The centroid distribution function

For a classical system at equihbrium, the canonical partition function is written 

The quantum version of the partition fimction is obtained by replacing the phase space integral and the classical Boltzmann distribution with the trace operation of the quantum Boltzmann operator, giving the usual expression 

This expression contains all the equilibrium information for the quantum ensemble as is in the classical case. 

One possible definition of a classical-like quantum density is given by 

For example, the classical-like phase space trace of this distribution function over the scalars x and p gives the quantum partition function in Eq. (5). However, in 

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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... 

For an arbitrary canonical density operator, the phase space centroid distribution fimction is imiquely defined. However, this function does not directly contain any dynamical information from the quantum ensemble because such information has been lost in the course of the trace operation. The lost information may be recovered by associating to each value of the centroid distribution function the following normalized operator ... 

The centroid distribution function and the effective potential for the CMD simulation can be obtained through the path integral simulation method, - but... 

The reactive flux method is also useful in calculating rate constants in quantum systems. The path integral formulation of the reactive flux together with the use of the centroid distribution function has proved very useful for the calculation of quantum transition-state rate constants [7]. In addition new methods, such as the Meyer-Miller method [8] for semiclassical dynamics, have been used to calculate the flux-flux correlation function and the reactive flux. 

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