Frequency-time correlation function

The main contributions to the frequency-time correlation function are assumed to be, as in the earlier works [123, 124], from the vibration-rotation coupling and the repulsive and attractive parts of the solvent-solute interactions. In several theories, the (faster) repulsive and the (slower) attractive contributions are assumed to be of widely different time scales and are treated separately. However, this may not be true in real liquids because the solvent dynamic interactions cover a wide range of time scales and there could be a considerable overlap of their contributions. The vibration-rotation coupling contribution takes place in a very short time scale and by neglecting the cross-correlation between this mechanism and the atom-atom forces, they... 

The frequency-time correlation function is dependent on the frequency and the force constants of the vibrational mode whose dephasing is being considered. They are determined by fitting the vibrational bond energies to a Morse potential of the following form ... 

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... 

An important ingredient of Oxtoby s work was the decomposition of the force on the normal coordinate, — (dV/dQ), in terms of the force on the atoms involved. Assuming that the forces on the different atoms of the diatomic are uncorrelated and that the area of contact of each atom with the solvent is a half-sphere, Oxtoby derived the following expression for the frequency-time correlation function ... 

The dynamics of fluctuating frequencies in an aqueous medium can be described in terms of the time correlation function of fluctuations of such frequencies around their average values. In fact, the frequency time correlation function serves as the key dynamical quantity in the studies of vibrational spectral diffusion. This correlation function is defined as... 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

Figure 9. Experimental [59] and theoretical values of the polarization anisotropy time correlation function at 100 fs, as a function of OD stretch frequency, for three different temperatures. See color insert.

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

As a new subject we have considered the effect of the frequency-dependence of the elastic moduli on dynamic light scattering. The resultant nonexponential decay of the time-correlation function seems to be observable ubiquitously if gels are sufficiently compliant. Furthermore, even if the frequency-dependent parts of the moduli are very small, the effect can be important near the spinodal point. The origin of the complex decay is ascribed to the dynamic coupling between the diffusion and the network stress relaxation [76], Further scattering experiments based on the general formula (6.34) should be very informative. 

Spectroscopic techniques have been applied most successfully to the study of individual atoms and molecules in the traditional spectroscopies. The same techniques can also be applied to investigate intermolecular interactions. Obviously, if the individual molecules of the gas are infrared inactive, induced spectra may be studied most readily, without interference from allowed spectra. While conventional spectroscopy generally emphasizes the measurement of frequency and energy levels, collision-induced spectroscopy aims mainly for the measurement of intensity and line shape to provide information on intermolecular interactions (multipole moments, range of exchange forces), intermolecular dynamics (time correlation functions), and optical bulk properties. 

The virial expansion of the time correlation functions is possible for times smaller than the mean time x between collisions. Accordingly, the spectral profiles may be expanded in powers of density, for angular frequencies much greater than the reciprocal mean time between collisions, co 1/r. Since at low density the mean time between collisions is inversely proportional to density, lower densities permit a meaningful virial expansion for a greater portion of the spectral profiles. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

This kind of investigation is becoming so common in infrared spectroscopy that investigators are becoming more concerned with the appropriate time-correlation function than with the frequency spectrum itself. 

The time-correlation functions, Cj.it), resonance frequencies, QJt, and memory functions, Kj.it), satisfy the reciprocal relations... 

There is an alternative approach to the theory of time-correlation functions. According to Eqs. (148), (156), and (157) the real and imaginary parts of the frequency dependent memory function... 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

The friction is given in terms of the force-force time correlation functions and in the frequency plane can be written as... 

The experimental observables are either the lineshape function 7(co), as in the classical experiments, or the normal coordinate time correlation function, (2(O)2(0) as in the time domain experiments of Tominaga and Yoshihara [126]. The normal coordinate time correlation is related to the frequency modulation time correlation function by... 

The calculated ln(j2(f)<2(0)) for the first three quantum levels for the CH3I case is shown in Fig. 12. The overtone normal coordinate time correlation function, Q(t)Q(0)), is obtained from the frequency-modulation time corre-... 

The sequence of the observed frequencies, resolved on the time scale, may be regrouped in a form giving a quantity S(t) which may be related to a time correlation function CAE(t) which represents the ensemble average of solvent fluctuations. 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

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