Velocity autocorrelation function water

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation function... 

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. 

Stillinger and Rahman have also considered the diffusion coefficient, velocity autocorrelation function and scattering function for simulated water. For discussion of these interesting calculations the reader is referred to their papers 3>. 

Figure 9 The integrated velocity-velocity autocorrelation function as described in Eqs. [122] for SPC/E water at 277.2 K and 1 atm as described in Ref. 43. Note the convergence of the integral over time. The experimental value of the diffusion constant is lO" m /s, whereas the calculated value from this curve is 1.6 0.08 x 10 m /s.

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.)...

A from any lipid head group atom were considered to be bound, and any water more than 4 A away from all lipid head groups was considered to be bulk. Because the bound/bulk status of waters can change during the course of a simulation, the nonbonded atom list was updated every picosecond. Of the 553 waters used in the simulation, on average there were only 160 bulk waters. The velocity autocorrelation functions (VAF), the mean square displacements (MSD), and the orientational correlational functions (OCF) for the bound and bulk waters were calculated. VAFs were calculated as ... 

From a conceptual standpoint, it is useful to have an understanding of the time scales for motions of particles near metal-water interfaces, to be able to better understand their nature, as well as how molecules and atoms near these interfaces differ from those of the bulk. The two most commonly calculated dynamic properties for metal surfaces are the mean-square displacement and the velocity autocorrelation functions, because these can be used to calculate diffusion constants and spectra. 

Figure 17 Velocity autocorrelation functions (VACF), (v I " water molecules...

As mentioned earlier, it is difficult to obtain a quantitative measure of entropy. By using the 2PT method (the method will be described later in Chapter 19), one can obtain the entropy of water molecules in both major and minor grooves of DNA. One can also get a measure of the translational diffijsivity of those water molecules from the mean-square displacement or velocity autocorrelation function - all these are fortunately easily available with computer simulations. 

Figure 5.18. The normalized velocity autocorrelation functions of cations and anions in water.

Molecular center-of-mass velocity autocorrelation functions for several supercritical states of water are shown in Figure 17. Obviously, the VACFs decay faster at the higher density. The density dependence of these functions is very similar to that for water at normal temperatures (Jancso et al. 1984), in agreement with the similarity in the pressure-induced changes of the structural properties discussed above. 

Figure 17. Normalized center of mass velocity autocorrelation functions for the water molecules under supercritical conditions.

Self-diffusion coefficients in supercritical water can be determined from MD simulations through the molecular mean-square displacement analysis, or through the velocity autocorrelation functions (Eqn. 20) with the help of the Green-Kubo relation (Allen and Tildesley 1987) ... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

Residence Times. The dynamic behavior of water is frequently characterized by the self diffusion coefficient (sdc) D, which can be calculated from the particle mean square displacements via the Einstein relation or from the velocity autocorrelation functions (acf) via the Kubo relation. Near an interface this quantity D is not the self diffusion coefficient, since there are no free boundary conditions for the surface layer. Sonnenschein and Heinzinger [52] calculate a property called residence autocorrelation function... 

Results with a certain degree of reliability from MD simulations of aqueous solutions reported up to now are restricted to structural properties of such solutions. In the section on aqueous solutions below very preliminary velocity autocorrelation functions are calculated from an improved simulation of a 9.55 molal NaCl solution. The problem connected with the stability of the system and the different cut-off parameters for ion-ion, ion-water and water-water interactions are discussed. Necessary steps in order to achieve quantitative results for various dynamical properties of aqueous electrolyte solutions are considered. 

Figure 8. Normalized translational (dashed) and rotational (solid) velocity autocorrelation functions

The rotational relaxation of water molecules is often discussed in terms of angular momentum autocorrelation functions (e g., Stillinger and Rahman 1972 Yoshii et al. 1998). For a flexible water model, a slightly different approach can also be used. In order to separate the various modes of molecular librations (hindered rotations) and intramolecular vibrations, the scheme proposed by Bopp (1986) and Spohr et al. (1988) can be employed. The instantaneous velocities of the two hydrogen atoms of every water molecule in the molecular center-of-mass system are projected onto the instantaneous unit vectors i) in the direction of the corresponding OH bond (ui and U2) ... 

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