Linear-Response and Nonlinear-Response Formulations of the Instantaneous Marcus Theory for Nonequilibrium Photoinduced Charge Transfer

Simulating Non-Markovian Quantum Dynamics on NISQ Computers Using the Hierarchical Equations of Motion

Quantum computing offers promising new avenues for tackling the long-standing challenge of simulating the quantum dynamics of complex chemical systems, particularly open quantum systems coupled to external baths. However, simulating such nonunitary dynamics on quantum computers is challenging since quantum circuits are specifically designed to carry out unitary transformations. Furthermore, chemical systems are often strongly coupled to the surrounding environment, rendering the dynamics non-Markovian and beyond the scope of Markovian quantum master equations like Lindblad or Redfield. In this work, we introduce a quantum algorithm designed to simulate non-Markovian dynamics of open quantum systems. Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers. We illustrate the method as applied in conjunction with the numerically exact hierarchical equations of motion (HEOM) method. The effectiveness of the resulting quantum HEOM algorithm is demonstrated as applied to simulations of the non-Lindbladian electronic energy and charge transfer dynamics in models of the carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran and the Fenna-Matthews-Olson complex.

Reduced density matrix dynamics in multistate harmonic models via time-convolution and time-convolutionless quantum master equations with quantum-mechanical and semiclassical kernels

In this work, we explore the electronic reduced density matrix (RDM) dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) that are based on perturbative electronic couplings within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonian consistently incorporates the electronic-vibrational correlations between all pairs of states by satisfying the pairwise reorganization energies directly obtained from all-atom simulations, representing the globally heterogeneous environments that couple to the multiple states differently. We derive the exact quantum-mechanical and a hierarchy of semiclassical approximate expressions for the kernels in TC and TCL QMEs that project the full RDM for general shifted harmonic systems, including the MSH model. These QMEs are applied to simulate RDM dynamics of photoinduced charge transfer (PICT) in organic photovoltaic carotenoid-porphyrin-fullerene triad solvated in tetrahydrofuran solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes from C. tepidum and P. aestuarii. Our results show that while both TC and TCL QMEs capture similar phenomena in PICT and EET processes, TC QME generally provides more accurate results than TCL QME, particularly in the initial oscillation of EET population dynamics. This study highlights the effectiveness of the TC and TCL QMEs in modeling RDM dynamics of nonadiabatic processes, offering insights for realistic condensed phase systems.

PyCTRAMER: A Python package for charge transfer rate constant of condensed-phase systems from Marcus theory to Fermi's golden rule

In this work, we introduce PyCTRAMER, a comprehensive Python package designed for calculating charge transfer (CT) rate constants in disordered condensed-phase systems at finite temperatures, such as organic photovoltaic (OPV) materials. PyCTRAMER is a restructured and enriched version of the CTRAMER (Charge-Transfer RAtes from Molecular dynamics, Electronic structure, and Rate theory) package [Tinnin et al. J. Chem. Phys. 154, 214108 (2021)], enabling the computation of the Marcus CT rate constant and the six levels of the linearized semiclassical approximations of Fermi's golden rule (FGR) rate constant. It supports various types of intramolecular and intermolecular CT transitions from the excitonic states to CT state. Integrating quantum chemistry calculations, all-atom molecular dynamics (MD) simulations, spin-boson model construction, and rate constant calculations, PyCTRAMER offers an automatic workflow for handling photoinduced CT processes in explicit solvent environments and interfacial CT in amorphous donor/acceptor blends. The package also provides versatile tools for individual workflow steps, including electronic state analysis, state-specific force field construction, MD simulations, and spin-boson model construction from energy trajectories. We demonstrate the software's capabilities through two examples, highlighting both intramolecular and intermolecular CT processes in prototypical OPV systems.

All-Atom Photoinduced Charge Transfer Dynamics in Condensed Phase via Multistate Nonlinear-Response Instantaneous Marcus Theory

Photoinduced charge transfer (CT) in the condensed phase is an essential component in solar energy conversion, but it is challenging to simulate such a process on the all-atom level. The traditional Marcus theory has been utilized for obtaining CT rate constants between pairs of electronic states but cannot account for the nonequilibrium effects due to the initial nuclear preparation. The recently proposed instantaneous Marcus theory (IMT) and its nonlinear-response formulation allow for incorporating the nonequilibrium nuclear relaxation to electronic transition between two states after the photoexcitation from the equilibrium ground state and provide the time-dependent rate coefficient. In this work, we extend the nonlinear-response IMT method for treating photoinduced CT among general multiple electronic states and demonstrate it in the organic photovoltaic carotenoid-porphyrin-fullerene triad dissolved in explicit tetrahydrofuran solvent. All-atom molecular dynamics simulations were employed to obtain the time correlation functions of energy gaps, which were used to generate the IMT-required time-dependent averages and variances of the relevant energy gaps. Our calculations show that the multistate IMT could capture the significant nonequilibrium effects due to the initial nuclear state preparation, and this is corroborated by the substantial differences between the population dynamics predicted by the multistate IMT and the Marcus theory, where the Marcus theory underestimates the population transfer. The population dynamics by multistate IMT is also shown to have a better agreement with the all-atom nonadiabatic mapping dynamics than the Marcus theory does. Because the multistate nonlinear-response IMT is straightforward and cost-effective in implementation and accounts for the nonequilibrium nuclear effects, we believe this method offers a practical strategy for studying charge transfer dynamics in complex condensed-phase systems.

Instantaneous Marcus theory for photoinduced charge transfer dynamics in multistate harmonic model systems

Modeling the dynamics of photoinduced charge transfer (CT) in condensed phases presents challenges due to complicated many-body interactions and the quantum nature of electronic transitions. While traditional Marcus theory is a robust method for calculating CT rate constants between electronic states, it cannot account for the nonequilibrium effects arising from the initial nuclear state preparation. In this study, we employ the instantaneous Marcus theory (IMT) to simulate photoinduced CT dynamics. IMT incorporates nonequilibrium structural relaxation following a vertical photoexcitation from the equilibrated ground state, yielding a time-dependent rate coefficient. The multistate harmonic (MSH) model Hamiltonian characterizes an organic photovoltaic carotenoid-porphyrin-fullerene triad dissolved in explicit tetrahydrofuran solvent, constructed by mapping all-atom inputs from molecular dynamics simulations. Our calculations reveal that the electronic population dynamics of the MSH models obtained with IMT agree with the more accurate quantum-mechanical nonequilibrium Fermi's golden rule. This alignment suggests that IMT provides a practical approach to understanding nonadiabatic CT dynamics in condensed-phase systems.

Interlayer Charge Transport in 2D Lead Halide Perovskites from First Principles

We report on the implementation of a versatile projection-operator diabatization approach to calculate electronic coupling integrals in layered periodic systems. The approach is applied to model charge transport across the saturated organic spacers in two-dimensional (2D) lead halide perovskites. The calculations yield out-of-plane charge transfer rates that decay exponentially with the increasing length of the alkyl chain, range from a few nanoseconds to milliseconds, and are supportive of a hopping mechanism. Most importantly, we show that the charge carriers strongly couple to distortions of the Pb-I framework and that accounting for the associated nonlocal dynamic disorder increases the thermally averaged interlayer rates by a few orders of magnitude compared to the frozen-ion 0 K-optimized structure. Our formalism provides the first comprehensive insight into the role of the organic spacer cation on vertical transport in 2D lead halide perovskites and can be readily extended to functional π-conjugated spacers, where we expect the improved energy alignment with the inorganic layout to speed up the charge transfer between the semiconducting layers.

Imaginary-Time Open-Chain Path-Integral Approach for Two-State Time Correlation Functions and Applications in Charge Transfer

Quantum time correlation functions (TCFs) involving two states are important for describing nonadiabatic dynamical processes such as charge transfer. Based on a previous single-state method, we propose an imaginary-time open-chain path-integral (OCPI) approach for evaluating the two-state symmetrized TCFs. Expressing the forward and backward propagation on different electronic potential energy surfaces as a complex-time path integral, we then transform the path variables to average and difference variables such that the integration over the difference variables up to the second order can be performed analytically. The resulting expression for the symmetrized TCF is equivalent to sampling the open-chain configurations in an effective potential that corresponds to the average surface. Using importance sampling over the extended OCPI space via open path integral molecular dynamics, we tested the resulting path-integral approximation by calculating the Fermi's golden rule charge transfer rate constant within a widely-used spin-boson model. Comparing with the real-time linearized semiclassical method and analytical result, we show that the imaginary-time OCPI provides an accurate two-state symmetrized TCF and rate constant in the typical turnover region. It is shown that the first bead of the open chain corresponds to physical zero-time, and the endpoint bead corresponds to final time t; oscillations of the end-to-end distance perfectly match the nuclear mode frequency. The two-state OCPI scheme is seen to capture the tested model's electronic quantum coherence and nuclear quantum effects accurately.

Charge-Transfer Landscape Manifesting the Structure-Rate Relationship in the Condensed Phase Via Machine Learning

In this work, we develop a machine learning (ML) strategy to map the molecular structure to condensed phase charge-transfer (CT) properties including CT rate constants, energy levels, electronic couplings, energy gaps, reorganization energies, and reaction free energies which are called CT fingerprints. The CT fingerprints of selected landmark structures covering the conformation space of an organic photovoltaic molecule dissolved in an explicit solvent are computed and used to train ML models using kernel ridge regression. The ML models show high predictive power with R² > 0.97 and both mean absolute error and root-mean-square error within chemical accuracy. The CT landscape for millions of molecular dynamics sampled structures is thus constructed, which allows for instant prediction of CT rate properties, given any conformation of the molecule. We demonstrate some immediate utilities of the CT landscape such as calculating the ensemble-averaged CT rate constant and interpreting the effects of molecular structural features on the CT rate. The unprecedented CT landscape will be useful for investigating real-time CT dynamics in nanoscale- and mesoscale-condensed phase systems and for the optimal fabrication design for homogeneous and heterogeneous optoelectronic devices.

Multi-state harmonic models with globally shared bath for nonadiabatic dynamics in the condensed phase

Model Hamiltonians constructed from quantum chemistry calculations and molecular dynamics simulations are widely used for simulating nonadiabatic dynamics in the condensed phase. The most popular two-state spin-boson model could be built by mapping the all-atom anharmonic Hamiltonian onto a two-level system bilinearly coupled to a harmonic bath using the energy gap time correlation function. However, for more than two states, there lacks a general strategy to construct multi-state harmonic (MSH) models since the energy gaps between different pairs of electronic states are not entirely independent and need to be considered consistently. In this paper, we extend the previously proposed approach for building three-state harmonic models for photoinduced charge transfer to the arbitrary number of electronic states with a globally shared bath and the system-bath couplings are scaled differently according to the reorganization energies between each pair of states. We demonstrate the MSH model construction for an organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad dissolved in explicit tetrahydrofuran solvent. Nonadiabatic dynamics was simulated using mixed quantum-classical techniques, including the linearized semiclassical and symmetrical quasiclassical dynamics with the mapping Hamiltonians, mean-field Ehrenfest, and mixed quantum-classical Liouville dynamics in two-state, three-state, and four-state harmonic models of the triad system. The MSH models are shown to provide a general and flexible framework for simulating nonadiabatic dynamics in complex systems.

Three-state harmonic models for photoinduced charge transfer

A widely used strategy for simulating the charge transfer between donor and acceptor electronic states in an all-atom anharmonic condensed-phase system is based on invoking linear response theory to describe the system in terms of an effective spin-boson model Hamiltonian. Extending this strategy to photoinduced charge transfer processes requires also taking into consideration the ground electronic state in addition to the excited donor and acceptor electronic states. In this paper, we revisit the problem of describing such nonequilibrium processes in terms of an effective three-state harmonic model. We do so within the framework of nonequilibrium Fermi's golden rule (NE-FGR) in the context of photoinduced charge transfer in the carotenoid-porphyrin-C₆₀ (CPC₆₀) molecular triad dissolved in explicit tetrahydrofuran (THF). To this end, we consider different ways for obtaining a three-state harmonic model from the equilibrium autocorrelation functions of the donor-acceptor, donor-ground, and acceptor-ground energy gaps, as obtained from all-atom molecular dynamics simulations of the CPC₆₀/THF system. The quantum-mechanically exact time-dependent NE-FGR rate coefficients for two different charge transfer processes in two different triad conformations are then calculated using the effective three-state model Hamiltonians as well as a hierarchy of more approximate expressions that lead to the instantaneous Marcus theory limit. Our results show that the photoinduced charge transfer in CPC₆₀/THF can be described accurately by the effective harmonic three-state models and that nuclear quantum effects are small in this system.