Frequency correlation function

The quality of the fit is excellent as can be judged from Fig. 1(b). The following parameters of the frequency correlation function (Fig.lb, solid line) were obtained in the simulations 1/Afast 2 130 fs, Afas, = 90 cm 1, l/Asiow = 700 100 fs, Asiow = 65 cm 1. These results are in very good agreement with our previous findings from heterodyne-detected two-pulse photon echo experiments [19]. 

Our experiments and numerical simulations have proven that interference between chromophore and solvent responses greatly obscures the experimental observables in IR spectroscopy on water at waiting times >0.5 ps. However, the water dynamics can still be obtained if the thermal effects are carefully characterized and self-consistently included in the model. This results in the longest time scale for the frequency correlation function of 700 fs. 

Figure 18. The normalized electronic transition frequency correlation function M(t) 1= S(i)] obtained from the experimental three-pulse photon echo peak shifts and transient grating data for IR144 in ethanol (—) total W(t) ( ) ultrafast Gaussian component in M(t) ( ) oscillatory component that arises from intramolecular vibrational motion.

V To examine photodissociation given this field requires, as shown below, the S t frequency-frequency correlation function (( (a X aq)) where T(m) is the Fourier V transform of eJt) for Jt) equal to a constant <5,.. Given Gaussian pulses [Eqs. (5.28) and (5.29)1 we have 11891... 

Note that despite the excitation of multiple levels the only correlation function5 required is between at two frequencies. Assuming the phase diffusion model r described above, then the frequency-frequency correlation function is given by, Eq. (5.30). The probability P(q) of forming product in channel q is then obtained by integrating over the pulse width ... 

In each limit, the experimental decay is characterized by a single parameter, whereas the underlying frequency correlation function is characterized by at least two parameters. This problem exists for other models as well. In general, the problem of inverting C, d(t) to find C0J(i) is underdetermined and requires model-dependent assumptions to complete an analysis. This problem reflects the loss of information caused by the time averaging in Equation (4). 

The echo and FID decays derive from the same frequency correlation function, but the echo dissects C, (t) with a more complex and informative operator. 

Dynamical quantities are harder to obtain, since the QMC representations only give access to imaginary-time correlation function. With the exception of measurements of spin gaps, which can be obtained from an exponential decay of the spin-spin correlation function in imaginary time, the measurement of real-time or real-frequency correlation functions requires an ill-posed analytical continuation of noisy Monte Carlo data, for example using the Maximum Entropy Method [46-48]. 

Nonlinear infrared spectroscopy can in principle provide knowledge of all the relaxation processes of oscillators, including those that do not manifest themselves in the linear spectral line shapes. The v = 0 v = I transition line shape is determined by the overall rotation of the molecule, population relaxation time T and by the vibrational frequency correlation function. The experimental line-shape is not a very useful determinant of this correlation function [40, 49] because it provides experimental data only along one axis, either frequency or time, and the line-shape function is usually too complex to... 

Note that these two approximations, Eqs. (4.72) and (4.73), are not limited to the two-state jump model but are generally valid in the fast modulation regime. The frequency correlation function is given by... 

Ti is the excited state lifetime. Unlike the spectral diflusion kernel or frequency correlation function, which involve only two different times, (t) depends on co(z) at all times between 0 and t. The seemingly formidable problem of calculating (t) has been solved in general by Kubo and Anderson [31-34], and the solution for the current model is [35],... 

In the second type of experiment that measures single molecule spectral dynamics one performs repeated fluorescence excitation scans of the same molecule. In each scan the line shape is described as above, but now there is the possibility that the center frequency of the line will change from scan to scan because of slow fluctuations. Thus one can measure the center frequency as a function of time, producing what has been called a spectral diffusion trajectory. This trajectory can, in principle, be characterized completely by the spectral diffusion kernel of Eqs. (16) and (19), but of course it must be understood that only the slow Kj < 1 /t) TLSs contribute. In fact, the experimental trajectories are really too short to be analyzed with this spectral diffusion kernel. Instead, it is useful [11, 12] to consider three simpler characterizations of the spectral diffusion trajectories the frequency-frequency correlation function in Eq. (14), the distribution of frequencies from Eq. (15), and the distribution of spectral jumps from Eq. (21). For this application of the theoretical results, in all three of these formulas j should be replaced by s, the labels for the slow TLSs. 

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