Short-time decay

Our calculations are only in qualitative agreement with experiment the most salient points of discrepancy are that our overall decays are significantly faster than experiment [46], and the theory shows that most of the frequency dependence comes in the initial (less than 100 fs) short-time decay, whereas the experiment shows significant frequency dependence out to 1 ps. 

Orientational relaxation measurements have also been performed on this system. Earlier papers showed no frequency dependence to the decay rate [48, 57, 58], but a recent study by the Fayer group [59] showed that the very short-time decay has a frequency dependence, in agreement with our theoretical calculations for... 

It is evident from the above equation and Eq. (26) that only static intermolecular correlations contribute to (Usoiv and therefore to the short-time decay of C(t) Identifying solvent-pair contributions to C(f) is straightforward for pairwise-additive potentials such as the site-site Coulombic form of Eq. (7). For such potentials. 

Figure 14.7 (a) 2D 1H-NOESY surface spectrum of SBR acquired for a mixing time of 2.7 ms and under 8 kHz MAS [47]. Already at this short mixing time pronounced crosspeaks are visible, (b) Short time decay of the crosspeak intensity evaluated from a series of 2D spectra. Such curves can be analysed, providing information on internuclear distances and molecular dynamics (correlation times)... 

For the example of TOL-d5, the temperature dependence of Fcos(tm tp = 80 ps) is presented in Fig. 22a, i.e., a large tv is used to probe highly hindered dynamics.1,99 When the temperature is increased, the amplitude of the short-time decay first increases and then decreases, where the amplitude is a maximum near Tx 97 K. Such peculiar behavior was also reported for PB-d6, cf. Fig. 22b. Further, the behavior resembles that found in simulations of a restricted reorientation governed... 

Fig. 23. Temperature-dependent amplitude 1-Ctp of the short-time decay of Lcos( m tPi f°r the type B glasses (a) TOL-d5 and (b) PB-d6. Evolution times tv — 30 and 80 ps were used. For comparison, results for the type A glasses GL-d5 and PS-d3 are included, indicating the absence of a short-time decay well below Tt. The dashed lines represent the respective fractions of the distribution (7(lg xp) lying in the time window of the experiments. They were calculated based on the distributions determined for TOL and PB in DS,12,19 using Eq. (12). The fractions were multiplied with temperature-independent constants atp to match the tp dependent values of 1-Ctp. (Adapted from Ref. 99.)...

Figure 7 The time auto correlation function and the corresponding spectrum for a Gaussian wave packet propagating on an excited harmonic potential energy surface, (a) The short time decay of C(/) (cf. Eq. (17)) and the broad spectrum (= the Franck Condon envelope (cf. (18)). (b)The longer time dependence of C(r) and the corresponding, vibrationally resolved, spectrum.

As shown in figure 4.9, the time evolution P f) shows three distinct time dependencies, each characterized by either 7, F, or e (1) The Lorentzian-like envelope, with width F for the complete absorption spectrum transforms into an exponential decay with rate constant F/ft, which is for the short-time decay of PXt)- (2) The set of resonances, separated on average by an energy e, transform into a set of oscillations (i.e., recurrences) whose periods are approximately elh. (3) The envelope of each individual resonance also transforms into an exponential-like decay, characterized by the rate 7/ft, which corresponds to leakage from the sparse i) — /) subspace into the quasi-continuum ). The recurrences described above in (2) are damped out by this slow decay. 

The above analysis pertains to a particular time-scale for F/r). If only the initial decay of P t) is measured with a rate F h, the resulting absorption spectrum will be broad and featureless with a width F. Thus, a low-resolution spectrum corresponds to only observing the short-time decay of PJlf)- On the other hand, if P/jt) were followed for a sufficiently long time, structure in the individual resonance envelopes will begin to appear in the spectrum. Finally, as discussed above, if F (r) is evaluated to infinite time /(co) will become a stick spectrum of the individual eigenstates n) in Eq. (4.27) for which cj 5 0. A projection of 1 ) onto eigenstates is illustrated in figure 4.7. 

Figure 7. The self intermediate scattering function for GB particles in the T range of 1050-1150K (defined in the text). The dashed curves are a fit of the stretched exponential relation, Fs q, t) 0 exp[—(t/r) ] to the long-time data, where the short-time decay arises from the inertial atomic dynamics. The inset shows a power fit of r to 7"— 7, where Tq and y are adjustable parameters as in previous measurements and simulations. Figure 7 was originally published in [16], National Academy of Sciences.

Moreover, short time decay rate constants can be estimated very accurately by linearization of the dynamics(8) around the HC periodic orbit(14). The eigenvalues of the Jacobian matrix propagator of the return map are = exp(ia), =exp(--ia) for elliptic (stable) fixed... 

The top plot in Figure 17 contains the stress-stress autocorrelation function for [Cimim][Cl] at 425 K calculated by Bhargava and Balasubrama-nian. ° The rapid oscillations are due to high-frequency intramolecular motions of the cation. The correlation function shows a rapid short-time decay but a very slow long-time decay, as can be seen in the bottom graph in... 

The stochastic dynamics takes its simplest form when the correlation time of the noise events is long compared to the periods of the limit cycles LI and L2, In this circumstance, the phase points are largely confined to the limit cycles, with infrequent hops between them. The short-time decay depends on the initial preparation of the system if the system is initially in L2, then those phase points in the vulnerable region will decay directly to F provided a second noise event does not act before the phase points cross B2. If the system is initially in LI, then L2 must be populated before escape can occur. The longtime decay, which is independent of the system preparation, yields the rate coefficient of the process L F. The results of simulations shown in Fig. 2 verify the existence of a simple macroscopic rate law for the decay of the fraction of phase points in L ... 

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