Rate, autocorrelation

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

FIGURE 9.4. The autocorrelation function of the time-dependent energy gap Q(t) = (e3(t) — 2(0) for the nucleophilic attack step in the catalytic reaction of subtilisin (heavy line) and for the corresponding reference reaction in solution (dotted line). These autocorrelation functions contain the dynamic effects on the rate constant. The similarity of the curves indicates that dynamic effects are not responsible for the large observed change in rate constant. The autocorrelation times, tq, obtained from this figure are 0.05 ps and 0.07ps, respectively, for the reaction in subtilisin and in water. 

Examine the autocorrelation function. The high autocorrelations will indicate the order of the autoregressive part if any. The rate of decay of the autocorrelations will indicate a need for differencing. 

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

The character of an FCS autocorrelation function for a chemical reaction system depends on the relative rates of reaction and diffusion. It is useful to illustrate this dependence by calculating the autocorrelation functions to be expected for a simple one-step reaction system (Elson and Magde, 1974). We take as an example the simplest possible isomerization within the unfolded state, a single-step isomerization ... 

Fig. 4. Variation of autocorrelation function with changes in the equilibrium constant in the fast reaction limit. A and B have the same diffusion coefficients but different optical (fluorescence) properties. A difference in the fluorescence of A and B serves to indicate the progress of the isomerization reaction the diffusion coefficients of A and B are the same. The characteristic chemical reaction time is in the range of 10 4-10-5 s, depending on the value of the chemical relaxation rate that for diffusion is 0.025 s. For this calculation parameter values are the same as those for Figure 3 except that DA = Z)B = lO"7 cm2 s-1 and QA = 0.1 and <9B = 1.0. The relation of CB/C0 to the different curves is as in Figure 3. 

When l l, the above gives the so-called cross-correlation functions and the associated cross-correlation rates (longitudinal and transverse). Crosscorrelation functions arise from the interference between two relaxation mechanisms (e.g., between the dipole-dipole and the chemical shielding anisotropy interactions, or between the anisotropies of chemical shieldings of two nuclei, etc.).40 When l = 1=2, one has the autocorrelation functions G2m(r) or simply... 

The divergence between the rate of conformational transitions and the decay of the torsional autocorrelation functions (and hence local relaxations)... 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

When translational diffusion and chemical reactions are coupled, information can be obtained on the kinetic rate constants. Expressions for the autocorrelation function in the case of unimolecular and bimolecular reactions between states of different quantum yields have been obtained. In a general form, these expressions contain a large number of terms that reflect different combinations of diffusion and reaction mechanisms. 

As the salt concentration continues to decrease, however, matters change dramatically Q). The total scattering intensity decreases more abruptly, and the QLS autocorrelation function, which has been a simple single-exponential decay, becomes markedly two-exponential. The two decay rates differ by as much as two orders of magnitude. The faster continues the upward trend of D pp from higher salt, and is thus assigned the term "ordinary . The slower, which is about 1/10 of Dapp high salt, and appears to reflect a new mode of solution dynamics, is termed "extraordinary . 

The first relevant quantity required to obtain the rates is the autocorrelation function which are shown in Fig.8 for the ground vibrational level of the two excited electronic states. The two cases present a very similar behavior. Simply, for the A case its decay seems much faster. What is notorious is the large difference between the EP halfwidths as a function of the energy for the two electronic states, of approximately 2-3 orders of magnitude, as shown in Fig.9. This is explain by the norm of the initial wavepackets, which is much smaller for the B state, because its well is at larger R and shorter r, where the non-adiabatic couplings are much smaller. 

Measurements at low Qs At low Qs, because of the large magnitudes of s , the measured intensity and its autocorrelation function are dominated by the cumulative diffusion of the particles. The measured decay rate thus represents the cumulative or mutual diffusion coefficient Dm given by... 

The rate coefficient kernel of eqn. (369) and the initial condition term of eqn. (370) are those given by Northrup and Hynes [ 103]. These expressions are cast into the form of correlation functions (cf. the velocity autocorrelation function) and have a close similarity to the matrix elements in quantum mechanic applications. While they are quite easy to derive and... 

It has recently been pointed out by Gordon1 that the root-mean-square fluctuations in the sampled values of the autocorrelation function of a dynamical variable do not necessarily relax to their equilibrium values at the same rate as the autocorrelation function itself relaxes. It is the purpose of this paper to investigate the relative rates of relaxation of autocorrelation functions and their fluctuations in certain systems that can be described by Smoluchowski equations,2 i.e., Fokker-Planck equations in coordinate space. We exhibit the fluctuation and autocorrelation functions for several simple systems, and show that they usually relax at different rates. 

Comparison of Rates of Relaxation of Scaled rms Fluctuation Functions U(f t) and Autocorrelation Functions p(f t) at Short, Intermediate, and Long Times, for the Functions /Shown and the Systems Described in Table I... 

W. H. Miller The expression for the reaction rate (in terms of a flux-flux autocorrelation function) obtained by myself, Schwartz, and Tromp in 1983 is very similar (though not identical) to the one given earlier by Yamamoto. It is also an example of Green-Kubo relations. 

For a theoretical calculation of relaxation times one. must write the temporal autocorrelation functions of several functions Fn of the interparticle coordinates riS(t), 0y(O, and interparticle distance and where 0,/O and external magnetic field Ho (here particle refers to magnetic nuclei and atoms). The relaxation rates are proportional to the Fourier intensities of these autocorrelation functions at selected frequencies. For example, Torrey (16) has written for this autocorrelation function the equivalent ensemble average... 

Equation (5.47) shows that the velocity autocorrelation function , v(t )-v(t), decays exponentially with time. The rate of decay is determined by the friction coefficient / (= 1 /b-m), that is, by particle mass and mobility. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

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