Time autocorrelation function dynamics

A powerful analytical tool is the time correlation function. For any dynamic variable A (it), such as bond lengths or dihedral angles, the time autocorrelation function Cy) is defined... 

Dynamics of Reorientation. The approach of Impey, Madden and McDonald is used to consider the dynamics of reorientation for water molecules. The time autocorrelation function of the second Legrendre polynomial P2 of the angle subtended by the intramolecular H-H bond vector at time t with respect to its position at time t = 0 is calculated ... 

Silvestrelli et al. (1997) used Car-Parrinello molecular dynamics to obtain the IR spectrum of D2O. They did so by determining the time autocorrelation function for the dipole moment of their cell and then relating this function to the absorption coefficient as a function of frequency. Their computed spectrum and comparison to experiment are displayed in Fig. 11, taking into account corrections introduced by the authors. The essential features of the experimental spectrum, particularly the low-frequency peak, are reproduced well. The authors were then able to assign specific modes of the spectrum. 

The time autocorrelation function can be written as a transition dipole correlation function, a form that is equally useful for an inhomogeneously broadened spectrum. This is the form that is extensively used to discuss the spectral effects of the environment (32-34). The dipole correlation function also provides for the novice an intuitively clear prescription as to how to compute a spectrum using classical dynamics. For the expert it points out limitations of this, otherwise very useful, approximation. The required transformation is to rewrite the spectrum so that the time evolution is carried by the dipole operator rather than by the bright state wave packet. The conceptual advantage is that it is easier to imagine what the classical limit will be because what is readily provided by classical mechanics trajectory computations is the time dependence of the coordinates and momenta and hence, of functions thereof. In other words, in our mind it is easier to... 

The dynamics of the collective excitations can be determined conveniently by photon autocorrelation spectroscopy, which is also called self-beating, time-resolved Rayleigh or quasi-elastic light scattering spectroscopy [29]. Here the time autocorrelation function of the scattered light intensity G (t) is measured, which, in the heterodyne detection regime, is given by ... 

The methods for calculating A/Zexc( ). an equilibrium property of the system, have already been discussed. The dynamic aspects of the permeation process are captured in D (2 ). The fluctuation-dissipation theorem provides the connection between the local diffusion constant and the time autocorrelation function of the random force acting on the solute. 

MD simulation can gain insight into the viscoelastic behavior of nanopartide-polymer composites. The shear stress relaxation modulus can be calculated using the time autocorrelation function of the stress tensor, while the viscosity is calculated based on the Einstein relations. Compared to conventional composites, the viscoelastic properties are strongly perturbed by the nanopartides and depend upon the nature of nanopartide-polymer interactions. The viscosity and dynamic shear modulus can be dramatically increased for composites with attractive... 

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by... 

Chain dynamics are described in terms of time autocorrelation functions. These describe the rate of decay of the probability that a chain with a specific configuration at time t has that same configuration at a later time (t +. The two important correlation functions in the... 

Here, 1 will show that the measurement of the dynamic heterogeneity revealed the characteristics of the random structure in the B4 phase [64]. Figure 10.30a shows the time autocorrelation function of the BX phase (at 27 °C) and the B4 phase (at 34 °C) of a mixed system of 80 wt% 5CB. Figure 10.30b shows the nematic and isotropic phases of pure 5CB for comparison. Large fluctuations are present around 0.1 ms in the Bx phase, but the corresponding fluctuation in the B4 phase has disappeared. In addition, the Bx-B4 transition temperature of the mixed system is almost the same as the phase transition temperature of pure 5CB, 34.10 °C. In the phase diagram, Bx-B4 transition temperature corresponds to the extrapolated isotropic-nematic phase transition temperature phase of 5CB. From these two experiments we reason that the fluctuations that had been observed in the Bx phase come from the orientation fluctuations of the 5CB nematic... 

Another form of the fluctuation-dissipation theorem involves time correlation functions. If the external perturbation is applied at t = —oo and turned off at t = 0, the system will move from a steady state back to equilibrium. The dynamic relaxation of property A from its original steady state, (A(0)), to equilibrium, A)o, can be shown to be congruent to the normalized time autocorrelation function of equilibrium fluctuations. 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

An examination of the autocorrelation function (0(0) <2(0) annucleophilic attack step in the catalytic reaction of subtilisin is presented in Fig. 9.4. As seen from the figure, the relaxation times for the enzymatic reaction and the corresponding reference reaction in solution are not different in a fundamental way and the preexponential factor t 1 is between 1012 and 1013 sec-1 in both cases. As long as this is the case, it is hard to see how enzymes can use dynamical effects as a major catalytic factor. 

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. 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

NMR 13C spin-lattice relaxation times are sensitive to the reorientational dynamics of 13C-1H vectors. The motion of the attached proton(s) causes fluctuations in the magnetic field at the 13C nuclei, which results in decay of their magnetization. Although the time scale for the experimentally measured decay of the magnetization of a 13C nucleus in a polymer melt is typically on the order of seconds, the corresponding decay of the 13C-1H vector autocorrelation function is on the order of nanoseconds, and, hence, is amenable to simulation. 

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