Real-time path integral calculations for the propagation of a system in contact with a harmonic dissipative environment often employ the iterative quasi-adiabatic propagator path integral (i-QuAPI) methodology. We compare two simple ways of applying this methodology to a bath initially in equilibrium with the localized state of the system (e.g., the donor in the case of charge transfer). The first way involves modifying the phase of the system via a time-local phase given in terms of integrals of the spectral density or in terms of the coefficients entering the QuAPI-discretized influence functional. In the iterative decomposition of the path integral, this approach requires consistent memory truncation to avoid extremely slow convergence. The second, alternative approach involves shifting the coordinate of the system, to bring the donor state in equilibrium with the bath, and requires no further modification of the i-QuAPI algorithm.
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