Assessing the Quality of Approximate Quantum Dynamics in Condensed Phase via Sum Rules

In this work, we discuss a general protocol for analyzing the quality of approximate quantum time correlation functions of non-trivial systems in many dimensions. This approach is based on the generalized deconvolution of the Kubo transformed quantum time correlation function onto an ensemble averaged quantum correlation function, at a given value τ in imaginary time (such that 0≤ τ ≤ β), which leads to a series of sum rules linking derivatives of different order in the corresponding pair of convoluted correlation functions. We focus on the case when τ = β/2 for which all deconvolution kernels become real valued functions and their asymptotic behavior at long times exhibit a polynomial divergence. It is then shown that thermally symmetrized static averages, and the static averages of the corresponding time derivatives, are ideally suited to investigate the quality of approximate quantum time correlation functions at successively larger, and up to arbitrarily long, times. This overall strategy is illustrated analytically for a harmonic system, and numerically for a multidimensional double-well potential and a Lennard-Jones fluid representing liquid neon at 30 K.