Velocity autocorrelation function VACF

Let us now consider the velocity autocorrelation function (VACF) obtained from the MCYL potential, (namely, with the inclusion of vibrations). Figure 3 shows the velocity autocorrelation function for the oxygen and hydrogen atoms calculated for a temperature of about 300 K. The global shape of the VACF for the oxygen is very similar to what was previously determined for the MCY model. Very notable are the fast oscillations for the hydrogens relative to the oxygen. 

The numerical calculations show that this theory can describe the velocity autocorrelation function (VACF) for liquid argon and rubidium [47] fairly well, and the agreements with the computer simulation studies [48, 49] were satisfactory. However, the numerical calculations needed the VACF as an input which was obtained from the respective computer simulation... 

The diffusion of a tagged particle or solute is given in terms of the time-dependent velocity autocorrelation function (VACF) Cv(t) and is given by... 

MD calculations readily relate the velocity, t), of a given atom, at time r = 0, to that of the same atom at some latter time, t. The development of this correlation as time passes provides access to the single-particle velocity autocorrelation function, VACF t). 

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

In MD simulations, the dynamical behavior of a molecular fluid can be monitored in terms of velocity autocorrelation functions (VACF), which are calculated as... 

One method for computing vibrational spectra relies on the results of MD simulations (see Cygan, Parker, or Garofalini, this volume) is to take the power spectrum of the velocity autocorrelation function (VACF). The VACF is represented by (e.g.,... 

The second used the velocity autocorrelation function (VACF) and Green-Kubo relations [3] ... 

As stated in the introduction, molecular simulations can be used as complementary tools to characterize ionic liquids from a modelling approach. The transport coefficients can be calculated through the corresponding Green-Kubo relations (Allen Tildesley, 1987 Frenkel Smit, 2002). Within this formalism, the self-diffusion coefficient (D) of each ion is calculated from its velocity autocorrelation function (vacf) through the following expression ... 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

There have been various approaches in the mode coupling theory [9, 37, 57, 176]. All these theories have exhibited the presence of t 3/2 of the velocity autocorrelation function in the asymptotic limit in three dimensions. Extending each of these theories for studies in two dimensions we can show that the velocity autocorrelation function has r1 tail in the asymptotic limit. Since the diffusion coefficient is related to Cv(t) through Eq. (337), it can be shown that D diverges in the long time due to the presence of this t l tail in the VACF. 

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. 

The calculation of the vibrational spectrum from an (AI)MD trajectory involves Fourier-transforming the time-dependent velocity autocorrelation function [60] an alternative approach involves calculating the phonon frequencies by diagonalizing the Hessian matrix of a model obtained by structural optimization of the classical MD structure [53]. The AIMD-VACF approach naturally include finite-temperature anharmonic effects missing in the Hessian-harmonic approximation, but it does not produce accurate IR intensities (for which an autocorrelation function based on the exact dipole moments would be needed [61-63]). Despite these issues, it turns out that, in the case of 45S5 Bioglass , the two methods give similar frequencies of the individual modes [53]. 

This section presents method of numerical determination of diflfiision D and friction C coefficients of Brownian particles from velocity (VACF) and force (FACF) autocorrelation functions in molecular dynamic simulations. Electrostatic parameters of particles in simulation were obtained using modem DFT methods of quantum chemistry. The calculations were carried out with PCS (Patch Clamp Simulation) program package, designed for simulation of neurotransmission processes, using Brownian and molecular dynamic methods, with GPU acceleration support, based on NVIDIA CUBA. 

---