Displacement autocorrelation functions

Expressions of the type of the right-hand side of Eq. 2.65 can be expressed in terms of the charge displacement autocorrelation function,... 

The simulations of BPTI in vacuum and in a van der Waals solvent, described above (this chapter, Sect. A), were analyzed to determine the effect of solvation on the time scales of the atomic motions.199 Given the displacement autocorrelation function, C(t), for a fluctuation, the relaxation time is defined... 

Figure 43. Solvent effects on local protein motions. The normalized displacement autocorrelation functions are plotted versus time for residues near the active site in lysozyme. Vacuum simulation results are plotted as dashed lines and solvent simulation results are plotted as solid lines for (a) Trp-62 N 1 and (b) Asn-46 Cfl.

The difference in the spectral density between the displacement and velocity autocorrelation functions can be understood from a normal-mode description (see Chapt. IV.F). Using Eq. 23 for the displacement autocorrelation function and differentiating it to obtain the velocity autocorrelation function, one finds that the terms in the latter are weighted by the square of the mode frequency relative to the former. Thus higher-frequency contributions are more important in the spectral density associated with the velocity autocorrelation function than the displacement autocorrelation function.153,332... 

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]. 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

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. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

Some qualitative comments were made about the velocity autocorrelation function in Chap. 8, Sect. 2.1. In this section, it is considered in more quantitative detail. One of the simplest expressions for the diffusion coefficient is that due to Einstein [514]. He found that a particle executing a random walk has an average mean square displacement of (r2 > after a time t, such that... 

We have already seen that even in the case of strong intermolecular interactions neither nor decay initially as exponentials. Gordon has been able to reproduce the decay of these latter functions in liquid CO and N2 by allowing for large angular displacements between interactions.58 However, Gordon s model incorrectly predicts the angular momentum autocorrelation function. 

The features discussed above can be readily interpreted in terms of the time-dependence of the autocorrelation function < t) >. The most important factor governing the changes in the widths of the spectra for the positive versus negative displacement is the initial decrease in < (t)>. When transforming from the time domain to the frequency domain, a slow decrease in corresponds to a narrow progression and a fast decrease in < m> corresponds to a broader progression. 

The decrease in < 0 < (t) > depends on the slope of the potential surface at the point at which the wavepacket is initially placed. The slope of the potential in the Qy direction is steeper on the positive Qx side of the surface than on the negative Qx side. When the slope in the Qy dimension is large, the Qy part of the two-dimensional wavepacket will rapidly change its shape and < 010 (t) > will decrease rapidly. Therefore, for a positive Qx displacement in the coupled potential, the autocorrelation function will decrease more rapidly than it would for a negative displacement. 

The magnitudes of < (j>(t)> versus time are shown in Fig. 4. The autocorrelation function for the positive displacement along Qx in the coupled potential (lowest curve in Fig. 4), starts out at 1 and drops to 0 over a shorter period of time than in the uncoupled potential (middle line). The Fourier transforms of these < (t) > give the spectra shown in Fig. 3. This reasoning explains why a positive displacement results in a broader progression and a negative displacement results in a narrower progression. 

Fig. 4. Autocorrelation functions plotted versus time for the spectra described in Fig. 3. The autocorrelation function shown by the middle curve corresponds to the reference spectrum (Fig. 3a), k.j, = Ocm-1, the autocorrelation function shown by the lowest curve corresponds to a positive displacement in the quadratically coupled potential surface, and the autocorrelation function shown by the top curve corresponds to a negative displacement in the quadratically coupled potential surface...

The rapid decay of the autocorrelation function at very short times is mainly due to a dephasing of the wavepacket rather than a displacement in coordinate space. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

However, the displacement of the particle x is in reality a function of time and therefore can also be expressed in terms of an autocorrelation function similar to that presented... 

Figure 13 (a) Mean-square displacement (Eq. [63]) and (b) CoM velocity autocorrelation function (Eq. [65]) for methanol molecules adsorbed in silica faujasite... 

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. 

Figure 45. Time-scale matching for protein-solvent motions. The normalized spectral density for the (a) displacement and (6) the velocity autocorrelation functions of Trp-62 N 1, and (c) for the velocity autocorrelation function of ST2 water. (Note the differences in the timescales.)...

The correlation function, <-P2[am(0) ( )]>. provides a measure of the internal motions of particular residues in the protein.324 333 Figure 46 shows the results obtained for Trp-62 and Trp-63 from the stochastic boundary molecular dynamics simulations of lysozyme used to analyze the displacement and velocity autocorrelation functions. The net influence of the solvent for both Trp-62 and Trp-63 is to cause a slower decay in the anisotropy than occurs in vacuum. In vacuum, the anisotropy decays to a plateau value of 0.36 to 0.37 (relative to the initial value of 0.4) for both residues within a picosecond. In solution there is an initial rapid decay, corresponding to that found in vacuum, followed by a slower decay (without reaching a plateau value) that continues beyond the period (10 ps) over which the correlation function is ex-... 

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