Distance autocorrelation function

Fig. 1.3. Distance autocorrelation function, C(t) see text, (a) Calculated values at 300 K and 50 K (b) Experimental values from [6]...

We now move to a dynamical analysis of the time series for these distances. In the left panel of Fig. 17, we show the Fourier transforms of the 0-5 -0-4 distance autocorrelation function, and of the 0-5 -0-4, 0-4 -0P distance-distance correlation function. The spectra are very similar, indicating that 0-5 —0-4 and 0-4 Op... 

Fig. 17 Left Comparison of the spectra of the 0-5 —0-4 distance autocorrelation function and the 0-5 —0-4 and 0-4 —Op distance-distance correlation function. Right The similarity of the spectra of the 0-5 —0-4 distance autocorrelation function for the E---S complex based on the classical and quantum/classical MD simulation, shows an agreement to the classical simulations.

Fig. 18 Left spectra of the 0-5 —0-4 distance autocorrelation function for hPNP and unsolvated substrates. Note that the natural vibration of the oxygen centers, i.e. 285 cm-1, is altered in the presence of the enzyme. Right the power spectrum of the H257G mutant of the E---S complex shows a distinct peak at 333cm-1, very similar to the result for the solvated substrate.

Next we examine whether these vibrations are unique in the enzymatic environment or they are inherent in the substrates. In the left panel of Fig. 18 we compare the calculation in the enzyme with a simulation of the substrates in aqueous solution, in the absence of hPNP. The spectrum of the 0-5 —0-4 distance autocorrelation function of the classical MD of solvated substrates showed a peak at 330 cm-1, and of the unsolvated substrates at 285 cm-1, i.e. distinct from the peaks in the presence of the enzyme, revealing that hPNP is directly affecting the way in which these oxygens naturally vibrate. 

In these expressions key is obtained from a rather complicated integral equation which depends upon the end-distance autocorrelation function (t) contained in the z(t) term, and a sink term Y which treats cyclization in terms of a spherical reaction volume with a capture radius a. Note that jf depends upon Rf, so that at a constant value of a, Y depends on chain length. 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

Thus, effects of the surfaces can be studied in detail, separately from effects of counterions or solutes. In addition, individual layers of interfacial water can be analyzed as a function of distance from the surface and directional anisotropy in various properties can be studied. Finally, one computer experiment can often yield information on several water properties, some of which would be time-consuming or even impossible to obtain by experimentation. Examples of interfacial water properties which can be computed via the MD simulations but not via experiment include the number of hydrogen bonds per molecule, velocity autocorrelation functions, and radial distribution functions. 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

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... 

The autocorrelation function (Fig. 2) shows clearly that a representative chain segment embracing about 5-7 CH2-units should be sufficiently long to describe the intramolecular interference modulation in the WAXS-pattern of a PE-melt completely up to distances of approx. 30 A. 

The autocorrelation functions of the investigated features calculated according to Eq. 6-22 and smoothed according to Eq. 9-4 are represented in Fig. 9-5. The increasing distance l of the measuring points results in steadily decaying autocorrelation functions for all features, with that for lead decaying most conspicuously. 

The autocorrelation function of lead shows the most conspicuous decay in the investigated area. Consequently it follows that the estimated value of the lower limit of the confidence interval for the critical distance between the measuring points for lead is the limiting sampling step. 

Another piece of information that we wanted to extract from our experiments was connected with the dynamic behavior of spatial variables. If we consider three successive particles in the chain and we denote by the distance of the middle one from the center of mass of the other two and by the distance between these two, we can compute the normalized autocorrelation function of these two variables. They are shown in Fig. 9 as can be immediately observed, they decay to zero on a time scale which is much greater than that of the velocity variable. Also, the center of mass decays faster than R . In the next section we shall argue that this suggests that the virtual potential characterizing the itinerant oscillator model has to be assumed to be fluctuating around a mean shape, which, moreover, will be shown to be nonlinear and softer than its harmonic approximation. 

The variable x, which is an interatomic distance r, is divided into elementary distance intervals of 0.5 A. All interatomic distances falling in the same interval are considered identical. The autocorrelation function values AQ are obtained by summing all the products w, wj of all pairs of atoms / and for which the interatomic distance r,y falls within the considered interval [x, x + 0.5]/. 

The Geary coefficient is a distance-type function varying from zero to infinity. Strong autocorrelation produces low values of this index moreover, positive autocorrelation translates into values between 0 and 1 whereas negative autocorrelation produces values larger than 1 therefore, the reference no correlation is c = 1. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Such scale invariant spectra reveal that the dislocation density variation is scale invariant, and put forward spatial correlations over very large distances with a correlation length of the order of the system (sample) size. This is confirmed with the autocorrelation function of the records shown in figure 4, in which is also represented a signal taken from a... 

Because the autocorrelation function uses the number of bonds to describe the distance between two atoms in a molecule, autocorrelation coefficients are topological descriptors. In addition to the topology information, however, they include the atom properties considered. 

Figure 3.4 The autocorrelation function gQQ of liquid heavy water at 11 °C as a function of interatomic distance R. The value of R at maximum of gQQ is the value of Qq, the average equilibrium O - O distance of O-D -O bonds in this liquid, defined in Fignre 2.1. Reprodnced from Figure lb of reference (24), with permission. Copyright (2006) by the American Physical Society.

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