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The reduced density matrix of excitons coupled to a phonon bath at a finitetemperature is studied using the path integral Monte Carlo method. Appropriatechoices of estimators and importance sampling schemes are crucial to theperformance of the Monte Carlo simulation. We show that by choosing thepopulation-normalized estimator for the reduced density matrix, an efficientand physically-meaningful sampling function can be obtained. In addition, thenonadiabatic phonon probability density is obtained as a byproduct during thesampling procedure. For importance sampling, we adopted the Metropolis-adjustedLangevin algorithm. The analytic expression for the gradient of the targetprobability density function associated with the population-normalizedestimator cannot be obtained in closed form without a matrix power series. Anapproximated gradient that can be efficiently calculated is explored to achievebetter computational scaling and efficiency. Application to a simpleone-dimensional model system from the previous literature confirms thecorrectness of the method developed in this manuscript. The displaced harmonicmodel system within the single exciton manifold shows the numerically exacttemperature dependence of the coherence and population of the excitonic system.The sampling scheme can be applied to an arbitrary anharmonic environment, suchas multichromophoric systems embedded in the protein complex. The result ofthis study is expected to stimulate further development of real timepropagation methods that satisfy the detailed balance condition for excitonpopulations.

Path integral Monte Carlo with importance sampling for excitons interacting with an arbitrary phonon bath