Generalized Langevin theory for many-body problems in chemical dynamics: General formulation and the equivalent harmonic chain representation

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Abstract
A comprehensive theoretical framework for classical trajectory simulations of many‐body chemical processes is presented. This framework generalizes previous theoretical methods designed to treat the many‐body problems arising in gas molecule collisions off perfect harmonic solids [S. A. Adelman and J. D. Doll, J. Chem. Phys. 64, 2374 (1976); S. A. Adelman and B. J. Garrison, J. Chem. Phys. 65, 3571 (1976)]. The present version of the theory is not restricted to the harmonic systems and thus provides a formal framework for treating liquid state as well as solid state chemical phenomena. Basic to the theory is the molecular time scale generalized Langevin equation (MTGLE), a formally exact representation of the dynamics of a chemical system coupled to an arbitrary heat bath in an arbitrary manner. The MTGLE is equivalent to but distinct from the conventional [Mori–Kubo] generalized Langevin representation. It, however, is more natural for chemical dynamics simulation work because in the MTGLE the apparent vibrational frequencies of the chemical system are those which govern the molecular time scale response of the system. In the conventional Langevin representation, the apparent vibrational frequencies are those which govern the static reponse of the chemical system. The MTGLE is next rigorously transformed into an effective equation of motion for a fictitious nearest neighbor collinear harmonic chain. The parameters characterizing the chain are calculable from the velocity autocorrelation function of the chemical system. The rigorous chain representation of many‐body dynamics: (i) provides a simple quasiphysical picture of energy transfer between the heat bath and chemical system during a chemical process; (ii) provides an equivalent Hamiltonian for the chemical system; (iii) permits one to construct a sequence of heat bath models whose response converges to that of the true heat bath in a rapid and systematic manner. The heat bath models are sets of collinear nearest neighbor harmonic chains composed of N=1,2,3,... fictitious atoms. Atom N in the chain is subject to time‐local functional damping and is driven by a white noise stochastic force. The remaining atoms 1,2,...N‐1 are undamped. The heat bath models (i) produce effective equations of motion for the chemical process which do not involve time nonlocal kernels and which may, hence, be readily solved by standard classical trajectory methods; (ii) allow one to rigorously generalize the Fokker–Planck theory of stochastic dynamics to non‐Markovian systems. Finally, the treatment of selected chemical processes via the theory is briefly outlined.