Simulation of Nonlinear Femtosecond Signals at Finite Temperature via a Thermo Field Dynamics-Tensor Train Method: General Theory and Application to Time- and Frequency-Resolved Fluorescence of the Fenna-Matthews-Olson Complex

Ultrafast Excited-State Energy Transfer in Phenylene Ethynylene Dendrimer: Quantum Dynamics with the Tensor Network Method

Photoinduced excited-state energy transfer (EET) processes play an important role in solar energy conversions. Owing to their excellent photoharvesting and exciton-transport properties, phenylene ethynylene (PE) dendrimers display great potential for improving the efficiency of solar cells. In this work, we investigated the intramolecular EET dynamics in a dendrimer composed of two linear PE units (2-ring and 3-ring) using a fully quantum description based on the tensor network method. We first constructed a diabatic model Hamiltonian based on the electronic structure calculations. Using this diabatic vibronic coupling model, we tried to obtain the main features of the EET dynamics in terms of the several diabatic models with different numbers of vibrational modes (from 4 modes to 129 modes) and to explore the corresponding vibronic coupling interactions. The results show that the EET in this PE dendrimer is ultrafast. Four modes of A' symmetry play dominant roles in the dynamics; the remaining 86 modes of A' symmetry can dampen the electronic coherence; and the modes of A″ symmetry do not exhibit significant influence on the EET process. Overall, the first-order intrastate vibronic coupling terms show the dominant role in the EET dynamics, while the second-order intrastate vibronic coupling terms cause damping of the electronic coherence and slow down the overall EET process. This work provides a microscopic understanding of the EET dynamics in PE dendrimers.

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.

Two-dimensional fluorescence excitation spectroscopy: A novel technique for monitoring excited-state photophysics of molecular species with high time and frequency resolution

We propose a novel UV/Vis femtosecond spectroscopic technique, two-dimensional fluorescence-excitation (2D-FLEX) spectroscopy, which combines spectral resolution during the excitation process with exclusive monitoring of the excited-state system dynamics at high time and frequency resolution. We discuss the experimental feasibility and realizability of 2D-FLEX, develop the necessary theoretical framework, and demonstrate the high information content of this technique by simulating the 2D-FLEX spectra of a model four-level system and the Fenna-Matthews-Olson antenna complex. We show that the evolution of 2D-FLEX spectra with population time directly monitors energy transfer dynamics and can thus yield direct qualitative insight into the investigated system. This makes 2D-FLEX a highly efficient instrument for real-time monitoring of photophysical processes in polyatomic molecules and molecular aggregates.

Ab initio simulation of peak evolutions and beating maps for electronic two-dimensional signals of a polyatomic chromophore

By employing the doorway-window (DW) on-the-fly simulation protocol, we performed ab initio simulations of peak evolutions and beating maps of electronic two-dimensional (2D) spectra of a polyatomic molecule in the gas phase. As the system under study, we chose pyrazine, which is a paradigmatic example of photodynamics dominated by conical intersections (CIs). From the technical perspective, we demonstrate that the DW protocol is a numerically efficient methodology suitable for simulations of 2D spectra for a wide range of excitation/detection frequencies and population times. From the information content perspective, we show that peak evolutions and beating maps not only reveal timescales of transitions through CIs but also pinpoint the most relevant coupling and tuning modes active at these CIs.

Exciton Dynamics and Time-Resolved Fluorescence in Nanocavity-Integrated Monolayers of Transition-Metal Dichalcogenides

We have developed an ab initio-based, fully quantum, numerically accurate methodology for the simulation of the exciton dynamics and time- and frequency-resolved fluorescence spectra of the cavity-controlled two-dimensional materials at finite temperatures and applied this methodology to the single-layer WSe₂ system. Specifically, the multiple Davydov D₂ Ansatz has been employed in combination with the method of thermofield dynamics for the finite-temperature extension of accurate time-dependent variation. This allowed us to establish dynamical and spectroscopic signatures of the polaronic and polaritonic effects as well as uncover their characteristic time scales in the relevant range of temperatures. Our study reveals the pivotal role of multidimensional conical intersections in controlling the many-body dynamics of highly intertwined excitonic, phononic, and photonic modes.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Ultrafast Charge Relocation Dynamics in Enol-Keto Tautomerization Monitored with a Local Soft-X-ray Probe

Proton-coupled electron transfer (PCET) is the underlying mechanism governing important reactions ranging from water splitting in photosynthesis to oxygen reduction in hydrogen fuel cells. The interplay of proton and electronic charge distribution motions can vary from sequential to concerted schemes, with elementary steps occurring on ultrafast time scales. We demonstrate with a simulation study that femtosecond soft-X-ray spectroscopy provides key insights into the PCET mechanism of a photoinduced intramolecular enol* → keto* tautomerization reaction. A full quantum treatment of the electronic and nuclear dynamics of 2-(2'-hydroxyphenyl)benzothiazole upon electronic excitation reveals how spectral signatures of local excitations from core to frontier orbitals display the distinctly different stages of charge relocation for the H atom, donating, and accepting sites. Our findings indicate that ultraviolet/X-ray pump-probe spectroscopy provides a unique way to probe ultrafast electronic structure rearrangements in photoinduced chemical reactions essential to understanding the mechanism of PCET.

Effect of mixed Frenkel and charge transfer states in time-gated fluorescence spectra of perylene bisimides H-aggregates: Hierarchical equations of motion approach

We theoretically investigated the effect of mixed Frenkel (F) and charge transfer (CT) states on the spectral properties of perylene bisimide (PBI) derivatives, focusing on the role of strong electron-phonon interactions. The model consists of a four-level system described by the Holstein Hamiltonian coupled to independent local heat-baths on each site, described by Brownian spectral distribution functions. We employ the reduced hierarchical equations of motion (HEOM) approach to calculate the time evolution of the system and compare it to the pure F exciton cases. We compute the absorption and time-gated fluorescence (TGF) spectra for different exciton transfer integrals and F-CT band gap conditions. The coherence length of excitons ($N_{coh}$) is evaluated employing two different definitions. We observe the presence of an excited hot state peak whose intensity is associated with the delocalization of the excited species and ultrafast dynamics that are solely dependent on the frequency of the local bath. The results indicate that the inclusion of CT states promotes localization of the excitons which is manifested in a decrease in the intensity of the hot state peak and the 0--1 peak, and an increase in the intensity of the 0--0 emission peak in TGF spectrum, leading to a decrease of $N_{coh}$.

Matrix Product State Formulation of the MCTDH Theory in Local Mode Representations for Anharmonic Potentials

The matrix product state formulation of the multiconfiguration time-dependent Hartree theory, MPS-MCTDH, reported previously [Kurashige, J. Chem. Phys. 2018, 19, 194114] is extended to realistic anharmonic potentials with n-mode representations beyond the linear vibronic coupling model. For realistic vibrational potentials, the local mode representation should give a more compact representation of the potentials, i.e., lowering the dimensionality of the entanglements, than the normal coordinates, and the MPS-MCTDH formulation should work more efficiently and maintain the accuracy with a small bond dimension of the MPS ansatz. In fact, it was confirmed that the use of the local coordinates made the interaction matrices diagonal dominant and the number of terms in the n-body expansion of the potentials was significantly reduced. The method was applied to the IR spectrum of the CH₂O molecule, the zero-point energies, and the vibrational energy redistribution dynamics of polyenes C_2nH_2n+2. The results showed that the efficiency of the MPS-MCTDH method is significantly accelerated by the use of local coordinates even if the long-range interactions are included in the potential.

Accurate Simulation of Spectroscopic Signatures of Cavity-Assisted, Conical-Intersection-Controlled Singlet Fission Processes

A numerically accurate, fully quantum methodology has been developed for the simulation of the dynamics and nonlinear spectroscopic signals of cavity-assisted, conical-intersection-controlled singlet fission systems. The methodology is capable of handling several molecular systems strongly coupled to the photonic mode of the cavity and treats the intrinsic conical intersection and cavity-induced polaritonic conical intersections in a numerically exact manner. Contributions of higher-lying molecular electronic states are accounted for comprehensively. The intriguing process of cavity-modified fission dynamics, including all of its electronic, vibrational, and photonic degrees of freedom, together with its two-dimensional spectroscopic manifestation, is simulated for two rubrene dimers strongly coupled to the cavity mode.

Two-dimensional vibronic spectroscopy with semiclassical thermofield dynamics

Thermofield dynamics is an exactly correct formulation of quantum mechanics at finite temperature in which a wavefunction is governed by an effective temperature-dependent quantum Hamiltonian. The optimized mean trajectory (OMT) approximation allows the calculation of spectroscopic response functions from trajectories produced by the classical limit of a mapping Hamiltonian that includes physical nuclear degrees of freedom and other effective degrees of freedom representing discrete vibronic states. Here, we develop a thermofield OMT (TF-OMT) approach in which the OMT procedure is applied to a temperature-dependent classical Hamiltonian determined from the thermofield-transformed quantum mapping Hamiltonian. Initial conditions for bath nuclear degrees of freedom are sampled from a zero-temperature distribution. Calculations of two-dimensional electronic spectra and two-dimensional vibrational–electronic spectra are performed for models that include excitonically coupled electronic states. The TF-OMT calculations agree very closely with the corresponding OMT results, which, in turn, represent well benchmark calculations with the hierarchical equations of motion method.

Stochastic Adaptive Single-Site Time-Dependent Variational Principle

In recent years, the time-dependent variational principle (TDVP) method based on the matrix product state (MPS) wave function formulation has shown its great power in performing large-scale quantum dynamics simulations for realistic chemical systems with strong electron-vibration interactions. In this work, we propose a stochastic adaptive single-site TDVP (SA-1TDVP) scheme to evolve the bond-dimension adaptively, which can integrate the traditional advantages of both the high efficiency of the single-site TDVP (1TDVP) variant and the high accuracy of the two-site TDVP (2TDVP) variant. Based on the assumption that the level statistics of entanglement Hamiltonians, which originate from the reduced density matrices of the MPS method, follows a Poisson or Wigner distribution, as generically predicted by random-matrix theory, additional random singular values are generated to expand the bond-dimension automatically. Tests on simulating the vibrationally resolved quantum dynamics and absorption spectra in the pyrazine molecule and perylene bisimide (PBI) J-aggregate trimer as well as a spin-1/2 Heisenberg chain show that it can be automatic and as accurate as 2TDVP but reduce the computational time remarkably.

Variational Squeezed Davydov Ansatz for Realistic Chemical Systems with Nonlinear Vibronic Coupling

Chemical systems normally possess strong nonlinear vibronic couplings at both zero and finite temperature. For the lowest-order quadratic couplings, here, we introduce a squeezing operator into a variational coherent-state-based method, Davydov ansatz, to simulate the quantum dynamics and the respective spectroscopy. Two molecular systems, pyrazine and the 2-pyridone dimer, are taken as calculated model systems, both of which involve nontrivial quadratic vibronic couplings in high- and low-frequency regions, respectively. Upon a comparison with the benchmarks, the method manifests its advantage for nonlinear couplings. The squeezed bases are also proven to be applicable for the finite temperature by adapting with the thermofield dynamics.

Signatures of coherent vibronic exciton dynamics and conformational control in two-dimensional electronic spectroscopy of conjugated polymers

Two-dimensional electronic spectroscopy (2DES) signals for homo-oligomer J-aggregates are computed, with a focus on the role of structural change induced by low-frequency torsional modes along with quasi-stationary trapping effects induced...

Computational and data driven molecular material design assisted by low scaling quantum mechanics calculations and machine learning

Electronic structure methods based on quantum mechanics (QM) are widely employed in the computational predictions of the molecular properties and optoelectronic properties of molecular materials. The computational costs of these QM methods, ranging from density functional theory (DFT) or time-dependent DFT (TDDFT) to wave-function theory (WFT), usually increase sharply with the system size, causing the curse of dimensionality and hindering the QM calculations for large sized systems such as long polymer oligomers and complex molecular aggregates. In such cases, in recent years low scaling QM methods and machine learning (ML) techniques have been adopted to reduce the computational costs and thus assist computational and data driven molecular material design. In this review, we illustrated low scaling ground-state and excited-state QM approaches and their applications to long oligomers, self-assembled supramolecular complexes, stimuli-responsive materials, mechanically interlocked molecules, and excited state processes in molecular aggregates. Variable electrostatic parameters were also introduced in the modified force fields with the polarization model. On the basis of QM computational or experimental datasets, several ML algorithms, including explainable models, deep learning, and on-line learning methods, have been employed to predict the molecular energies, forces, electronic structure properties, and optical or electrical properties of materials. It can be conceived that low scaling algorithms with periodic boundary conditions are expected to be further applicable to functional materials, perhaps in combination with machine learning to fast predict the lattice energy, crystal structures, and spectroscopic properties of periodic functional materials.

Quantum dynamics simulation of intramolecular singlet fission in covalently linked tetracene dimer

In this work, we study singlet fission in tetracene para-dimers, covalently linked by a phenyl group. In contrast to most previous studies, we account for the full quantum dynamics of the combined excitonic and vibrational system. For our simulations, we choose a numerically unbiased representation of the molecule's wave function, enabling us to compare with experiments, exhibiting good agreement. Having access to the full wave function allows us to study in detail the post-quench dynamics of the excitons. Here, one of our main findings is the identification of a time scale t₀ ≈ 35 fs dominated by coherent dynamics. It is within this time scale that the larger fraction of the singlet fission yield is generated. We also report on a reduced number of phononic modes that play a crucial role in the energy transfer between excitonic and vibrational systems. Notably, the oscillation frequency of these modes coincides with the observed electronic coherence time t₀. We extend our investigations by also studying the dependency of the dynamics on the excitonic energy levels that, for instance, can be experimentally tuned by means of the solvent polarity. Here, our findings indicate that the singlet fission yield can be doubled, while the electronic coherence time t₀ is mainly unaffected.

Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue

Modeling linear absorption spectra of solvated chromophores is highly challenging as contributions are present both from coupling of the electronic states to nuclear vibrations and from solute-solvent interactions. In systems where excited states intersect in the Condon region, significant non-adiabatic contributions to absorption line shapes can also be observed. Here, we introduce a robust approach to model linear absorption spectra accounting for both environmental and non-adiabatic effects from first principles. This model parameterizes a linear vibronic coupling (LVC) Hamiltonian directly from energy gap fluctuations calculated along molecular dynamics (MD) trajectories of the chromophore in solution, accounting for both anharmonicity in the potential and direct solute-solvent interactions. The resulting system dynamics described by the LVC Hamiltonian are solved exactly using the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA). The approach is applied to the linear absorption spectrum of methylene blue in water. We show that the strong shoulder in the experimental spectrum is caused by vibrationally driven population transfer between the bright S₁ and the dark S₂ states. The treatment of the solvent environment is one of many factors that strongly influence the population transfer and line shape; accurate modeling can only be achieved through the use of explicit quantum mechanical solvation. The efficiency of T-TEDOPA, combined with LVC Hamiltonian parameterizations from MD, leads to an attractive method for describing a large variety of systems in complex environments from first principles.

Efficient quantum dynamics simulations of complex molecular systems: A unified treatment of dynamic and static disorder

We present a unified and highly numerically efficient formalism for the simulation of quantum dynamics of complex molecular systems, which takes into account both temperature effects and static disorder. The methodology is based on the thermo-field dynamics formalism, and Gaussian static disorder is included into simulations via auxiliary bosonic operators. This approach, combined with the tensor-train/matrix-product state representation of the thermalized stochastic wave function, is applied to study the effect of dynamic and static disorders in charge-transfer processes in model organic semiconductor chains employing the Su-Schrieffer-Heeger (Holstein-Peierls) model Hamiltonian.

A thermofield-based multilayer multiconfigurational time-dependent Hartree approach to non-adiabatic quantum dynamics at finite temperature

We introduce a thermofield-based formulation of the multilayer multiconfigurational time-dependent Hartree (MCTDH) method to study finite temperature effects on non-adiabatic quantum dynamics from a non-stochastic, wave function perspective. Our approach is based on the formal equivalence of bosonic many-body theory at zero temperature with a doubled number of degrees of freedom and the thermal quasi-particle representation of bosonic thermofield dynamics (TFD). This equivalence allows for a transfer of bosonic many-body MCTDH as introduced by Wang and Thoss to the finite temperature framework of thermal quasi-particle TFD. As an application, we study temperature effects on the ultrafast internal conversion dynamics in pyrazine. We show that finite temperature effects can be efficiently accounted for in the construction of multilayer expansions of thermofield states in the framework presented herein. Furthermore, we find our results to agree well with existing studies on the pyrazine model based on the ρMCTDH method.