Real-time correlation functions

The real-time correlation function at this level of theory is seen to be the... 

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]. 

The operator representations in Eq. (2.50) or (2.62) are expressed at the level of second-order cumulant expansions. Although this approximation is an excellent one for imaginary-time calculations, real-time correlation functions are more sensitive to nonlinear interactions and hence less predictable in their behavior. In principle, however, the cumulant averages could be carried out to higher order. 

To begin, it is useful to revisit the effective harmonic result of Section III.A. In doing this it is informative to introduce a different effective harmonic real-time correlation function, given in terms of the centroid variable by [4, 8]... 

The cumulant expansion with CMD approach for general correlation functions can be extended to calculate correlation functions having momentum-dependent operators via the phase-space perspective of Paper III. Only the final expressions will be given here, so the reader is referred to the original paper for the details [5]. The approximate result for the real-time correlation function C g(t) in this approach is given by [5]... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Craig IR, Manolopoulos DE (2004a) Quantum statistics and classical mechanics Real time correlation functions from ring polymer molecular dynamics. J Chem Phys 121 3368... 

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