the system Hamiltonian is divided into components for the vibration Hvib, solvent Hsoi, and solvent-vibration coupling V, which depend on the vibrational coordinate q and the set of solvent coordinates Q ... 

In this case, because the bath includes intramolecular coordinates, the evaluation of this formula, which involves products of translational and vibrational time-correlation functions, is quite complicated (12). For a polyatomic solute with a large enough number of vibrational modes, we argue that the time-correlation functions for translations decay on the time scale of the inverse of the characteristic frequencies of the translational bath, which is much slower than the decay of time-correlation functions for the intramolecular vibrations. Therefore we can replace the translational time-correlation functions by their initial values, which we evaluate classically. The upshot is that the rate constant can be written as (12)... 

They are collective coordinates and, therefore, provide a way to describe phenomena which may show high collectivity like solvation" or vibrational relaxation processes. Collective velocity correlation functions, e.g., have been discussed as indicators for the onset of glass formation. ... 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted ... 

Figure 9.4. The time correlation function of the solvent coordinate for water with immersed donor and acceptor molecules of radii 3.5 A at different separation. The insert magnifies the short-time behavior. One can distinguish an initial rapid Gaussian stage due to inertial motions, the subsequent oscillatory behavior due to the hhration of water, and the final exponential diffusive tail. Very weak high-frequency components come from intramolecular vibrations. The lower curves starting from the origin are the estimated simulation uncertainties. (Reproduced from [41c] with permission. Copyright (1997) by the American Institute of Physics.)...

Independently, the vibrational lifetime can be estimated by using classical equilibrium MD to approximate the quantum first-order perturbation theory expression for the vibrational relaxation rate (inverse of the lifetime). The quantum relaxation rate of a vibrational mode coupled to a bath is proportional to the Fourier transform of a force along the vibrational coordinate correlation function. The idea is to approximate this correlation function using classical equilibrium MD trajectories. " " ... 

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