Centroid methods time correlation functions

In Paper IV, the self-diffusion process in fluid neon was also studied with CMD using the pairwise pseudopotential method. In Fig. 17 the centroid velocity time correlation function is plotted for quantum neon using the pseudopotential method and for classical neon. When the quantum mechanical nature of the Ne atoms is taken into account, the diffusion constant is reduced by a small fraction. In the gas phase and to some degree in liquids, the diffusion process can be viewed as a sequence of two-body collisions, the frequency of which depends on the collision cross section. Because the quantum centroid cross section is larger than the corresponding classical value, the quantum diffusion constant is found... 

General Time Correlation Functions Centroid Molecular Dynamics Method... 

It should be noted that the imaginary time correlation function in Eq. (2.4) provides a measure of the localization of quantum particles in condensed media [17-19,52]. From this point on, the notation denotes an averaging by integrating some centroid-dependent function over the centroid position q weighted by the normalized centroid density p q ) Z. An alternative method for defining the correlation function... 

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

The CMD method is based on the propagation of quasiclassical centroid trajectories q t) derived from the mean force on the centroid as a function of position [cf. Eqs. (3.11) and (3.12)]. This method, combined with Eq. (3.13), generally provides an accurate representation of the exact quantum real-time position or velocity correlation functions. There is a significant theoretical question, however, on how CMD might be used to compute time correlation functions of the form (A(f)5(0)), where A and B are general quantum operators [4,5]. Two approaches to this problem will now be described. 

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