Gaussian approximation dynamics

Gaussian approximation, 61 Monte Carlo simulations, 67-81 dynamics, 75-81 Metropolis algorithms, 70-71 nonequilibrium molecular dynamics, 71-74 structure profiles, 74-75 system details, 67-70... 

Free and Lombardi (FL) models, Renner-Teller effect, triatomic molecules, 618-621 Free electrons, electron nuclear dynamics (END), time-dependent variational principle (TDVP), 333-334 Frozen Gaussian approximation direct molecular dynamics ... 

Independent Gaussian approximation (IGA), direct molecular dynamics, Gaussian wavepacket propagation, 379-383... 

In the Gaussian approximation (Eq. 4.12) the mean squared displacement is given by (r (t))=3/[2a(t)], and a2(t) is zero of course. In the light of the above results obtained by neutron scattering (summarized in Eq. 4.14), the values of the non-Gaussian parameter for this process should be very small. However, this result is in apparent contradiction to recent molecular dynamics (MD)... 

On Laplace inversion and then inserting the rate kernel into the Noyes expression for the rate coefficient [eqn. (191)], the rate coefficient is seen to be exactly that of the Collins and Kimball [4] analysis [eqn. (25)]. It is a considerable achievement. What is apparent is the relative ease of incorporating the dynamics of the hard sphere motion. The competitive effect comes through naturally and only the detailed static structure of the solvent is more difficult to incorporate. Using the more sophisticated Gaussian approximation to the reactant propagators, eqn. (304), Pagistas and Kapral calculated the rate kernel for the reversible reaction [37]. These have already been shown in Fig. 40 (p. 219) and are discussed in the next section. 

At this time the only experimental method available for determining to what extent the Gaussian approximation is realistic is molecular dynamics studies of polyatomic liquids such as the ones we have been discussing. We have therefore tested this approximation by... 

The results of these computations are presented in Figures 14, 15, 16, 17, and 18. These first few calculated moments indicate that the Gaussian transition probabilities for the linear and angular momentum may represent the dynamics fairly well, However, it may not yet be concluded that the Gaussian approximation is actually correct, since this same test must... 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

In order to have a closer look at the observed phenomena we derive the moment dynamics for the system 1.44 in Gaussian approximation ... 

Equation (18) is a very simple example of how the very short time dynamics determines the low-resolution (i.e., the envelope) spectrum. The details of the intensities of individual transitions require a longer time propagation and result in the resolved spectrum shown in Fig. 7. The envelope is however given by Eq. (18) and what the longer time dynamics reveals are the details which make up this envelope. A subsidiary lesson is that a broad envelope is not necessarily a signature of IVR. In the present example, and this carries over to the polyatomic case as well, the broad envelope is determined by inertia i.e., by the acceleration of the position of the oscillator. (To quantitatively show this, take a in Eq. (13) to be a function of time. It is the first deviation of x(t) from its initial value that gives rise to the Gaussian approximation. We reiterate that this can be shown in the multidimensional case as well (53).)... 

The time-dependent mean square displacement can be obtained from the incoherent scattering measurements. It is also pointed out that the frequency-dependent mean square displacement can be obtained from the incoherent dynamic scattering law within the Gaussian approximation. 

Brodeck et al., 2010). In the Gaussian approximation for displacements, the segmental self-correlation function relates directly to (r (r)), resulting in the intermediate self-dynamic structure factor having the Gaussian form (Niedzwiedz et al., 2007),... 

This expression can be regarded as the generalization of Eq. (5.159) so as to exhibit the proper short-time dynamics. For an adequate description of the short-time dynamics of the solute, one also has to generalize Fu k,t) given in Eq. (5.162) which is valid only in the long-time regime. For this purpose, we adopt the Gaussian approximation for Fu k,t),... 

The semiclassical approach to the problem of atom-crystal inelastic scattering is very attractive due to its relative simplicity, analytical nature and wide applicability. This approach allows one to obtain a simple Gaussian approximation (Brako and Newns 1982 Manson 1991) to the dynamic structural feictor of inelastic phonon scattering and the intensities of diffraction peaks (Billing 1975). The effect of umklapp processes on the dynamic structural factor hcis been considered only in the hard-wall approximation (Berry 1975 Bogdanov 1980) or numerically (Manson 1991). 

The Gaussian approximation can therefore be seen to be equivalent to a quadratic potential or linear elastic restoring force. Deviations from the Gaussian distribution will correspondingly yield nonlinear force terms in the dynamics. The Gaussian approximation should therefore be an appropriate simplification for describing systems close to equilibrium or at most linearly displaced from the equilibrium state. 

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