Transient time correlation function

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

GK = Green-Kubo LIT = linear irreversible thermodynamics LRT = linear response theory NEMD = nonequilibrium molecular dynamics NESS = nonequilibrium steady state TTC = thermal transport coefficient TTCF = transient time correlation function. 

The development of NEMD simulations has inevitably led to questions regarding the nonlinear response of the system. Evans and Morriss, in particular, concentrated their efforts on deriving useful expressions for this purpose. On a microscopic level these authors developed two principally equivalent formulae the Kawasaki distribution function and the transient time correlation function (TTCF). 

In this paper we investigate the time dependent ground state hole spectrum of cresyl violet in polar solvents by means of subpicosecond transient absorption spectroscopy. The time correlation function expressed by eq (7) showed large difference in time profiles compared with the reported one expressed by eq (6). Possible mechanisms will be discussed. 

Femtosecond spectroscopic investigations in the spectral range 400-880 ran have permitted to discriminate specific OH effects on the dynamics of short lived UV excited CTTS states and transient near-IR (HO e )H20 pairs. The complex nature of ultrafast prehydration elementary redox reactions with nascent OH radical (strong acid) must be contemplated in the framework of ion-pairs dynamics, ion-solvent correlation function, short-range ordering water molecules, solvent screening or anisotropic electric field effects and short-time vibronic couplings. 

Sample preparation was given elsewhere [2]. Femtosecond fluorescence upconversion and picosecond time-correlated single-photon-counting set-ups were employed for the measurement of the fluorescence transients. The system response (FWHM) of the femtosecond fluorescence up-conversion and time-correlated single-photon-counting setups are 280 fs and 16 ps, respectively [3] The measured transients were fitted to multiexponential functions convoluted with the system response function. After deconvolution the time resolution was 100 fs. In the upconversion experiments, excitation was at 350 nm, the transients were measured from 420 nm upto 680 nm. Experiments were performed under magic angle conditions (to remove the fluorescence intensity effects of rotational motions of the probed molecules), as well as under polarization conditions in order to obtain the time evolution of the fluorescence anisotropy. 

Solution of the linear approximation (4.1.48) reveals the transient region Ar 1, in which the correlation function Yo(r,t) increases from zero to unity. For the auto-model variable r/ = r/ o in equation (4.1.48) the transient region width Arj = 1 / o —> 0, which corresponds to the step-like function. The function Y (r, i) in the superposition approximation reveals the transient region Ar increasing in the time which is confirmed by computer simulations shown in Figs 5.5 and 5.6. 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

For alcohol solvents, measurements were made with time-correlated single photon counting. The remaining measurements were made with the fluorescence upconversion system. The transients in alcohol solvents were fitted with a single exponential kinetic function. The kinetics in acetone is also well described by a single exponential, but in benzonitrile, dimethyl-sulfoxide, and propylene carbonate the kinetics were modeled with a biexponential decay. 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

Solvation dynamics are measured using the more reliable energy relaxation method after a local perturbation [83-85], typically using a femtosecond-resolved fluorescence technique. Experimentally, the wavelength-resolved transients are obtained using the fluorescence upconversion method [85], The observed fluorescence dynamics, decay at the blue side and rise at the red side (Fig. 3a), reflecting typical solvation processes. The molecular mechanism is schematically shown in Fig. 5. Typically, by following the standard procedures [35], we can construct the femtosecond-resolved emission spectra (FRES, Stokes shifts with time) and then the correlation function (solvent response curve) ... 

Figure 19 shows the typical fluorescence transients of TBE from more than 10 gated emission wavelengths from the blue to the red side. At the blue side of the emission maximum, all transients obtained from four Trp-probes in the cubic phase aqueous channels drastically slow down compared with that of tryptophan in bulk water. The transients show significant solvation dynamics that cover three orders of magnitude on time scales from sub-picosecond to a hundred picoseconds. These solvation dynamics can be represented by three distinct decay components The first component occurs in about one picosecond, the second decays in tens of picoseconds, and the third takes a hundred picoseconds. The constmcted hydration correlation functions are shown in Fig. 20a with anisotropy dynamics in Fig. 20b. Surprisingly, three similar time scales (0.56-1.431 ps, 9.2-15 ps, and 108-140 ps) are obtained for all four Trp-probes, but their relative amplitudes systematically change with the probe positions in the channel. Thus, for the four Trp-probes studied here, we observed a correlation between their local hydrophobicity and the relative contributions of the first and third components from Trp, melittin, TME to TBE, the first components have contributions of 40%, 35%, 26%, and 17%, and the third components vary from 32%, to 38%, 43%, and 53%, respectively. The...

Through time autocorrelation and cross-correlation functions at equilibrium. These can be computed for and among a variety of vectors associated with the molecular motion. They are equiUbrium properties, and the fiuctuation-dissipation theorem relates them to transient properties such as... 

As expected from examinations of the individual fluorescence transients, the peak of the reconstructed spectrum shifts to red with time. In Figure 3, the spectral shift correlation functions C(t) are shown with the results of bi-exponential fitting. The observed solvent relaxation times are 7.9 ps (19 %) and 18.7 ps (81 %) for DMA and 6.7 ps (81 %) and 13.3 ps (19 %) for AN. These are much longer than the fluorescence lifetimes of NB, i.e., 0.1 ps... 

Transient species are characterized by their spectra. If the spectra are not known, assignment can be made, for example, by comparison with analogues or from solvent effects. Reaction times are extracted from the analysis of AA(A, t) as a single exponential function of time or as the sum of exponential functions. If some reaction times are very close to the time-resolution of the set-up the analysis is made by fitting the convolution product ofthe pump-probe cross correlation function G t ) by a sum of exponential functions... 

Fig. 14.11 The reactive flux correlation function, Eq. (14.99) plotted against time. After the initial transient period this function becomes essentially constant on the time scale shown.

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