The Linear Response Approximation and its Lowest Order Corrections
The mapping of large-amplitude translation or intramolecular motion onto effective harmonic bath models is often deemed to be a reasonable approximation which can correctly capture qualitative behaviors. This mapping, which forms the basis or Marcus’ electron transfer theory, is understood to be valid only for the purpose of extracting the dynamics of observables pertaining to the subsystem of interest and implies Gaussian behavior of the medium. It is not clear a priori how good this approximation is for a given system, although this question can in principle be addressed within the classical mechanical framework. Most importantly, no methodology is available for estimating corrections to the linear response approximation in a systematic fashion.
Recent work employed an influence functional approach to analyze and quantify the validity of linear response theory for processes in complex environments. A cumulant expansion of the influence functional was obtained, in which the nth term is given by an n-dimensional time-integral involving the n-time correlation function of the environment. The lowest order (quadratic) term in this expansion maps the environment onto an effective harmonic medium and corresponds to the linear response approximation. It was shown that this effective harmonic bath model is exact for arbitrary values of the overall coupling strength if the system-bath coupling is diluted over an infinite number of degrees of freedom. Numerical calculations on a bath of two-level systems were performed which provided a quantitative feel for the magnitude of the deviations from the above behavior. It was found that the errors incurred by the linear response approximation are considerably more significant at low temperatures where quantum mechanical effects are more important. Finally, a systematic way of correcting the linear response model by including low-order anharmonic contributions from classical trajectory information was proposed. It is hoped that the ability to estimate and systematically evaluate anharmonic corrections in the influence functional will increase the utility of harmonic bath models in chemical dynamics.
Further, the cumulant expansion of the influence functional, where all terms are related to multi-time correlation functions of the environment (which are subject to decoherence that truncates the extent of nonlocality), suggests a decomposition of the path integral that leads to a general iterative methodology similar in sprit to our path integral methodology for a system coupled to a harmonic bath. In the general case of an anharmonic environment the influence functional is not known analytically but must be evaluated numerically for each given pair of forward and backward system path segments. The forward-backward approximation to the influence functional provides a convenient approximation for this purpose.
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