Understanding the free energy barrier and multiple timescale dynamics of charge separation in organic photovoltaic cells

Hierarchical Equations of Motion for Quantum Chemical Dynamics: Recent Methodology Developments and Applications

ConspectusQuantum effects are critical to understanding many chemical dynamical processes in condensed phases, where interactions between molecules and their environment are usually strong and non-Markovian. In this Account, we review recent progress from our group in development and application of the hierarchical equations of motion (HEOM) method, highlighting its ability to address some challenging problems in quantum chemical dynamics.In the HEOM method, the bath degrees of freedom are represented using effective modes from exponential decomposition of the bath correlation function. Complex spectral densities and low temperature simulations often require a larger number of modes, making the simulations very expensive. Recent advances, such as the barycentric spectral decomposition (BSD) technique, can significantly reduce the number of effective modes, allowing to handle complex spectral densities and enabling simulations at very low temperatures, including near-zero temperature dynamics.Another key improvement in the computational efficiency is the use of tensor network methods like matrix product states and hierarchical tensor networks. These techniques allow for efficient HEOM propagation with thousands of effective modes, crucial for simulating large molecular systems interacting with multiple baths. This combination enables simulations of excitation energy transfer (EET) in systems like the Fenna-Matthews-Olson (FMO) complex and even larger systems with experimentally determined spectral densities.The versatility of the HEOM method is demonstrated through applications to a wide range of chemical dynamics problems. Simulations of EET and related ultrafast spectroscopy are first briefly covered. Applications of the HEOM to quantum tunneling effects in proton transfer reactions are then presented. Early works have studied the non-Kramers dependence of the rate constant as a function of bath friction due to deep tunneling and revealed vibrationally nonadiabatic dynamics within the so-called nontraditional view of proton transfer reactions. A recent work on the large kinetic isotope effects in soybean lipoxygenase also indicated that many quantum correction approximations to classical transition-state theory may fall short in describing deep tunneling effects.Charge transport and separation dynamics in organic semiconductors are another area where the HEOM method has been instrumental. We first demonstrate that the HEOM provides a unified description of both band-like and thermally assisted charge carrier transport in organic materials. The effect of non-nearest neighbor transitions is then investigated by combining generalized master equations with exact memory kernels. The HEOM method also enables simulation of charge separation in organic photovoltaics (OPVs) and reveals how factors such as external electric fields, entropy, and charge delocalization influence the charge separation barrier and dynamics.Moreover, HEOM has been applied to investigate hydrogen atom scattering on the Au(111) surface and vibrational energy relaxation at molecule-metal interfaces. These studies provide deeper insights into how electron-hole pair excitations and temporary charge transfer states influence the nuclear motion, offering a new framework for simulating nonadiabatic dynamics on metal surfaces.In summary, the HEOM method has developed into a robust tool for simulating quantum effects in condensed phases. Future developments in algorithm efficiency and computational power will likely expand its applicability to even more complex systems.

Mori generalized master equations offer an efficient route to predict and interpret polaron transport

Predicting how a material's microscopic structure and dynamics determine its transport properties remains a fundamental challenge. To alleviate this task's often prohibitive computational expense, we propose a Mori-based generalized quantum master equation (GQME) to predict the frequency-resolved conductivity of small-polaron forming systems described by the dispersive Holstein model. Unlike previous GQME-based approaches to transport that scale with the system size and only give access to the DC conductivity, our method requires only one calculation and yields both the DC and AC mobilities. We further show how to easily augment our GQME with numerically accessible derivatives of the current to increase computational efficiency, collectively offering computational cost reductions of up to 90%, depending on the transport regime. Finally, we leverage our exact simulations to demonstrate the limited applicability of the celebrated and widely invoked Drude-Smith model in small-polaron forming systems. We instead introduce a cumulant-based analysis of experimentally accessible frequency data to infer the microscopic Hamiltonian parameters. This approach promises to provide valuable insights into material properties and facilitate guided design by linking macroscopic terahertz measurements to the microscopic details of small polaron-forming systems.

Even a little delocalization produces large kinetic enhancements of charge-separation efficiency in organic photovoltaics

In organic photovoltaics, charges can separate efficiently even if their Coulomb attraction is an order of magnitude greater than the available thermal energy. Delocalization has been suggested to explain this fact, because it could increase the initial separation of charges in the charge-transfer (CT) state, reducing their attraction. However, understanding the mechanism requires a kinetic model of delocalized charge separation, which has proven difficult because it involves tracking the correlated quantum-mechanical motion of the electron and the hole in large simulation boxes required for disordered materials. Here, we report the first three-dimensional simulations of charge-separation dynamics in the presence of disorder, delocalization, and polaron formation, finding that even slight delocalization, across less than two molecules, can substantially enhance the charge-separation efficiency, even starting with thermalized CT states. Delocalization does not enhance efficiency by reducing the Coulomb attraction; instead, the enhancement is a kinetic effect produced by the increased overlap of electronic states.

The role of entropy gains in the exciton separation in organic solar cells

In organic solar cell (OSC), the lower dielectric constant of organic semiconductor material induces a strong Coulomb attraction between electron-hole pairs, which leads to a low exciton separation efficiency, especially the charge transfer (CT) state. The CT state formed at the electron-donor (D) and electron-acceptor (A) interface is regarded as an unfavorable property of organic photovoltaic devices. Since the OSC works in a nonzero temperature condition, the entropy effect would be one of the main reasons to overcome the Coulomb energy barrier and must be taken into account. In this review, we review the present understanding of the entropy-driven charge separation and describe how factors such as the dimensionality of the organic semiconductor, energy disorder effect, the morphology of the active layer, and the nonequilibrium effect affect the entropy contribution in compensating the Coulomb dissociation barrier for CT exciton separation and charge generation process. We focus on the investigation of the entropy effect on exciton dissociation mechanism from both theoretical and experimental aspects, which provides pathways for understanding the underlying mechanisms of exciton separation and further enhancing the efficiency of OSCs. This article is protected by copyright. All rights reserved.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 10⁴ effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

Vibronic coherence contributes to photocurrent generation in organic semiconductor heterojunction diodes

Charge separation dynamics after the absorption of a photon is a fundamental process relevant both for photosynthetic reaction centers and artificial solar conversion devices. It has been proposed that quantum coherence plays a role in the formation of charge carriers in organic photovoltaics, but experimental proofs have been lacking. Here we report experimental evidence of coherence in the charge separation process in organic donor/acceptor heterojunctions, in the form of low frequency oscillatory signature in the kinetics of the transient absorption and nonlinear two-dimensional photocurrent spectroscopy. The coherence plays a decisive role in the initial ~200 femtoseconds as we observe distinct experimental signatures of coherent photocurrent generation. This coherent process breaks the energy barrier limitation for charge formation, thus competing with excitation energy transfer. The physics may inspire the design of new photovoltaic materials with high device performance, which explore the quantum effects in the next-generation optoelectronic applications.

Although coherent vibrational motion in donor-acceptor blends may contribute to photogeneration generation in organic solar cells (OSCs), proof of a direct correlation is still lacking. Here, the authors report the role of vibrational coherence on photocurrent generation in ternary OSC blends.

Exciton dissociation and charge separation at donor–acceptor interfaces from quantum-classical dynamics simulations

Nonadiabatic dynamics simulations based on the quantum-classical Liouville equation are employed to study the real-time dynamics of exciton dissociation and charge separation at a model donor–acceptor interface.

Coupled excitation energy and charge transfer dynamics in reaction centre inspired model systems

Functional operation conditions of reaction centre core complexes require the tight coupling of exciton states to concomitant charge separation. Rigorous theoretical treatment of such integrated excitation energy transfer (EET) and charge transfer (CT) dynamics is particularly challenging due to (i) appreciable system sizes, (ii) inter-site and system-bath couplings of similar magnitude that render the Born-Markov approximation invalid, (iii) substantial reorganization energies of CT states, and (iv) the presence of complex structured spectral densities due to vibrational modes of the surroundings. We present numerical simulations on bacterial reaction centre (bRC) inspired model systems that utilize the recently developed MACGIC-iQUAPI method [Richter et al., J. Chem. Phys., 2017, 146, 214101]. The simulations demonstrate that the method provides a rigorous framework for the investigation of such integrated EET-CT dynamics. First, the applicability of the MACGIC-iQUAPI method is explored for a transition from monotonically decaying to oscillatory system-bath influence coefficients, a behavior inherently imposed by structured bath spectral densities. Tightly coupled EET and CT dynamics is further addressed for an excitonic subsystem that resembles strong coupling of special pair states and serves as donor towards a generic bridge-acceptor system. By solving the dissipative quantum dynamics of such bRC inspired model systems, the quenching of excitonic coherence on the hundreds of femtoseconds timescale is explored via a variation of the bridge state energetics, resembling a continuous transition from sequential to superexchange mediated CT regimes. Further, the simulations explore the influence of resonant vibrational modes on the quenching of excitonic coherence via CT. The results reveal a moderate influence of vibrational mode on charge separation dynamics in regimes of biologically relevant EET and CT dynamics.