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We derive a semi-analytical form for the Wigner transform for the canonicaldensity operator of a discrete system coupled to a harmonic bath based on thepath integral expansion of the Boltzmann factor. The introduction of thissimple and controllable approach allows for the exact rendering of thecanonical distribution and permits systematic convergence of static propertieswith respect to the number of path integral steps. In additions, theexpressions derived here provide an exact and facile interface with quasi- andsemi-classical dynamical methods, which enables the direct calculation ofequilibrium time correlation functions within a wide array of approaches. Wedemonstrate that the present method represents a practical path for thecalculation of thermodynamic data for the spin-boson and related systems. Weillustrate the power of the present approach by detailing the improvement ofthe quality of Ehrenfest theory for the correlation function$\mathcal{C}_{zz}(t) = \mathrm{Re}\langle \sigma_z(0)\sigma_z(t)\rangle$ forthe spin-boson model with systematic convergence to the exact samplingfunction. Importantly, the numerically exact nature of the scheme presentedhere and its compatibility with semiclassical methods allows for the systematictesting of commonly used approximations for the Wigner-transformed canonicaldensity.

Path integral approach to the Wigner representation of canonical density operators for discrete systems coupled to harmonic baths