Finite temperature dynamics in a polarized sub-Ohmic heat bath: A hierarchical equations of motion-tensor train study.

The dynamics of the sub-Ohmic spin-boson model under polarized initial conditions at finite temperatures is investigated by employing both analytical tools and the numerically accurate hierarchical equations of motion-tensor train method. By analyzing the features of nonequilibrium dynamics, we discovered a bifurcation phenomenon, which separates two regimes of the dynamics. It is found that before the bifurcation time, increasing temperature slows down the population dynamics, while the opposite effect occurs after the bifurcation time. The dynamics is highly sensitive to both initial preparation of the bath and thermal effects.

High accuracy exponential decomposition of bath correlation functions for arbitrary and structured spectral densities: Emerging methodologies and new approaches.

This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.

Current-induced bond rupture in single-molecule junctions: Effects of multiple electronic states and vibrational modes.

Current-induced bond rupture is a fundamental process in nanoelectronic architectures, such as molecular junctions, and scanning tunneling microscopy measurements of molecules at surfaces. The understanding of the underlying mechanisms is important for the design of molecular junctions that are stable at higher bias voltages and is a prerequisite for further developments in the field of current-induced chemistry. In this work, we analyze the mechanisms of current-induced bond rupture employing a recently developed method, which combines the hierarchical equations of motion approach in twin space with the matrix product state formalism and allows accurate, fully quantum mechanical simulations of the complex bond rupture dynamics. Extending previous work [Ke et al. J. Chem. Phys. 154, 234702 (2021)], we consider specifically the effect of multiple electronic states and multiple vibrational modes. The results obtained for a series of models of increasing complexity show the importance of vibronic coupling between different electronic states of the charged molecule, which can enhance the dissociation rate at low bias voltages profoundly.

Hierarchical equations of motion approach to hybrid fermionic and bosonic environments: Matrix product state formulation in twin space.

We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation [R. Borrelli, J. Chem. Phys. 150, 234102 (2019)] to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schrödinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-rank matrices. The new approach facilitates accurate simulations of non-equilibrium quantum dynamics in larger and more complex open quantum systems. The performance of the method is demonstrated for a model of a molecular junction exhibiting current-induced mode-selective vibrational excitation.

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 bandgap conditions. The coherence length of excitons (Ncoh) 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 the TGF spectrum, leading to a decrease of Ncoh.

Reliable Predictions of Benzophenone Singlet-Triplet Transition Rates: A Second-Order Cumulant Approach.

Fermi golden rule and second-order cumulant expansion of the time-dependent density matrix have been used to compute from first principles the rate of intersystem crossing in benzophenone, using minimum-energy geometries and normal modes of vibrations computed at the TDDFT/CAM-B3LYP level. Both approaches yield reliable values of the S1 decay rate, the latter being almost in quantitative agreement with the results of time-dependent spectroscopic measurements (0.154 ps-1 observed vs 0.25 ps-1 predicted). The Fermi golden rule slightly overestimates the decay rate of S1 state (kd = 0.45 ps-1) but provides better insights into the chemico-physical parameters, which govern the transition from a thermally equilibrated population of S1, showing that the indirect mechanism is much faster than the direct one because of the vanishingly small Franck-Condon weighted density of states at ΔE of transition.

Probing photoinduced proton coupled electron transfer process by means of two-dimensional resonant electronic-vibrational spectroscopy.

We develop a detailed theoretical model of photo-induced proton-coupled electron transfer (PPCET) processes, which are at the basis of solar energy harvesting in biological systems and photovoltaic materials. Our model enables us to analyze the dynamics and the efficiency of a PPCET reaction under the influence of a thermal environment by disentangling the contribution of the fundamental electron transfer and proton transfer steps. In order to study quantum dynamics of the PPCET process under an interaction with the non-Markovian environment, we employ the hierarchical equations of motion. We calculate transient absorption spectroscopy (TAS) and a newly defined two-dimensional resonant electronic-vibrational spectroscopy (2DREVS) signals in order to study the nonequilibrium reaction dynamics. Our results show that different transition pathways can be separated by TAS and 2DREVS.

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.

Proton tunneling in a two-dimensional potential energy surface with a non-linear system-bath interaction: Thermal suppression of reaction rate.

We consider a proton transfer (PT) system described by a proton transfer reaction (PTR) coordinate and a rate promoting vibrational (RPV) coordinate interacting with a non-Markovian heat bath. While dynamics of PT processes has been widely discussed using two-dimensional potential energy surfaces, the role of the heat bath, in particular, in a realistic form of the system-bath interaction has not been well explored. Previous studies are largely based on a one-dimensional model and linear-linear system-bath interaction. In the present study, we introduce an exponential-linear (EL) system-bath interaction, which is derived from the analysis of a PTR-RPV system in a realistic situation. This interaction mainly causes vibrational dephasing in the PTR mode and population relaxation in the RPV mode. Numerical simulations were carried out using the hierarchical equations of motion approach. We analyze the role of the heat bath interaction in the chemical reaction rate as a function of the system-bath coupling strength at different temperatures and for different values of the bath correlation time. A prominent feature of the present result is that while the reaction rate predicted from classical and quantum Kramers theory increases as the temperature increases, the present EL interaction model exhibits opposite temperature dependence. The Kramers turn-over profile of the reaction rate as a function of the system-bath coupling is also suppressed in the present EL model, turning into a plateau-like curve for larger system-bath interaction strength. Such features arise from the interplay of the vibrational dephasing process in the PTR mode and the population relaxation process in the RPV mode.

Density matrix dynamics in twin-formulation: An efficient methodology based on tensor-train representation of reduced equations of motion.

The twin-formulation of quantum statistical mechanics is employed to describe a new methodology for the solution of the equations of motion of the reduced density matrix in their hierarchical formulation. It is shown that the introduction of tilde operators and of their algebra in the dual space greatly simplifies the application of numerical techniques for the propagation of the density matrix. The application of tensor-train representation of a vector to solve complex quantum dynamical problems within the framework of the twin-formulation is discussed. Next, applications of the hierarchical equations of motion to a dissipative polaron model are presented showing the validity and accuracy of the new approach.

Dynamics of Coupled Electron-Boson Systems with the Multiple Davydov D1 Ansatz and the Generalized Coherent State.

The dynamics of a coupled electron-boson system is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron-nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.

Efficient orientational averaging of nonlinear optical signals in multi-chromophore systems.

The simulation of nonlinear optical signals measured in isotropic media requires averaging over the orientations of the chromophores. Herein, we discuss the evaluation of orientationally averaged four-wave-mixing, six-wave-mixing, etc., signals in terms of contracted transition dipole moment operators, which were introduced earlier for the orientational averaging of linear absorption signals. We demonstrate that the contracted transition dipole moment operators substantially facilitate the orientational averaging of nonlinear signals in multi-chromophore systems. We consider both the perturbative evaluation of signals (through nonlinear response functions) as well as the nonperturbative evaluation (through the numerical calculation of the nonlinear polarization of driven systems).

Absorption Band Shapes of a Push-Pull Dye Approaching the Cyanine Limit: A Challenging Case for First Principle Calculations.

The absorption band shapes of a solvent tunable donor-acceptor dye have been theoretically investigated by using Kubo's generating function approach, with minimum energy geometries and normal coordinates computed at the DFT level of theory. The adopted computational procedure allows us to include in the computation of Franck-Condon factors the whole set of normal modes, without any limitation on excitation quanta, allowing for an almost quantitative reproduction of the absorption band shape when the equilibrium geometries of the ground and the excited states are well predicted by electronic computations. Noteworthy, the functionals that yield more accurate band shapes also provide good prediction of the moment variations upon excitation; because the latter quantities are rarely available, theoretical simulation of band shapes could be a powerful tool for choosing the most appropriate computational method for predictive purposes.

Quantum dynamics of electronic transitions with Gauss-Hermite wave packets.

A new methodology based on the superposition of time-dependent Gauss-Hermite wave packets is developed to describe the wave function of a system in which several interacting electronic states are coupled to a bath of harmonic oscillators. The equations of motion for the wave function parameters are obtained by employing the Dirac-Frenkel time-dependent variational principle. The methodology is applied to study the quantum dynamical behaviour of model systems with two interacting electronic states characterized by a relatively large reorganization energy and a range of energy biases. The favourable scaling properties make it a promising tool for the study of the dynamics of chemico-physical processes in molecular systems.

Quantum electron-vibrational dynamics at finite temperature: Thermo field dynamics approach.

Quantum electron-vibrational dynamics in molecular systems at finite temperature is described using an approach based on the thermo field dynamics theory. This formulation treats temperature effects in the Hilbert space without introducing the Liouville space. A comparison with the theoretically equivalent density matrix formulation shows the key numerical advantages of the present approach. The solution of thermo field dynamics equations with a novel technique for the propagation of tensor trains (matrix product states) is discussed. Numerical applications to model spin-boson systems show that the present approach is a promising tool for the description of quantum dynamics of complex molecular systems at finite temperature.

Vibronic couplings and coherent electron transfer in bridged systems.

A computational strategy to analyze the dynamics of coherent electron transfer processes in bridged systems, involving three or more electronic states, is presented. The approach is based on partitioning of the Hilbert space of the time independent basis functions in subspaces of increasing dimensionality, which allows us to easily check the convergence of the time dependent wave function. Vibronic couplings are determined by Duschinsky's analysis of the equilibrium position displacements, carried out using the equilibrium geometries and normal modes of the redox partners obtained at the DFT computational level.

Theoretical and experimental determination of the absorption and emission spectra of a prototypical indolenine-based squaraine dye.

The spectroscopy of a prototypical indolenine-based squaraine dye is analysed theoretically using state-of-the-art methodologies for the simulation of spectral lineshapes, and experimentally using optical absorption and emission spectroscopies. Density functional theory and its time-dependent extension are used to determine the stability of several conformers, to compute their excitation energies, equilibrium geometries and vibrational frequencies, both in the ground and in their first excited singlet state. Finally the generating function approach is used to simulate the vibronic lineshape of the low energy valence ππ* excitation and emission spectra. Solvent effects are also computed and discussed by using the polarizable continuum model. The developed model correctly reproduces the main spectral features of the squaraine, and allows us to identify the vibrational motions that mainly contribute to the observed lineshape.

Computational investigation of the photoinduced homolytic dissociation of water in the pyridine-water complex.

The photochemistry of the hydrogen-bonded pyridine-water complex has been investigated with ab initio computational methods. Vertical excitation energies, excited-state reaction paths for proton transfer as well as structures and energies of conical intersections and reaction barriers have been determined with multi-configuration self-consistent-field and multi-reference perturbation methods, as well as with single-reference coupled-cluster and propagator methods. In the pyridine-water complex, the energies of two charge-separated excited states of (1)nπ* and (1)ππ* character are connected to the energies of the locally excited (1)nπ* and (1)ππ* states of the pyridine chromophore via a low energy barrier. The charge-separated excited states are strongly stabilized by the transfer of a proton from water to pyridine. The energies of the resulting biradical states intersect the potential-energy surface of the closed-shell ground state as a function of the proton-transfer coordinate. The resulting radical pair may dissociate to yield pyridinium and hydroxyl radicals. The photochemistry of the hypervalent pyridinium radical has been explored with the same computational methods. It has been found that a low-lying dissociative (2)πσ* excited state exists in pyridinium, which can lead to the photodetachment of the hydrogen atom. Overall, the H2O molecule can thus be dissociated into H˙ and OH˙ radicals by the absorption of two ultraviolet photons. The relevance of these results for solar water splitting and solar carbon dioxide reduction is discussed.

Thermo-Field Dynamics Approach to Photo-induced Electronic Transitions Driven by Incoherent Thermal Radiation.

The effects of thermal light-matter interaction on the dynamics of photo-induced electronic transitions in molecules are investigated using a novel first principles approach based on the thermo-field dynamics description of both the molecular vibrational modes and of the radiation field. The developed approach permits numerically accurate simulations of quantum dynamics of electronic/excitonic systems coupled to nuclear and photonic baths kept at different temperatures. The baths can be described by arbitrary spectral densities and can have any system-bath coupling strengths. In agreement with the results obtained previously by less rigorous methods, we show that the excitation process obtained by the continuous interaction with the suddenly turned-on thermal radiation field creates a mixed ensemble having a nonnegligible component consisting of a superposition of vibronic eigenstates which can sustain coherent oscillations for relatively long times. The results become especially relevant for the dynamics of electronic transitions upon sunlight excitation. Analytical results based on time-dependent perturbation theory support the numerical simulations and provide a simple interpretation of the time evolution of quantum observables.

Generating function approach to the calculation of spectral band shapes of free-base chlorin including Duschinsky and Herzberg-Teller effects.

A generating function approach to the calculation of spectral band shapes including Duschinsky and Herzberg-Teller effects is proposed and applied to the computation of the free-base chlorin Q absorption bands, using molecular geometries and normal vibrations obtained by density functional theory computations. The results clearly show that non-Condon effects can significantly affect the relative intensities of the weakest Q(y) and, to a lesser extent, Q(x) bands. The proposed approach is extremely powerful and can be used in the cases where the molecular size makes the direct calculation of Franck-Condon integrals by recurrence formulas prohibitive.