Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach

Abstract

Multiple displaced oscillators coupled to an Ohmic heat bath are used to describe electron transfer (ET) in a dissipative environment. By performing a canonical transformation, the model is reduced to a multilevel system coupled to a heat bath with the Brownian spectral distribution. A reduced hierarchy equations of motion approach is introduced for numerically rigorous simulation of the dynamics of the three-level system with various oscillator configurations, for different nonadiabatic coupling strengths and damping rates, and at different temperatures. The time evolution of the reduced density matrix elements illustrates the interplay of coherences between the electronic and vibrational states. The ET reaction rates, defined as a flux-flux correlation function, are calculated using the linear response of the system to an external perturbation as a function of activation energy. The results exhibit an asymmetric inverted parabolic profile in a small activation regime due to the presence of the intermediate state between the reactant and product states and a slowly decaying profile in a large activation energy regime, which arises from the quantum coherent transitions.

Energy transfer in structured and unstructured environments: Master equations beyond the Born-Markov approximations

“…We shall show that over the broad regimes considered, the RC model captures all important system-environment correlations for EET in the presence of both underdamped and overdamped environments, agreeing perfectly with numerically exact data generated using the HEOM. [42][43][44][45] Furthermore, we demonstrate that the RC model significantly outperforms the closely related Zusman equations, 70,73,77 a set of drift-diffusion equations often used to describe tunnelling processes in molecular systems, which we derive from the RCME in a semiclassical limit. We also examine the role that a structured environment may play in the dimer energy transfer dynamics.…”

“…25,37,39,40 A multitude of powerful computational methods have been developed to deal with the difficulties faced in modelling strongly dissipative quantum systems. Examples include the hierarchical equations of motion (HEOM), [41][42][43][44][45][46] density matrix renormalisation group (and related) techniques, 25,36,47,48 and those based on the path integral formalism. [49][50][51][52] All can converge to numerically exact results in specific circumstances.…”

Reduced hierarchy equations of motion approach with Drude plus Brownian spectral distribution: Probing electron transfer processes by means of two-dimensional correlation spectroscopy

“…63,64 The present approach can also be applied to a system driven by pulses of arbitrary number, shape, and strength, as well as a system with time-dependent ET couplings. 62 The present formulation can also be extended to multimode Brownian oscillator systems by introducing a higher dimensional hierarchy.…”

“…[80][81][82][83] While most research with the HEOM approach assumed the Drude spectral distribution, we have shown that the ET problem can be handled in a nonperturbative manner for both the system-bath and ET couplings by applying the hierarchy formalism to the Brownian oscillator (BO) spectral distribution that arises from the canonical transformation of ET system. [62][63][64] Because realistic environments in many cases involve both the overdamped Drude and underdamped Brownian modes as shown by molecular dynamics simulations, [84][85][86][87][88] an extension to the multimode case is necessary. In this paper, we demonstrate a way to deal with the Drude+BO spectral distribution in the framework of HEOM formalism.…”

“…Thanks to the truncation schemes for higher-order hierarchy elements [44][45][46][47] one can numerically integrate HEOM for variety of systems expressed as Wigner distributions [48][49][50][51][52] and energy eigen states [53][54][55][56][57] as well as when a time-dependent external perturbation such as laser 49 or magnetic excitation 58 is present. By generalizing hierarchy structures, one can deal with a low temperature system 45,59,60 as well as general spectral distributions 61 including Brownian spectral distributions [62][63][64][65] and a Lorentzian distribution. 66,67 This formalism is valuable since it can handle not only strong system-bath coupling, but also quantum coherence between the system and bath, which plays important roles in multidimensional spectroscopy, [50][51][52][53][54][55][67][68][69] energy transfer processes in photosynthetic antenna systems [70][71][72][73][74][75][76][77] and DNA systems, …”