Centroid density

The quantity is the Feynman path integral centroid density [43] that is understood to be expressed asymptotically as... 

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. 

To show that this guess is actually consistent with the Im F approach and to see what happens to the velocity factor at low temperatures let us study the statistics of centroids. We introduce the centroid density... 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

Cao, J. Voth, G.A., The formulation of quantum statistical mechanics based on Feynman path centroid density, J. Chem. Phys. 1994,100, 5093-5105... 

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., ... 

Aeeording to Eq. (10), (x 0(Xc,Pc) x") is aphase spaee path integral representation for the operator 27t/iexp —pA, where all the paths run from x to x", but their eentroids are eonstrained to the values of Xc and po. Integration over the diagonal element, whieh corresponds to the trace operation, leads to the usual definition of the phase space centroid density multiplied by 2nH. In this review and in Refs. 9,10 this multiplicative factor is included in the definition of the centroid distribution function, pc (xc, pc). Equation (6) thus becomes equivalent to... 

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

The quantum partition function in Eq. (1.1) is then obtained by the integration of the centroid density over all possible configurations of the... 

It should be noted that the centroid density is distinctly different from the coordinate (or particle) density p( ) = ((jlexp(-)3//) (5f). The particle density function is the diagonal element of the equilibrium density matrix in the coordinate representation, while the centroid density does not have a similar physical interpretation. However, the integration over either density yields the quantum partition function. 

The broad topics of this article are organized into sections as follows In Section II the centroid density-based formulation for calculating equilibrium properties in quantum mechanical systems is described. Then... 

One conclusion that can be reached from the early work on effective potentials [1,21-23], the work of Cao and Voth [3-8], as well as the centroid density-based formulation of quantum transition-state theory [42-44,49] is that the path centroid is a particularly useful variable in statistical mechanics about which to develop approximate, but quite accurate, quantum mechanical expressions and to probe the quantum-classical correspondence principle. It is in this spirit that a general centroid density-based formulation of quantum Boltzmann statistical mechanics is presented in the present section. This topic is the subject of Paper I, and the emphasis in this section is on analytic theory as opposed to computational approaches (cf. Sections III and IV). 

The correlation function in Eq. (2.1) differs from the usual Euclidean time position correlation function C(t) because only paths with centroids at q contribute to the centroid-constrained propagator C (t, q ). However, one can obtain C(t) by averaging the centroid-constrained propagator over the normalized centroid density pXqc) that is. 

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... 

C (t, qj is through functional differentiation of a generating functional based on the centroid density [3]. 

With the quadratic reference system in hand, the centroid density can be expressed as... 

The first step in the formulation of the complete diagrammatic theory for the centroid density is to Taylor expand the average in Eq. (2.7), that... 

The diagrammatic representation of the centroid density enhances one s ability to approximately evaluate the full perturbation series [3]. For example, one can focus on a class of diagrams with the same topological characteristics. The sum of such a class results in a compact analytical expression that includes infinite terms in the summation. A very useful technique in such cases is the renormalization of diagrams [57,58]. This procedure can be applied to the vertices to define the effective potential theory diagrammatically [3, 21-23] and, in doing so, an accurate approximation to the centroid density [3]. 

It should be noted that these equations are to be solved for each position of the centroid q. The frequency in Eq. (2.27) is the same as the effective frequency obtained for the optimized LHO reference system using the path-integral centroid density version of the Gibbs-Bogoliubov variational method [1, pp. 303-307 2, pp. 86-96], Correspondingly, Eqs. (2.27) and (2.28) are exactly the same as those in the quadratic effective potential theory [1,21-23], The derivation above does not make use of the variational principle but, instead, is the result of the vertex renormalization procedure. The diagrammatic analysis thus provides a method of systematic identification and evaluation of the corrections to the variational theory [3],... 

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