Nonadiabatic dynamics of molecules interacting with metal surfaces: Extending the hierarchical equations of motion and Langevin dynamics approach to position-dependent metal-molecule couplings.

Electronic friction and Langevin dynamics is a popular mixed quantum-classical method for simulating the nonadiabatic dynamics of molecules interacting with metal surfaces, as it can be computationally more efficient than fully quantum approaches. In this work, we extend the theory of electronic friction within the hierarchical equations of motion formalism to models with a position-dependent metal-molecule coupling. We show that the addition of a position-dependent metal-molecule coupling adds new contributions to the electronic friction and other forces, which are highly relevant for many physical processes. Our expressions for the electronic forces within the Langevin equation are valid both in and out of equilibrium and for molecular models containing strong interactions. We demonstrate the approach by applying it to different models of interest.

Intramolecular singlet fission: Quantum dynamical simulations including the effect of the laser field.

In the previous work [Reddy et al., J. Chem. Phys. 151, 044307 (2019)], we have analyzed the dynamics of the intramolecular singlet fission process in a series of prototypical pentacene-based dimers, where the pentacene monomers are covalently bonded to a phenylene linker in ortho, meta, and para positions. The results obtained were qualitatively consistent with the experimental data available, showing an ultrafast population of the multiexcitonic state that mainly takes place via a mediated (superexchange-like) mechanism involving charge transfer and doubly excited states. Our results also highlighted the instrumental role of molecular vibrations in the process as a sizable population of the multiexcitonic state could only be obtained through vibronic coupling. Here, we extend these studies and investigate the effect of the laser field on the dynamics of intramolecular singlet fission by explicitly including the coupling to the laser field in our model. In this manner, and by selectively tuning the laser field to the different low-lying absorption bands of the systems investigated, we analyze the wavelength dependence of the intramolecular singlet fission process. In addition, we have also analyzed how the nature of the initially photoexcited electronic state (either localized or delocalized) affects its dynamics. Altogether, our results provide new insights into the design of intramolecular singlet fission-active molecules.

Nonadiabatic dynamics of molecules interacting with metal surfaces: A quantum-classical approach based on Langevin dynamics and the hierarchical equations of motion.

A novel mixed quantum-classical approach to simulating nonadiabatic dynamics of molecules at metal surfaces is presented. The method combines the numerically exact hierarchical equations of motion approach for the quantum electronic degrees of freedom with Langevin dynamics for the classical degrees of freedom, namely, low-frequency vibrational modes within the molecule. The approach extends previous mixed quantum-classical methods based on Langevin equations to models containing strong electron-electron or quantum electronic-vibrational interactions, while maintaining a nonperturbative and non-Markovian treatment of the molecule-metal coupling. To demonstrate the approach, nonequilibrium transport observables are calculated for a molecular nanojunction containing strong interactions.

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.

Protein charge transfer far from equilibrium: a theoretical perspective.

Potential differences for protein-assisted electron transfer across lipid bilayers or in bio-nano setups can amount to several 100 mV; they lie far outside the range of linear response theory. We describe these situations by Pauli-master equations that are based on Marcus theory of charge transfer between self-trapped electrons and that obey Kirchhoff's current law. In addition, we take on-site blockade effects and a full non-linear response of the local potentials into account. We present analytical and numerical current-potential curves and electron populations for multi-site model systems and biological electron transfer chains. Based on these, we provide empirical rules for electron populations and chemical potentials along the chain. The Pauli-master mean-field results are validated by kinetic Monte Carlo simulations. We briefly discuss the biochemical and evolutionary aspects of our findings.

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.

Nonequilibrium reaction rate theory: Formulation and implementation within the hierarchical equations of motion approach.

The study of chemical reactions in environments under nonequilibrium conditions has been of interest recently in a variety of contexts, including current-induced reactions in molecular junctions and scanning tunneling microscopy experiments. In this work, we outline a fully quantum mechanical, numerically exact approach to describe chemical reaction rates in such nonequilibrium situations. The approach is based on an extension of the flux correlation function formalism to nonequilibrium conditions and uses a mixed real and imaginary time hierarchical equations of motion approach for the calculation of rate constants. As a specific example, we investigate current-induced intramolecular proton transfer reactions in a molecular junction for different applied bias voltages and molecule-lead coupling strengths.

Unraveling current-induced dissociation mechanisms in single-molecule junctions.

Understanding current-induced bond rupture in single-molecule junctions is both of fundamental interest and a prerequisite for the design of molecular junctions, which are stable at higher-bias voltages. In this work, we use a fully quantum mechanical method based on the hierarchical quantum master equation approach to analyze the dissociation mechanisms in molecular junctions. Considering a wide range of transport regimes, from off-resonant to resonant, non-adiabatic to adiabatic transport, and weak to strong vibronic coupling, our systematic study identifies three dissociation mechanisms. In the weak and intermediate vibronic coupling regime, the dominant dissociation mechanism is stepwise vibrational ladder climbing. For strong vibronic coupling, dissociation is induced via multi-quantum vibrational excitations triggered either by a single electronic transition at high bias voltages or by multiple electronic transitions at low biases. Furthermore, the influence of vibrational relaxation on the dissociation dynamics is analyzed and strategies for improving the stability of molecular junctions are discussed.

Quantum dynamical simulation of intramolecular singlet fission in covalently coupled pentacene dimers.

We analyze the dynamics of intramolecular singlet fission in a series of pentacene-based dimers consisting of two pentacene-like chromophores covalently bonded to a phenylene linker in ortho, meta, and para positions. The study uses a quantum dynamical approach that employs a model vibronic Hamiltonian whose parameters are obtained using multireference perturbation theory methods. The results highlight the different role of the direct and mediated mechanism in these systems, showing that the population of the multiexcitonic state, corresponding to the first step of the intramolecular singlet fission process, occurs mainly through a superexchange-like mechanism involving doubly excited or charge transfer states that participate in the process in a virtual way. In addition, the systems investigated provide insight into the roles that built-in geometrical constraints and the electronic structure of the spacer play in the intramolecular singlet fission process.

High-Voltage-Assisted Mechanical Stabilization of Single-Molecule Junctions.

Resonant tunneling is an efficient mechanism for charge transport through nanoscale conductance junctions due to the relatively high currents involved. However, continuous charging and discharging cycles of the nanoconductor during resonant tunneling often lead to mechanical instability. The realization of efficient nanoscale electronic components therefore depends to a large extent on the ability to mechanically stabilize them during resonant transport. In this work, we focus on single-molecule junctions, demonstrating that their mechanical stability during resonant transport can be increased by increasing the bias voltage. This counter-intuitive effect is attributed to the energy dependence of the molecule-lead coupling densities, which promote the rate of transport-induced cooling of molecular vibrations at higher voltages. The required energy dependence is characteristic of realistic electrodes (such as graphene), which cannot be modeled within the commonly invoked wide-band approximation. Our research provides new guidelines for the design of mechanically stable molecular devices operating in the regime of resonant charge transport and demonstrates these guidelines while considering realistic features of single-molecule junctions.

Perspective: Theory of quantum transport in molecular junctions.

Molecular junctions, where single molecules are bound to metal or semiconductor electrodes, represent a unique architecture to investigate molecules in a distinct nonequilibrium situation and, in a broader context, to study basic mechanisms of charge and energy transport in a many-body quantum system at the nanoscale. Experimental studies of molecular junctions have revealed a wealth of interesting transport phenomena, the understanding of which necessitates theoretical modeling. The accurate theoretical description of quantum transport in molecular junctions is challenging because it requires methods that are capable to describe the electronic structure and dynamics of molecules in a condensed phase environment out of equilibrium, in some cases with strong electron-electron and/or electronic-vibrational interaction. This perspective discusses recent progress in the theory and simulation of quantum transport in molecular junctions. Furthermore, challenges are identified, which appear crucial to achieve a comprehensive and quantitative understanding of transport in these systems.

A broadened classical master equation approach for treating electron-nuclear coupling in non-equilibrium transport.

We extend the broadened classical master equation (bCME) approach [W. Dou and J. E. Subotnik, J. Chem. Phys. 144, 024116 (2016)] to the case of two electrodes, such that we may now calculate non-equilibrium transport properties when molecules come near metal surfaces and there is both strong electron-nuclear and strong metal-molecule coupling. By comparing against a numerically exact solution, we show that the bCME usually works very well, provided that the temperature is high enough that a classical treatment of nuclear motion is valid. Finally, in the low temperature (quantum) regime, we suggest a means to incorporate broadening effects in the quantum master equation (QME). This bQME works well for fairly low temperatures.

On the memory kernel and the reduced system propagator.

We relate the memory kernel in the Nakajima-Zwanzig-Mori time-convolution approach to the reduced system propagator which is often used to obtain the kernel in the Tokuyama-Mori time-convolutionless approach. The connection provides a robust and simple formalism to compute the memory kernel for a generalized system-bath model circumventing the need to compute high order system-bath observables, thus streamlining the use of numerically exact solvers for calculating the memory kernel. We illustrate this for a model system with electron-electron and electron-phonon couplings, driven away from equilibrium.

Singlet fission in pentacene dimers.

Singlet fission (SF) has the potential to supersede the traditional solar energy conversion scheme by means of boosting the photon-to-current conversion efficiencies beyond the 30% Shockley-Queisser limit. Here, we show unambiguous and compelling evidence for unprecedented intramolecular SF within regioisomeric pentacene dimers in room-temperature solutions, with observed triplet quantum yields reaching as high as 156 ± 5%. Whereas previous studies have shown that the collision of a photoexcited chromophore with a ground-state chromophore can give rise to SF, here we demonstrate that the proximity and sufficient coupling through bond or space in pentacene dimers is enough to induce intramolecular SF where two triplets are generated on one molecule.

A multilayer multiconfiguration time-dependent Hartree simulation of the reaction-coordinate spin-boson model employing an interaction picture.

The multilayer multiconfiguration time-dependent Hartree method is applied in an interaction picture to simulate dynamics of the spin-boson model in the reaction-coordinate representation. The use of the interaction picture allows a more effective description of correlation effects, especially when the coupling strength between the reaction coordinate and the bath is very strong. Examples show that in most physical regimes the efficiency is improved significantly, in some cases up to several orders of magnitude. This opens up new avenues for studying quantum dynamical problems.

Employing an interaction picture to remove artificial correlations in multilayer multiconfiguration time-dependent Hartree simulations.

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is implemented in the interaction picture to allow a more effective description of correlation effects. It is shown that the artificial correlation present in the original Schrödinger picture can be removed with an appropriate choice of the zeroth-order Hamiltonian. Thereby, operators in the interaction picture are obtained through time-dependent unitary transformations, which have negligible computational cost compared with other parts of the ML-MCTDH algorithm. The efficiency of the method is demonstrated by application to a model of vibrationally coupled charge transport in molecular junctions.

Single-molecule junctions with epitaxial graphene nanoelectrodes.

On the way to ultraflat single-molecule junctions with transparent electrodes, we present a fabrication scheme based on epitaxial graphene nanoelectrodes. As a suitable molecule, we identified a molecular wire with fullerene anchor groups. With these two components, stable electrical characteristics could be recorded. Electrical measurements show that single-molecule junctions with graphene and with gold electrodes display a striking agreement. This motivated a hypothesis that the differential conductance spectra are rather insensitive to the electrode material. It is further corroborated by the assignment of asymmetries and spectral features to internal molecular degrees of freedom. The demonstrated open-access graphene electrodes and the electrode-insensitive molecules provide a model system that will allow for a thorough investigation of an individual single-molecule contact with additional probes.

Semiclassical Green's functions and an instanton formulation of electron-transfer rates in the nonadiabatic limit.

We present semiclassical approximations to Green's functions of multidimensional systems, extending Gutzwiller's work to the classically forbidden region. Based on steepest-descent integrals over these functions, we derive an instanton method for computing the rate of nonadiabatic reactions, such as electron transfer, in the weak-coupling limit, where Fermi's golden-rule can be employed. This generalizes Marcus theory to systems for which the environment free-energy curves are not harmonic and where nuclear tunnelling plays a role. The derivation avoids using the Im F method or short-time approximations to real-time correlation functions. A clear physical interpretation of the nuclear tunnelling processes involved in an electron-transfer reaction is thus provided. In Paper II [J. O. Richardson, J. Chem. Phys. 143, 134116 (2015)], we discuss numerical evaluation of the formulae.