Semiclassical approaches to perturbative time-convolution and time-convolutionless quantum master equations for electronic transitions in multistate systems.

Understanding the dynamics of photoinduced processes in complex systems is crucial for the development of advanced energy-conversion materials. In this study, we investigate the nonadiabatic dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) based on treating electronic couplings as perturbation within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonians are mapped from all-atom simulations such that all pairwise reorganization energies are consistently incorporated, leading to a heterogeneous environment that couples to the multiple electronic states differently. Our exploration encompasses the photoinduced charge transfer dynamics in organic photovoltaic carotenoid-porphyrin-C60 triad dissolved in liquid solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes. By systematically comparing the perturbative TC and TCL QME approaches with exact quantum-mechanical and various semiclassical approximate kernels, we demonstrate their efficacy and accuracy in capturing the essential features of photoinduced dynamics. Our calculations show that TC QMEs generally yield more accurate results than TCL QMEs, especially in EET, although both methods offer versatile approaches adaptable across different systems. In addition, we investigate various semiclassical approximations featuring the Wigner-transformed and classical nuclear densities as well as the governing dynamics during the quantum coherence period, highlighting the trade-off between accuracy and computational cost. This work provides valuable insights into the applicability and performance of TC and TCL QME approaches via the MSH model, offering guidance for realistic applications to condensed-phase systems on the atomistic level.

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.

Generalized nonequilibrium Fermi's golden rule and its semiclassical approximations for electronic transitions between multiple states.

The nonequilibrium Fermi's golden rule (NE-FGR) approach is developed to simulate the electronic transitions between multiple excited states in complex condensed-phase systems described by the recently proposed multi-state harmonic (MSH) model Hamiltonian. The MSH models were constructed to faithfully capture the photoinduced charge transfer dynamics in a prototypical organic photovoltaic carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran. A general expression of the fully quantum-mechanical NE-FGR rate coefficients for transitions between all pairs of states in the MSH model is obtained. Besides, the linearized semiclassical NE-FGR formula and a series of semiclassical approximations featuring Wigner and classical nuclear sampling choices and different dynamics during the quantum coherence period for the MSH model are derived. The current approach enables all the possible population transfer pathways between the excited states of the triad, in contrast to the previous applications that only addressed the donor-to-acceptor transition. Our simulations for two triad conformations serve as a demonstration for benchmarking different NE-FGR approximations and show that the difference between all levels of approximation is small for the current system, especially at room temperature. By comparing with nonadiabatic semiclassical dynamics, we observe similar timescales for the electronic population transfer predicted by NE-FGR. It is believed that the general formulation of NE-FGR for the MSH Hamiltonian enables a variety of applications in realistic systems.

Benchmarking various nonadiabatic semiclassical mapping dynamics methods with tensor-train thermo-field dynamics.

Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.

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 (CT). 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 CT 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 that 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.

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.

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.

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.

Multistate Reaction Coordinate Model for Charge and Energy Transfer Dynamics in the Condensed Phase.

Constructing multistate model Hamiltonians from all-atom electronic structure calculations and molecular dynamics simulations is crucial for understanding charge and energy transfer dynamics in complex condensed phases. The most popular two-level system model is the spin-boson Hamiltonian, where the nuclear degrees of freedom are represented as shifted normal modes. Recently, we proposed the general multistate nontrivial extension of the spin-boson model, i.e., the multistate harmonic (MSH) model, which is constructed by extending the spatial dimensions of each nuclear mode so as to satisfy the all-atom reorganization energy restrictions for all pairs of electronic states. In this work, we propose the multistate reaction coordinate (MRC) model with a primary reaction coordinate and secondary bath modes as in the Caldeira-Leggett form but in extended spatial dimensions. The MRC model is proven to be equivalent to the MSH model and offers an intuitive physical picture of the nuclear-electronic feedback in nonadiabatic processes such as the inherent trajectory of the reaction coordinate. The reaction coordinate is represented in extended dimensions, carrying the entire reorganization energies and bilinearly coupled to the secondary bath modes. We demonstrate the MRC model construction for photoinduced charge transfer in an organic photovoltaic caroteniod-porphyrin-C60 molecular triad dissolved in tetrahydrofuran as well as excitation energy transfer in a photosynthetic light-harvesting Fenna-Matthews-Olson complex. The MRC model provides an effective and robust platform for investigating quantum dissipative dynamics in complex condensed-phase systems since it allows a consistent description of realistic spectral density, state-dependent system-bath couplings, and heterogeneous environments due to static disorder in reorganization energies.

Effects of Heterogeneous Protein Environment on Excitation Energy Transfer Dynamics in the Fenna-Matthews-Olson Complex.

The Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria has been serving as a prototypical light-harvesting protein for studying excitation energy transfer (EET) dynamics in photosynthesis. The most widely used Frenkel exciton model for FMO complex assumes that each excited bacteriochlorophyll site couples to an identical and isolated harmonic bath, which does not account for the heterogeneous local protein environment. To better describe the realistic environment, we propose to use the recently developed multistate harmonic (MSH) model, which contains a globally shared bath that couples to the different pigment sites according to the atomistic quantum mechanics/molecular mechanics simulations with explicit protein scaffold and solvent. In this work, the effects of heterogeneous protein environment on EET in FMO complexes from Prosthecochloris aestuarii and Chlorobium tepidum, specifically including realistic spectral density, site-dependent reorganization energies, and system-bath couplings are investigated. Semiclassical and mixed quantum-classical mapping dynamics were applied to obtain the nonadiabatic EET dynamics in several models ranging from the Frenkel exciton model to the MSH model and their variants. The MSH model with realistic spectral density and site-dependent system-bath couplings displays slower EET dynamics than the Frenkel exciton model. Our comparative study shows that larger average reorganization energy, heterogeneity in spectral densities, and low-frequency modes could facilitate energy dissipation, which is insensitive to the static disorder in reorganization energies. The effects of the spectral densities and system-bath couplings along with the MSH model can be used to optimize EET dynamics for artificial light-harvesting systems.

Fabrication of Conjugated Amphiphilic Triblock Copolymer for Drug Delivery and Fluorescence Cell Imaging.

An elegant integration of light-emitting segments into the structure of polymeric delivery systems endows the resulting self-assembled nanovehicles with the diagnostic ability toward an enhanced therapeutic efficiency. A variety of polyfluorene (PF)-based binary delivery systems were designed and developed successfully, but PF-based ternary formulations remain rarely explored, likely due to the synthetic challenge. To develop a universal synthesis strategy toward linear conjugated amphiphilic triblock copolymer for cancer theranostics, herein we focused on the functionalization of the PF terminus for further chain extension and prepared well-defined PF-based amphiphilic triblock copolymers, PF-b-poly(ε-caprolactone)-b-poly(oligo(ethylene glycol) monomethyl ether methacrylate) (PF-b-PCL-b-POEGMA), by integrated state-of-the-art polymer chemistry techniques, including Suzuki reaction, ring-opening polymerization, atom transfer radical polymerization, and click coupling. The resulting conjugated amphiphilic triblock copolymers can self-assembe into core-shell-corona (CSC) micelles with PF block constructing the inner hydrophobic core for fluorescent tracking, PCL segment forming the hydrophobic middle shell for drug encapsulation, and POEGMA moiety building the hydrophilic outer corona for particulate stabilization. Interestingly, the CSC micelles with hydrophobic PCL middle layer show a greater drug loading capacity as well as a higher fluorescence quantum yield (Φ) relative to the core-shell micelles self-assembled from the control of PF-b-POEGMA diblock copolymers without PCL sequence due to having more hydrophobic spaces and better separation of PF sequence provided simultaneously by the PCL central block. The efficient cellular uptake of the anticancer drug doxorubicin-loaded CSC micelles together with the in vitro cytotoxicity against the HeLa cells makes the conjugated amphiphilic triblock copolymers developed herein a promising platform for simultaneous cell image and drug delivery, thus offering great potential for cancer theranostics.