Velocity auto-correlation

Physically this description corresponds to putting an atom (mass M) in an external time-dependent harmonic potential (frequency co0). The potential relaxes exponentially in time (time constant l/x0) so that eventually the atom experiences only a frictional force. Compared with other models2 which have been proposed for neutron scattering calculation, the present model treats oscillatory and diffusive motions of an atom in terms of a single equation. Both types of motion are governed by the shape of the potential and the manner in which it decays. The model yields the same velocity auto-correlation function v /(r) as that obtained by Berne, Boon, and Rice2 using the memory function approach. 

Not only can simulations predict distributions, dynamic properties can also be obtained using Molecular Dynamics. From I.App.ll it is recalled that the extent to which a molecule is able to retain its original velocity w(0), during a brief period t, is quantified in the velocity (auto) correlation /unction C (t). defined as (I.Al 1.1)... 

In underdamped motion, the popular functional form cos a>i t exp —at) differs only in phase from the true correlation function, whose zeros are at ( = tui tan (— 2o)i//3), so that the first zero occurs somewhat after the first quarter-cycle. The velocity auto-correlation function, which we shall need later, is obtained by dififerentiating y twice and normaliang, giving... 

A time correlation function that involves the same observable at two different times is called an autocorrelation function. We have found that the self-diffusion coefficient is the time integral of the velocity auto-correlation function... 

Noting that the spectral density of the ensemble-averaged velocity auto-correlation function is the diffusion tensor... 

The self-diffusion coefficient gives a metric quantifying the diffusion rate. It is often used as a way to compare the rate of phase space exploration between methods, and is typically calculated using the integral of the velocity auto-correlation function. However, in theory one can construct arbitrary methods to artificially scale the velocity auto-correlation function, hence giving inaccurate diffusion constants the momentum sampled may not be the momentum we use to propagate the position with. 

Very important differences emerge if we attempt to use the stochastic integrators to compute dynamics, e.g. a time-correlation function. Velocity auto-correlation functions are shown in Fig. 8.4 for various choices of the parameters. 

Figure 19. Velocity auto-correlation functions for the three intramolecular vibrations of water molecules at supercritical temperatures (a) 771 K, 1.284 g/cm (b) 673 K, 0.166 g/cm. Q, Q2, and Q3 denote the symmetric stretching, bending, and asyimnetric stretching modes, as illustrated on the inserts.

An accurate calculation of anharmonic infrared spectra is one goal to achieve, the assignment of the active bands into individual atomic displacements or vibrational modes is another. This issue is essential to the understanding of the underlying molecular structural and dynamical properties. In molecular dynamics simulations, interpretation of the infrared active bands into individual atomic displacements is traditionally and easily done using the vibrational density of states (VDOS) formalism. The VDOS is obtained through the Fourier transform of the atomic velocity auto-correlation function ... 

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