Centroid density formalism

The centroid density formally defines a classical-like effective potential that is [1, 3, 21-23]... 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

Feynman noted that the quantum mechanical centroid density, Pc(xc), can be defined for the path centroid variable which is the path integral over all paths having their centroids fixed at the point in space Xc. Specifically, the formal imaginary time path integral expression for the centroid density is given by... 

In this equation, Dx(x) and S [x(x)] are, respectively, the position space path measure and the Euclidean time action. The centroid density also formally defines a classical-like effective potential, i.e., ... 

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]... 

One significant feature of Eq. (3.2) is the factorization of the expression into the centroid density (i.e., the centroid statistical distribution) and the dynamical part, which depends on the centroid frequency u>. It is not obvious that such a factorization should occur in general. For example, a rather different factorization occurs when the conventional formalism for computing time correlation functions is used [i.e., a double integration in terms of the off-diagonal elements of the thermal density matrix and the Heisenberg operator q t) is obtained]. This result sheds light on the dynamieal role of the centroid variable in real-time correlation functions (cf. Section III.B) [4,8]. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

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