One of the major outstanding challenges in computational chemistry is the calculation of thermal quantum time correlation functions in condensed phases. Of the methods most commonly employed, semi-classical approaches generally require the generation of a very large number of trajectories with a high associated computational overhead. Alternatively, the popular centroid and ring-polymer MD algorithms are accurate for linear operators but degrade for non-linear operators, and routes to systematic improvement of these methods are not obvious. In this talk, I will introduce a formally exact formulation of quantum time correlation functions in terms of open-chain Feynman path integrals that can be sampled using a Monte Carlo or molecular dynamics algorithm. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, a change of integration variables, often used in the derivation of trajectory-based semiclassical methods, is introduced. In particular, a transformation to sum and difference variables between forward and backward complex-time propagation paths. It can be shown that a formal integration over the path-difference variables yields a function of the path-sum variables that can be shown to be positive definite, thereby allowing the problem to be formulated as a sampling problem in the path-sum variables. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. This approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Practical approximation schemes are considered for implementing the approach, and the scheme is compared to the ring-polymer MD and thermal-Gaussian LSC-IVR approaches for a handful of example problems. Other formal considerations for problems such as rate theory calculations and electronic excitation spectroscopy are also discussed.
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