Quantum Dissipative Dynamics of Electron Transfer Reaction System: Nonperturbative Hierarchy Equations Approach

Abstract

A multistate displaced oscillator system strongly coupled to a heat bath is considered a model of an electron transfer (ET) reaction system. By performing canonical transformation, the model can be reduced to the multistate system coupled to the Brownian heat bath defined by a non-ohmic spectral distribution. For this system, we have derived the hierarchy equations of motion for a reduced density operator that can deal with any strength of the system bath coupling at any temperature. The present formalism is an extension of the hierarchy formalism for a two-state ET system introduced by Tanimura and Mukamel into a low temperature and multistate system. Its ability to handle a multistate system allows us to study a variety of problems in ET and nonlinear optical spectroscopy. To demonstrate the formalism, the time-dependent ET reaction rates for a three-state system are calculated for different energy gaps. Electron transfer (ET) processes play an important role in many fields in physics, chemistry, and biology.1) Most ET processes occur in condensed phases where the surrounding molecules provide the energetic fluctuations and dissipations needed in the reactions. In the case of ET in a polar solvent, the interaction energy that consists of the solvent-solute interaction energy and solvent dipole-dipole energy is used as the ET reaction coordinate.2-4) The ET reaction is then characterized by parabolic free-energy surfaces for the reactant and product states as a function of the reaction coordinate. Although ET rates are commonly studied using free-energy surfaces, the ultrafast dynamics of ET processes is investigated using the potential energy surfaces as a function of molecular coordinates.5,6) This potential energy description is similar to that used in nonlinear ultrafast spectroscopy. 7,8) By adopting this description, one may study ET processes by nonlinear optical measurements. 9,10) A variety of approaches that have been developed to study not only ET processes but also nonlinear optical responses are used to investigate ET dynamics. [5][6][7][8][9][10][11][12][13][14] Unlike most of the above-mentioned approaches based on the perturbative treatment of nonadiabatic transitions, the reduced equation of motion approach that describes the dynamics of density matrix of an ET system coupled to the environment can handle any strength of nonadiabatic coupling. This approach is successful in a classical case characterized by free-energy surfaces, 5,6,15) but it has to employ crucial assumptions such as rotating wave approximation and perturbative system-bath interaction in a quantum case. This strongly limits its applicability. Meanwhile, it was found that the hierarchy equation approach is a powerful means of describing a system strongly coupled to a bath without using rotating wave approximation and white-noise (Born or Van Hove) approximation. [16][17][18][19][20][21] This approach was introduced to investigate the connection between the phenomenological stochastic Liouville equation theory and dynamical Hami...

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

“…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.…”

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

“…15,26) Then the 0th element is identical to^ð 0Þ 0;0;...;0 ðtÞ ¼ Tr B f^t ot ðtÞg. The HEOM has been used to study chemical reactions, 12,27) linear and nonlinear spectroscopy, 28,29) exciton and electron transfer, [30][31][32][33][34] and quantum information. 35,36) The HEOM is ideal for studying quantum transport systems when we employ the Wigner representation, because it allows us to treat continuous systems utilizing open boundary conditions and periodic boundary conditions.…”

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.…”

“…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, …”

“…[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.…”