Frequency dependence time correlation function

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. 

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. 

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 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 ... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

What has been presented here underscores the fact that the elastic scattering is the Fourier transform of the time-independent component of the intermediate scattering function. Naturally, the Fourier transform of a constant function produces a 5-function at > = 0, which is the elastic scattering, but this is not the same as the zero frequency component of the scattered intensity. If the time correlation function has a component that exhibits some time decay or relaxation, then the integral of the time dependant part of... 

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

One significant feature of Eq. (3.2) is the factorization of the expression into the centroid density (i.e., the centroid statistical distribution) and the dynamical part, which depends on the centroid frequency u>. It is not obvious that such a factorization should occur in general. For example, a rather different factorization occurs when the conventional formalism for computing time correlation functions is used [i.e., a double integration in terms of the off-diagonal elements of the thermal density matrix and the Heisenberg operator q t) is obtained]. This result sheds light on the dynamieal role of the centroid variable in real-time correlation functions (cf. Section III.B) [4,8]. 

In Paper IV, the self-diffusion process in fluid neon was also studied with CMD using the pairwise pseudopotential method. In Fig. 17 the centroid velocity time correlation function is plotted for quantum neon using the pseudopotential method and for classical neon. When the quantum mechanical nature of the Ne atoms is taken into account, the diffusion constant is reduced by a small fraction. In the gas phase and to some degree in liquids, the diffusion process can be viewed as a sequence of two-body collisions, the frequency of which depends on the collision cross section. Because the quantum centroid cross section is larger than the corresponding classical value, the quantum diffusion constant is found... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

We next need to find the frequency-dependent friction C(s) that we use in Eq. (3.16), to obtain the barrier frequency. This is obtained from the solvation time correlation function [22,23]. 

Let us summarize the steps quickly. First, we use the Marcus theory to obtain the reaction free-energy surface. Second, we adopt the Grote-Hynes theory to obtain the reaction rate. The latter needs frequency-dependent friction on the reactive motion, which is the solvent polarization. Third, we use the solvation time correlation function to obtain the frequency-dependent friction. 

There is an indirect way to detect intermittent local collective motions. In the case of depolarized Raman scattering, the depolarization ratio is sensitive to low-frequency fluctuations in water. Depolarization is the scattering of the polarization of the electric field of light in a direction perpendicular to the original direction of polarization. Each fluctuating state has a distinct depolarization ratio. The intermittent character of the dynamics is known to appear as a so-called 1,/ frequency (f) dependence in a power spectrum. The power spectmm is obtained by Fourier transforming a time correlation function. 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

The straightforward computation of the SE spectmm according to Eqs. 9.14 and 9.15 requires two time propagations, via G(t, t ) and G(t, 0), on a two-dimensional grid (t, t ). Since this procedure is time consuming, we directly calculate the ideal time- and frequency-resolved spectrum SsE(t, rather than the two-time correlation function Cse(, f)- Thi allows us to avoid the time integration over f in Eq. 9.14. The time dependence of the spectrum SseC, < s) at a fixed SE frequency cos is obtained by the propagation of just two auxiliary density matrices. 

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