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Chemical Physics

Path-integral treatment of the real-time dynamics of few-mode spin-boson models

H. -J. Plöhn, Stefan Krempl, Manfred Winterstetter, Wolfgang Domcke

Published Nov 1, 1995• 8 citations
Publisher
DOI
10.1016/0301-0104(95)00232-d

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Quantum mechanical effects play an important role in dynamics of condensed phases. It is also well known that the difficulty of solving the full Schrödinger equation grows exponentially with time. One approach to cope up with this difficulty is to restrict quantum treatment to a few particles (system) and treat the dynamics of the rest of particles (bath) using classical trajectories. Path integral methods due to their trajectory like nature provide excellent tool to develop quantum-classical methods, but to capture long time dynamics, the number of classical trajectories grow exponentially. A part of this work focuses on improving Quantum-Classical Path Integral (QCPI) treatments that allow for larger path integral time step by building “smarter” propagators. These improved solvent-driven reference propagators is developed by incorporating physically motivated approximations to the solvent. When used in QCPI expression they allow convergence with larger time steps leading to an exponential reduction of the number of trajectories required. Further advantages of these propagators include improvements in path filtering techniques, smaller number of path integral steps for achieving the memory decoherence time, and smoothing of the integrand which leads to convergence with fewer Monte Carlo sample points. These ideas have been validated on the spin boson model - a prototypical model to study condensed phase dynamics, which consists of a two-level system coupled to a harmonic bath. The new approach of building the reference propagators combined with their iterative evaluation and filtering is validated using parameters that mimic the first electron transfer in wild-type photosynthetic reaction centres. A real-world dynamical simulation contains various anharmonic effects, Ferrocene (donor) - Ferrocenium (acceptor) system in liquid Benzene was studied where the anharmonic effects of the bath of liquid Benzene on the dynamics of the system were treated using Ensemble Averaged Classical Path. The process of electron or proton transfer occurs between two species having different charges and fluctuations in the solvent plays a key role in determining the rates of such reaction dynamics. To correctly account for the effects of initial distribution of the solvent on the dynamics of the quantum system, the concept of equilibrating the bath to the donor state of the system is used. Two different approaches have been used within the influence functional framework.

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Quantum mechanical effects play an important role in dynamics of condensed phases. It is also well known that the difficulty of solving the full Schrödinger equation grows exponentially with time. One approach to cope up with this difficulty is to restrict quantum treatment to a few particles (system) and treat the dynamics of the rest of particles (bath) using classical trajectories. Path integral methods due to their trajectory like nature provide excellent tool to develop quantum-classical methods, but to capture long time dynamics, the number of classical trajectories grow exponentially. A part of this work focuses on improving Quantum-Classical Path Integral (QCPI) treatments that allow for larger path integral time step by building “smarter” propagators. These improved solvent-driven reference propagators is developed by incorporating physically motivated approximations to the solvent. When used in QCPI expression they allow convergence with larger time steps leading to an exponential reduction of the number of trajectories required. Further advantages of these propagators include improvements in path filtering techniques, smaller number of path integral steps for achieving the memory decoherence time, and smoothing of the integrand which leads to convergence with fewer Monte Carlo sample points. These ideas have been validated on the spin boson model - a prototypical model to study condensed phase dynamics, which consists of a two-level system coupled to a harmonic bath. The new approach of building the reference propagators combined with their iterative evaluation and filtering is validated using parameters that mimic the first electron transfer in wild-type photosynthetic reaction centres. A real-world dynamical simulation contains various anharmonic effects, Ferrocene (donor) - Ferrocenium (acceptor) system in liquid Benzene was studied where the anharmonic effects of the bath of liquid Benzene on the dynamics of the system were treated using Ensemble Averaged Classical Path. The process of electron or proton transfer occurs between two species having different charges and fluctuations in the solvent plays a key role in determining the rates of such reaction dynamics. To correctly account for the effects of initial distribution of the solvent on the dynamics of the quantum system, the concept of equilibrating the bath to the donor state of the system is used. Two different approaches have been used within the influence functional framework.
Strong coupling between a system and its environment leads to the emergence of non-Markovian dynamics, which cannot be described by a time-local master equation. One way to capture such dynamics is to use numerical real-time path integrals, where assuming a finite bath memory time enables manageable simulation scaling. However, by comparing to the exactly soluble independent boson model, we show that the presence of transient negative decay rates in the exact dynamics can result in simulations with unphysical exponential growth of density matrix elements when the finite memory approximation is used. We therefore reformulate this approximation in such a way that the exact dynamics are reproduced identically and then apply our new method to the spin-boson model with superohmic environmental coupling, commonly used to model phonon environments, but which cannot be solved exactly. Our new method allows us to easily access parameter regimes where we find revivals in population dynamics which are due to non-Markovian backflow of information from the bath to the system.
Strong coupling between a system and its environment leads to the emergence of non-Markovian dynamics, which cannot be described by a time-local master equation. One way to capture such dynamics is to use numerical real-time path integrals, where assuming a finite bath memory time enables manageable simulation scaling. However, by comparing to the exactly soluble independent boson model, we show that the presence of transient negative decay rates in the exact dynamics can result in simulations with unphysical exponential growth of density matrix elements when the finite memory approximation is used. We therefore reformulate this approximation in such a way that the exact dynamics are reproduced identically and then apply our new method to the spin-boson model with superohmic environmental coupling, commonly used to model phonon environments, but which cannot be solved exactly. Our new method allows us to easily access parameter regimes where we find revivals in population dynamics which are due to non-Markovian backflow of information from the bath to the system.