Momentum correlation function, centroid

The momentum correlation function can be calculated with CMD in two ways. The first method is to compute the centroid position correlation function and to use the usual Fourier relationships between position and velocty correlation functions to obtain the momentum correlation function. The second route is to calculate the centroid momentum correlation function and then use its Fourier relation with the momentum correlation function directly (see Ref. 5). It can be shown that the two methods are equivalent, although the latter is numericaly preferable because the former has a tendency to amplify the high-frequency noise in the transform. 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

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. 

Returning to the first term on the right-hand side of Eq. (3.31), the quantum fluctuations in momentum contribute a further deviation from the similar term in the centroid correlation function [Eq. (3.28)]. The difference between the two terms for all powers of n is found to be [5]... 

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