The diabatic picture of electron transfer, reaction barriers, and molecular dynamics

Imaging the Collision Energy-Dependent Charge-Transfer Dynamics between the Spin-Orbit Ground Ar+(2P3/2) Ion and CO

The collisional charge-transfer reaction between Ar+(2P3/2,1/2) and CO represents one of the most studied ion-molecule systems; many controversies persist among different studies, and the detailed quantum state-to-state charge-transfer dynamics remains unknown. Here, differential cross sections of the charge-transfer process between the spin-orbit ground Ar+(2P3/2) ion and CO are reported at three center-of-mass collision energies of 1.02, 0.72, and 0.40 eV using a home-built three-dimensional velocity-map imaging-based ion-molecule crossed beam setup. At all three collision energies, the direct energy resonant charge-transfer mechanism dominates the reaction, featuring predominantly forward scattering with the CO+ product population peaking at the v' = 6 and v' = 7 vibrational levels. Only at the lowest collision energy of 0.40 eV is the significant backward peaked scattering product observed, with CO+ populated from v' = 4 to v' = 8. There is no obvious evidence for the formation of CO+ in excited electronic state A2Π+, in qualitative accord with previous theoretical predictions.

Non-equilibrium quantum domain reconfiguration dynamics in a two-dimensional electronic crystal and a quantum annealer

Relaxation dynamics of complex many-body quantum systems trapped into metastable states is a very active field of research from both the theoretical and experimental point of view with implications in a wide array of topics from macroscopic quantum tunnelling and nucleosynthesis to non-equilibrium superconductivity and energy-efficient memory devices. In this work, we investigate quantum domain reconfiguration dynamics in the electronic superlattice of a quantum material using time-resolved scanning tunneling microscopy and unveil a crossover from temperature to noisy quantum fluctuation dominated dynamics. The process is modeled using a programmable superconducting quantum annealer in which qubit interconnections correspond directly to the microscopic interactions between electrons in the quantum material. Crucially, the dynamics of both the experiment and quantum simulation is driven by spectrally similar pink noise. We find that the simulations reproduce the emergent time evolution and temperature dependence of the experimentally observed electronic domain dynamics.

Universal Measure for the Impact of Adiabaticity on Quantum Transitions

Adiabaticity is crucial for our understanding of complex quantum dynamics and thus for advancing fundamental physics and technology, but its impact cannot yet be quantified in complex but common cases where dynamics is only partially adiabatic, several eigenstates are simultaneously populated and transitions between noneigenstates are of key interest. We construct a universally applicable measure that can quantify the adiabaticity of quantum transitions in an arbitrary basis. Our measure distinguishes transitions that occur due to the adiabatic change of populated system eigenstates from transitions that occur due to beating between several eigenstates and can handle nonadiabatic events. While all quantum dynamics fall within the scope of the measure, we demonstrate its usage and utility through two important material science problems-energy and charge transfer-where adiabaticity could be effected by nuclear motion and its quantification will aid not only in unraveling mechanisms but also in system design, for example, of light harvesting systems.

Comparison of Computational Strategies for the Calculation of the Electronic Coupling in Intermolecular Energy and Electron Transport Processes

Electronic couplings in intermolecular electron and energy transfer processes calculated by six different existing computational techniques are compared to nonorthogonal configuration interaction for fragments (NOCI-F) results. The paper addresses the calculation of the electronic coupling in diketopyrrolopyrol, tetracene, 5,5'-difluoroindigo, and benzene-Cl for hole and electron transport, as well as the local exciton and singlet fission coupling. NOCI-F provides a rigorous computational scheme to calculate these couplings, but its computational cost is rather elevated. The here-considered ab initio Frenkel-Davydov (AIFD), Dimer projection (DIPRO), transition dipole moment coupling, Michl-Smith, effective Hamiltonian, and Mulliken-Hush approaches are computationally less demanding, and the comparison with the NOCI-F results shows that the NOCI-F results in the couplings for hole and electron transport are rather accurately predicted by the more approximate schemes but that the NOCI-F exciton transfer and singlet fission couplings are more difficult to reproduce.

Simulating Electron Transfer Reactions in Solution: Radical-Polar Crossover

Single-electron transfer (SET) promotes a wide variety of interesting chemical transformations, but modeling of SET requires a careful treatment of electronic and solvent effects to give meaningful insight. Therefore, a combined constrained density functional theory and molecular mechanics (CDFT/MM) tool is introduced specifically for SET-initiated reactions. Mechanisms for two radical-polar crossover reactions involving the organic electron donors tetrakis(dimethylamino)ethylene (TDAE) and tetrathiafulvalene (TTF) were studied with the new tool. An unexpected tertiary radical intermediate within the TDAE system was identified, relationships between kinetics and substitution in the TTF system were explained, and the impact of the solvent environments on the TDAE and TTF reactions were examined. The results highlight the need for including solvent dynamics when quantifying SET kinetics and thermodynamics, as a free energy difference of >20 kcal/mol was observed. Overall, the new method informs mechanistic analysis of SET-initiated reactions and therefore is envisioned to be useful for studying reactions in the condensed phase.

Hexakis-TIPS-Alkynylated Nonacenes: Persistent and Processible

Four substituted nonacenes were prepared and characterized by UV-vis and EPR spectroscopy and X-ray crystallography. The compounds are the most stable and soluble nonacenes to date - due to six strategically placed triisopropylsilyl(TIPS)-ethynyl groups. They are stable for several weeks in the solid state. In dilute solution their half-life is 5-9 h. Crystal structure analyses of two nonacenes prove their structures. A nonacene derivative was tested in a solution-processed transistor and exhibits ambipolar charge transport (μe =0.007 cm2 /Vs; μh =0.023 cm2 /Vs).

Unified quantum theory of electrochemical kinetics by coupled ion-electron transfer

A general theory of coupled ion-electron transfer (CIET) is presented, which unifies Marcus kinetics of electron transfer (ET) with Butler-Volmer kinetics of ion transfer (IT). In the limit of large reorganization energy, the theory predicts normal Marcus kinetics of "electron-coupled ion transfer" (ECIT). In the limit of large ion transfer energies, the theory predicts Butler-Volmer kinetics of "ion-coupled electron transfer" (ICET), where the charge transfer coefficient and exchange current are connected to microscopic properties of the electrode/electrolyte interface. In the ICET regime, the reductive and oxidative branches of Tafel's law are predicted to hold over a wide range of overpotentials, bounded by the ion-transfer energies for oxidation and reduction, respectively. The probability distribution of transferring electron energies in CIET smoothly interpolates between a shifted Gaussian distribution for ECIT (as in the Gerischer-Marcus theory of ET) to an asymmetric, fat-tailed Meixner distribution centered at the Fermi level for ICET. The latter may help interpret asymmetric line shapes in x-ray photo-electron spectroscopy (XPS) and Auger electron spectroscopy (AES) for metal surfaces in terms of shake-up relaxation of the ionized atom and its image polaron by ICET. In the limit of large overpotentials, the theory predicts a transition to inverted Marcus ECIT, leading to a universal reaction-limited current for metal electrodes, dominated by barrierless quantum transitions. Uniformly valid, closed-form asymptotic approximations are derived that smoothly transition between the limiting rate expressions for ICET and ECIT for metal electrodes, using simple but accurate mathematical functions. The theory is applied to lithium intercalation in lithium iron phosphate (LFP) and found to provide a consistent description of the observed current dependence on overpotential, temperature and concentration. CIET theory thus provides a critical bridge between quantum electrochemistry and electrochemical engineering, which may find many other applications and extensions.

Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrodynamics

When molecules are coupled to an optical cavity, new light-matter hybrid states, so-called polaritons, are formed due to quantum light-matter interactions. With the experimental demonstrations of modifying chemical reactivities by forming polaritons under strong light-matter interactions, theorists have been encouraged to develop new methods to simulate these systems and discover new strategies to tune and control reactions. This review summarizes some of these exciting theoretical advances in polariton chemistry, in methods ranging from the fundamental framework to computational techniques and applications spanning from photochemistry to vibrational strong coupling. Even though the theory of quantum light-matter interactions goes back to the midtwentieth century, the gaps in the knowledge of molecular quantum electrodynamics (QED) have only recently been filled. We review recent advances made in resolving gauge ambiguities, the correct form of different QED Hamiltonians under different gauges, and their connections to various quantum optics models. Then, we review recently developed ab initio QED approaches which can accurately describe polariton states in a realistic molecule-cavity hybrid system. We then discuss applications using these method advancements. We review advancements in polariton photochemistry where the cavity is made resonant to electronic transitions to control molecular nonadiabatic excited state dynamics and enable new photochemical reactivities. When the cavity resonance is tuned to the molecular vibrations instead, ground-state chemical reaction modifications have been demonstrated experimentally, though its mechanistic principle remains unclear. We present some recent theoretical progress in resolving this mystery. Finally, we review the recent advances in understanding the collective coupling regime between light and matter, where many molecules can collectively couple to a single cavity mode or many cavity modes. We also lay out the current challenges in theory to explain the observed experimental results. We hope that this review will serve as a useful document for anyone who wants to become familiar with the context of polariton chemistry and molecular cavity QED and thus significantly benefit the entire community.

Extended Single-Electron Transfer Model and Dynamically Associated Energy Transfer Event in a Dual-Functional Catalyst System

Organic photocatalysis has been developed flourishingly to rely on bimolecular energy transfer (EnT) or oxidative/reductive electron transfer (ET), promoting a variety of synthetic transformations. However, there are rare examples to merge EnT and ET processes rationally within one chemical system, of which the mechanistic investigation still remains in its infancy. Herein, the first mechanistic illustration and kinetic assessments of the dynamically associated EnT and ET paths were conducted for realizing the C-H functionalization in a cascade photochemical transformation of isomerization and cyclization by using the dual-functional organic photocatalyst of riboflavin. An extended single-electron transfer model of transition-state-coupled dual-nonadiabatic crossings was explored to analyze the dynamic behaviors in the proton transfer-coupled cyclization. This can also be used to clarify the dynamic correlation with the EnT-driven E → Z photoisomerization that has been kinetically evaluated by using Fermi's golden rule with the Dexter model. The present computational results of electron structures and kinetic data contribute to a fundamental basis for understanding the photocatalytic mechanism of the combined operation of EnT and ET strategies, which will guide the design and manipulation for the implementation of multiple activation modes based on a single photosensitizer.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Calculation of exciton couplings based on density functional tight-binding coupled to state-interaction state-averaged ensemble-referenced Kohn-Sham approach

We introduce the combination of the density functional tight binding (DFTB) approach, including onsite correction (OC) and long-range corrected (LC) functional and the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method with extended active space involving four electrons and four orbitals [LC-OC-DFTB/SSR(4,4)], to investigate exciton couplings in multichromophoric systems, such as organic crystals and molecular aggregates. We employ the LC-OC-DFTB/SSR(4,4) method to calculate the excitonic coupling in anthracene and tetracene. As a result, the LC-OC-DFTB/SSR(4,4) method provides a reliable description of the locally excited (LE) state in a single chromophore and the excitonic couplings between chromophores with reasonable accuracy compared to the experiment and the conventional SSR(4,4) method. In addition, the thermal fluctuation of excitonic couplings from dynamic nuclear motion in an anthracene crystal with LC-OC-DFTB/SSR(4,4) shows a similar fluctuation of excitonic coupling and spectral density with those of first-principle calculations. We conclude that LC-OC-DFTB/SSR(4,4) is capable of providing reasonable features related to LE states, such as Frenkel exciton with efficient computational cost.

Subsystem density-functional theory: A reliable tool for spin-density based properties

Subsystem density-functional theory compiles a set of features that allow for efficiently calculating properties of very large open-shell radical systems such as organic radical crystals, proteins, or deoxyribonucleic acid stacks. It is computationally less costly than correlated ab initio wave function approaches and can pragmatically avoid the overdelocalization problem of Kohn-Sham density-functional theory without employing hard constraints on the electron-density. Additionally, subsystem density-functional theory calculations commonly start from isolated fragment electron densities, pragmatically preserving a priori specified subsystem spin-patterns throughout the calculation. Methods based on subsystem density-functional theory have seen a rapid development over the past years and have become important tools for describing open-shell properties. In this Perspective, we address open questions and possible developments toward challenging future applications in connection with subsystem density-functional theory for spin-dependent properties.

Diabatic Decomposition Perspective on the Role of Charge Transfer and Local Excitations in Thermally Activated Delayed Fluorescence

Thermally activated delayed fluorescence (TADF) is a phenomenon that relies on the upconversion of triplet excitons to singlet excitons by means of reverse intersystem crossing (rISC). It has been shown both experimentally and theoretically that the TADF mechanism depends on the interplay between charge transfer and local excitations. However, the difference between the diabatic and adiabatic character of the involved excited states is rarely discussed in the literature. Here we develop a diabatization procedure to implement a four-state model Hamiltonian to a set of TADF molecules. We provide physical interpretations of the Hamiltonian elements and show their dependence on the electronic state of the equilibrium geometry. We also demonstrate how vibrations affect the TADF efficiency by modifying the diabatic decomposition of the molecule. Finally, we provide a simple model that connects the diabatic Hamiltonian to the electronic properties relevant to TADF and show how this relationship translates into different optimization strategies for rISC, fluorescence, and overall TADF performance.

Ultrafast Symmetry-Breaking Charge Separation in a Perylene Bisimide Dimer Enabled by Vibronic Coupling and Breakdown of Adiabaticity

Perylene bisimides (PBIs) have received great attention in their applicability to optoelectronics. Especially, symmetry-breaking charge separation (SB-CS) in PBIs has been investigated to mimic the efficient light capturing and charge generation in natural light-harvesting systems. However, unlike ultrafast CS dynamics in donor-acceptor heterojunction materials, ultrafast SB-CS in a stacked homodimer has still been challenging due to excimer formation in the absence of rigidifying surroundings such as a special pair in the natural systems. Herein, we present the detailed mechanism of ultrafast photoinduced SB-CS occurring in a 1,7-bis(N-pyrrolidinyl) PBI dimer within a cyclophane. Through narrow-band and broad-band transient absorption spectroscopy, we demonstrate that ultrafast SB-CS in the dimer is enabled by the combination of (1) vibrationally coherent charge-transfer resonance-enhanced excimer formation and (2) breakdown of adiabaticity (formation of SB-CS diabats) in the excimer state via structural and solvent fluctuation. Quantum chemical calculations also underpin that the participation of strong electron-donating substituents in overall vibrational modes plays a crucial role in triggering the ultrafast SB-CS. Therefore, our work provides an alternative route to facilitate ultrafast SB-CS in PBIs and thereby establishes a novel strategy for the design of optoelectronic materials.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born-Huang formalism, covering different initial states, which either start on a single Born-Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential

Light-induced chemical processes are ubiquitous in nature and have widespread technological applications. For example, photoisomerization can allow a drug with a photo-switchable scaffold such as azobenzene to be activated with light. In principle, photoswitches with desired photophysical properties like high isomerization quantum yields can be identified through virtual screening with reactive simulations. In practice, these simulations are rarely used for screening, since they require hundreds of trajectories and expensive quantum chemical methods to account for non-adiabatic excited state effects. Here we introduce a diabatic artificial neural network (DANN), based on diabatic states, to accelerate such simulations for azobenzene derivatives. The network is six orders of magnitude faster than the quantum chemistry method used for training. DANN is transferable to azobenzene molecules outside the training set, predicting quantum yields for unseen species that are correlated with experiment. We use the model to virtually screen 3100 hypothetical molecules, and identify novel species with high predicted quantum yields. The model predictions are confirmed using high-accuracy non-adiabatic dynamics. Our results pave the way for fast and accurate virtual screening of photoactive compounds.

Cooperative molecular structure in polaritonic and dark states

The interaction of quantum light with matter is known to give rise to mixed light-matter states. An ensemble of identical molecules is discussed. The resulting hybrid light-matter states exhibit complex structure even if only a single vibrational coordinate per molecule is considered. Starting from the uniform situation where all molecules possess the same value of this coordinate, polaritons and dark states follow like in atoms, but are functions of this coordinate. It is proven that any point on a resulting polariton energy curve is a (local) minimum or maximum for distorting molecules perpendicular to this curve. It is shown how to explicitly compute the impact of distortion solely based on the data of a free molecule. The structure of the dark states and their behavior upon distortion is analyzed as well. Useful techniques are introduced and general results on, for example, minimum energy path, symmetry breaking and restoration, are obtained. The developed strategy is transferred to include several or even many nuclear degrees of freedom per molecule and it is demonstrated that the interplay of several vibrational degrees of freedom in a single molecule of the ensemble is expected to lead to qualitatively different physics. General consequences are discussed.

Spin–vibronic coupling in the quantum dynamics of a Fe(III) trigonal-bipyramidal complex

The presence of a high density of excited electronic states in the immediate vicinity of the optically bright state of a molecule paves the way for numerous photo-relaxation channels. In transition-metal complexes, the presence of heavy atoms results in a stronger spin–orbit coupling, which enables spin forbidden spin-crossover processes to compete with the spin-allowed internal conversion processes. However, no matter how effectively the states cross around the Franck–Condon region, the degree of vibronic coupling, of both relativistic and non-relativistic nature, drives the population distribution among these states. One such case is demonstrated in this work for the intermediate-spin Fe(III) trigonal-bipyramidal complex. A quantum dynamical investigation of the photo-deactivation mechanism in the Fe(III) system is presented using the multi-configurational time-dependent Hartree approach based on the vibronic Hamiltonian whose coupling terms are derived from the state-averaged complete active space self-consistent field/complete active space with second-order perturbation theory (CASPT2) calculations and spin–orbit coupling of the scalar-relativistic CASPT2 states. The results of this study show that the presence of a strong (non-relativistic) vibronic coupling between the optically bright intermediate-spin state and other low-lying states of the same spin-multiplicity overpowers the spin–orbit coupling between the intermediate-spin and high-spin states, thereby lowering the chances of spin-crossover while exhibiting ultrafast relaxation among the intermediate-spin states. In a special case, where the population transfer pathway via the non-relativistic vibronic coupling is blocked, the probability of the spin-crossover is found to increase. This suggests that a careful modification of the complex by incorporation of heavier atoms with stronger relativistic effects can enhance the spin-crossover potential of Fe(III) intermediate-spin complexes.

Importance of dynamical electron correlation in diabatic couplings of electron-exchange processes

We demonstrate the importance of the dynamical electron correlation effect in diabatic couplings of electron-exchange processes in molecular aggregates. To perform a multireference perturbation theory with large active space of molecular aggregates, an efficient low-rank approximation is applied to the complete active space self-consistent field reference functions. It is known that kinetic rates of electron-exchange processes, such as singlet fission, triplet-triplet annihilation, and triplet exciton transfer, are not sufficiently explained by the direct term of the diabatic couplings but efficiently mediated by the low-lying charge transfer states if the two molecules are in close proximity. It is presented in this paper, however, that regardless of the distance of the molecules, the direct term is considerably underestimated by up to three orders of magnitude without the dynamical electron correlation, i.e., the diabatic states expressed in the active space are not adequate to quantitatively reproduce the electron-exchange processes.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

The Seamless Connection of Local and Collective Excited States in Subsystem Time-Dependent Density Functional Theory

The theoretical understanding of photoinduced processes in multichromophoric systems requires, as an essential ingredient, the possibility of accurately describing their electronically excited states. However, the size of these systems often prohibits the usage of conventional electronic-structure methods, so that often multiscale approaches based on phenomenologically motivated models are employed. In contrast, subsystem time-dependent density functional theory (sTDDFT) allows for a subsystem-based ab initio description of multichromophoric systems and therefore allows for, in principle, an exact description of photoinduced processes. This Perspective aims to outline the theoretical foundations and commonly used practical realizations as well as to illustrate benefits of recent developments and open issues in the field of sTDDFT. Prospective, potential future applications and possible methodological developments are discussed.

Insightful understanding of charge transfer processes in metalated phthalocyanines

Marcus electron transfer theory coupling with quantum-mechanics (QM) calculations was applied to study the hole mobilities of a series of metalated phthalocyanine molecular crystals. The effect of metal on the...

Enabling Large-Scale Condensed-Phase Hybrid Density Functional Theory-Based Ab Initio Molecular Dynamics II: Extensions to the Isobaric-Isoenthalpic and Isobaric-Isothermal Ensembles

In the previous paper of this series [Ko, H.-Y. et al. J. Chem. Theory Comput. 2020, 16, 3757-3785], we presented a theoretical and algorithmic framework based on a localized representation of the occupied space that exploits the inherent sparsity in the real-space evaluation of the exact exchange (EXX) interaction in finite-gap systems. This was accompanied by a detailed description of exx, a massively parallel hybrid message-passing interface MPI/OpenMP implementation of this approach in Quantum ESPRESSO (QE) that enables linear scaling hybrid density functional theory (DFT)-based ab initio molecular dynamics (AIMD) in the microcanonical/canonical (NVE/NVT) ensembles of condensed-phase systems containing 500-1000 atoms (in fixed orthorhombic cells) with a wall time cost comparable to semi-local DFT. In this work, we extend the current capabilities of exx to enable hybrid DFT-based AIMD simulations of large-scale condensed-phase systems with general and fluctuating cells in the isobaric-isoenthalpic/isobaric-isothermal (NpH/NpT) ensembles. The theoretical extensions to this approach include an analytical derivation of the EXX contribution to the stress tensor for systems in general simulation cells with a computational complexity that scales linearly with system size. The corresponding algorithmic extensions to exx include optimized routines that (i) handle both static and fluctuating simulation cells with non-orthogonal lattice symmetries, (ii) solve Poisson's equation in general/non-orthogonal cells via an automated selection of the auxiliary grid directions in the Natan-Kronik representation of the discrete Laplacian operator, and (iii) evaluate the EXX contribution to the stress tensor. Using this approach, we perform a case study on a variety of condensed-phase systems (including liquid water, a benzene molecular crystal polymorph, and semi-conducting crystalline silicon) and demonstrate that the EXX contributions to the energy and stress tensor simultaneously converge with an appropriate choice of exx parameters. This is followed by a critical assessment of the computational performance of the extended exx module across several different high-performance computing architectures via case studies on (i) the computational complexity due to lattice symmetry during NpT simulations of three different ice polymorphs (i.e., ice Ih, II, and III) and (ii) the strong/weak parallel scaling during large-scale NpT simulations of liquid water. We demonstrate that the robust and highly scalable implementation of this approach in the extended exx module is capable of evaluating the EXX contribution to the stress tensor with negligible cost (<1%) as well as all other EXX-related quantities needed during NpT simulations of liquid water (with a very tight 150 Ry planewave cutoff) in ≈5.2 s ((H₂O)₁₂₈) and ≈6.8 s ((H₂O)₂₅₆) per AIMD step. As such, the extended exx module presented in this work brings us another step closer to routinely performing hybrid DFT-based AIMD simulations of sufficient duration for large-scale condensed-phase systems across a wide range of thermodynamic conditions.

Extended Mulliken-Hush Method with Applications to the Theoretical Study of Electron Transfer

A novel adiabatic-to-diabatic (ATD) transformation strategy, namely, the extended Mulliken-Hush (XMH) method, is proposed to evaluate diabatic properties including electronic couplings, potential energy surfaces, and their crossings. The XMH method is developed by adopting our recently proposed ATD transformation formula of a general vectorial physical observable, in which a useful ATD transformation is further determined by using an auxiliary dipole between localized frontier orbitals as a simple approximation of the diabatic transition dipole. The XMH method is simple and practical that provides a flexible way to construct diabatic states. To some extent, it can be regarded as an extension of the generalized Mulliken-Hush (GMH) method since the latter takes a stronger approximation, in which the diabatic transition dipole is assumed to be vanishing. Test calculations on the HeH₂⁺ system show that the electronic couplings predicted by the XMH method are closer to the ones calculated by the valence bond block-diagonalization approach than the GMH ones since the XMH method takes into account both the magnitude and direction of the diabatic transition dipole, which is consistent with the properties of this molecule. In the study of electron transfer in the two kinds of donor-bridge-acceptor systems, the XMH method maintains the simplicity of the GMH method and gives reasonable results even when the latter fails, wherein the diabatic transition dipole is nearly perpendicular to the difference of the initial and final adiabatic dipoles. More importantly, the XMH method can be easily combined with high-level electronic structure methods, in which the properties of the ground and excited states may be more accurately calculated, and hence, one may expect that further development of the XMH method would result in a general computational model for studying electron transfer reactions.

Machine Learning Approach to Calculate Electronic Couplings between Quasi-diabatic Molecular Orbitals: The Case of DNA

Diabatization of one-electron states in flexible molecular aggregates is a great challenge due to the presence of surface crossings between molecular orbital (MO) levels and the complex interaction between MOs of neighboring molecules. In this work, we present an efficient machine learning approach to calculate electronic couplings between quasi-diabatic MOs without the need of nonadiabatic coupling calculations. Using MOs of rigid molecules as references, the MOs that can be directly regarded to be quasi-diabatic in molecular dynamics are selected out, state tracked, and phase corrected. On the basis of this information, artificial neural networks are trained to characterize the structure-dependent onsite energies of quasi-diabatic MOs and the intermolecular electronic couplings. A representative sequence of DNA is systematically studied as an illustration. Smooth time evolution of electronic couplings in all base pairs is obtained with quasi-diabatic MOs. In particular, our method can calculate electronic couplings between different quasi-diabatic MOs independently, and thus, this possesses unique advantages in many applications.

N-Body Reduced Density Matrix-Based Valence Bond Theory and Its Applications in Diabatic Electronic-Structure Computations

Valence bond (VB) theory, as a helpful complement to the more popular molecular orbital theory, is a fundamental electronic-structure theory that aims at interpreting molecular structure and chemical reactions in a lucid way. Both theoretical and experimental chemists have shown great interest in VB theory because of its capability of providing intuitive insight into the nature of chemical bonding and the mechanism of chemical reaction in a clear and comprehensible language rooted in Lewis structure. Therefore, there is a great call for the renaissance of VB theory. Nevertheless, this is possible only after a series of methods and algorithms were developed and efficiently implemented in user-friendly programs so as to serve computational chemists for general applications. In the past three decades, we have devoted a great amount of scientific enthusiasm toward this goal. In this Account, we will concisely summarize and briefly but insightfully discuss recent developments in ab initio VB theory, especially the N-body reduced density matrices (RDM)-based approach and its applications in diabatic electronic-structure computations, which is very useful for the vivid interpretation of many fundamental chemical processes such as electron and energy transfers. Furthermore, because of the fundamentally important role that the diabatic state plays in electron and energy transfers, which are two frontier research topics in both molecular and biochemical sciences, there are a broad range of applications that VB theory can handle.We start by briefly reviewing the general feature of ab initio VB wave functions. In particular, we focus on the multistructural ab initio VB theory that uses strictly localized orbitals, including the fundamental VB self-consistent field (VBSCF) and two post-SCF methods, VBCI and VBPT2, that use the VBSCF wave function as reference. We then allot a section to describing the recent developments of the RDM-based VB approach in the second quantization language. In this section, the enhanced Wick theorem is first outlined, followed by a brief discussion of its applications in evaluating VBSCF energy gradients and a Hessian with respect to the orbital expansion coefficients, together with a short review of the implementation of an automatic formula and code generator (AFCG) designed for many-body methods with nonorthogonal orbitals. Then, we introduce the application of the RDM-based approach in implementing the post-SCF method that addresses dynamic electronic correlation via perturbation theory, viz., the icVBPT2 method that adopts an internal contraction technique naturally. We finish this section by incorporating VB theory with the concept of seniority number, in which the tensor analysis technique is carefully exploited with the RDM-based approach, resulting in significant improvements in both the number of the active electrons/orbitals and in the speedup of the computational efficiency, thus pushing VB theory to its new limit. With these achievements available, we present the applications of VB theory in diabatic electronic-structure computations by using the intuitive insight rendered by VB theory. Therefore, we believe that there is a bright future in VB theory with true opportunities and new challenges coexisting both for theoretical developments and computational applications.

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.

Solvation sheath reorganization enables divalent metal batteries with fast interfacial charge transfer kinetics

[Figure: see text].

Molecular Dynamics Simulations of Bimolecular Electron Transfer: the Distance-Dependent Electronic Coupling

Understanding the distance dependence of the parameters underpinning Marcus theory is imperative when interpreting the results of experiments on electron transfer (ET). Unfortunately, most of these parameters are difficult or impossible to access directly with experiments, necessitating the use of computer simulations to model them. In this work, we use molecular dynamics simulations in conjunction with constrained density functional theory calculations to study the distance dependence of the electronic coupling matrix element, |H_RP|, for bimolecular ET. Contrary to what is typically assumed for such intermolecular reactions, we find that the magnitude of |H_RP| does not decay exponentially with the center-of-mass separation of the reactants, r_COM. The addition of other simple measures of donor/acceptor (D/A) orientation did not improve the correlation of |H_RP| with r_COM. Using the minimum distance separation, r_min, of the reactants as the structural descriptor allowed the system to be partitioned into high-coupling/close-contact and low-coupling/non-contact regimes, but large fluctuations of |H_RP| were still found for the close-contact reactant pairs. Despite the persistent large fluctuations of |H_RP|, its mean value was found to decay piecewise exponentially with increasing r_min, which was attributed to significant changes in the average D/A pair structure. The results herein advise one to use caution when interpreting the experimental results derived from spherical reactant models of bimolecular ET.

Ab initio symmetric quasi-classical approach to investigate molecular Tully models

We perform on-the-fly non-adiabatic molecular dynamics simulations using the symmetrical quasi-classical (SQC) approach with the recently suggested molecular Tully models: ethylene and fulvene. We attempt to provide benchmarks of the SQC methods using both the square and triangle windowing schemes as well as the recently proposed electronic zero-point-energy correction scheme (the so-called γ correction). We use the quasi-diabatic propagation scheme to directly interface the diabatic SQC methods with adiabatic electronic structure calculations. Our results showcase the drastic improvement of the accuracy by using the trajectory-adjusted γ-corrections, which outperform the widely used trajectory surface hopping method with decoherence corrections. These calculations provide useful and non-trivial tests to systematically investigate the numerical performance of various diabatic quantum dynamics approaches, going beyond simple diabatic model systems that have been used as the major workhorse in the quantum dynamics field. At the same time, these available benchmark studies will also likely foster the development of new quantum dynamics approaches based on these techniques.

Nonadiabatic dynamics of cobalt tricarbonyl nitrosyl for ligand dissociation induced by electronic excitation

We utilize real-time time-dependent density functional theory and Ehrenfest dynamics scheme to investigate excited-state nonadiabatic dynamics of ligand dissociation of cobalt tricarbonyl nitrosyl, Co(CO)3NO, which is a precursor used for cobalt growth in advanced technologies, where the precursor's reaction is enhanced by electronic excitation. Based on the first-principles calculations, we demonstrate two dissociation pathways of the NO ligand on the precursor. Detailed electronic structures are further analyzed to provide an insight into dynamics following the electronic excitations. This study sheds light on computational demonstration and underlying mechanism of the electronic-excitation-induced dissociation, especially in molecules with complex chemical bonds such as the Co(CO)3NO.

Electron transfer pathways from quantum dynamics simulations

This work explores the possibility of simulating an electron transfer process between a donor and an acceptor in real time using time-dependent density functional theory electron dynamics. To achieve this objective, a central issue to resolve is the definition of the initial state. This must be a non-equilibrium electronic state able to trigger the charge transfer dynamics; here, two schemes are proposed to prepare such states. One is based on the combination of the density matrices of the donor and acceptor converged separately with appropriate charges (for example, -1 for the donor and +1 for the acceptor). The second approach relied on constrained DFT to localize the charge on each fragment. With these schemes, electron transfer processes are simulated in different model systems of increasing complexity: an atomic hydrogen dimer, a polyacetylene chain, and the active site of the T. cruzi hybrid type A heme peroxidase, for which two possible electron transfer paths have been postulated. For the latter system, the present methodology applied in a hybrid Quantum Mechanics - Molecular Mechanics framework allows us to establish the relative probabilities of each path and provides insight into the inhibition of the electron transfer provoked by the substitution of tryptophan by phenylalanine in the W233F mutant.

Tunneling and Nonadiabatic Effects on a Proton-Coupled Electron Transfer Model for the Qo Site in Cytochrome bc1

Cytochrome bc₁ is a fundamental enzyme for cellular respiration and photosynthesis. This dimeric protein complex catalyzes a proton-coupled electron transfer (PCET) from the reduced coenzyme-Q substrate (Q) to a bimetallic iron-sulfur cluster in the Q_o active site. Herein, we combine molecular dynamics simulations of the complete cytochrome bc₁ protein with electronic-structure calculations of truncated models and a semiclassical tunneling theory to investigate the electron-proton adiabaticity of the initial reaction catalyzed in the Q_o site. After sampling possible orientations between the Q substrate and a histidine side chain that functions as hydrogen acceptor, we find that a truncated model composed by ubiquinol-methyl and imidazole-iron(III)-sulfide captures the expected changes in oxidation and spin states of the electron donor and acceptor. Diabatic electronic surfaces obtained for this model with multiconfigurational wave function calculations demonstrate that this reaction is electronic nonadiabatic, and proton tunneling is faster than mixing of electronic configurations. These results indicate the formalism that should be used to calculate vibronic couplings and kinetic parameters for the initial reaction in the Q_o site of cytochrome bc₁. This framework for molecular simulation may also be applied to investigate other PCET reactions in the Q-cycle or in various metalloproteins that catalyze proton translocation coupled to redox processes.

Chirality-Induced Spin Coherence in Electron Transfer Reactions

Recently, there has been much interest in the chirality-induced spin selectivity effect, whereby electron spin polarization, which is dependent on molecular chirality, is produced in electrode-molecule electron transfer processes. Naturally, one might consider if a similar effect can be observed in simple molecular charge transfer reactions, for example, in light-induced electron transfer from an electron donor to an electron acceptor. In this work, I explore the effect of electron transfer on spins in chiral single radicals and chiral radical pairs using Nakajima-Zwanzig theory. In these cases, chirality, in conjuction with spin-orbit coupling, does not lead to spin polarization, but instead, the electron transfer generates quantum coherence between spins states. In principle, this chirality-induced spin coherence could manifest in a range of experiments, and in particular, I demonstrate that the out of phase electron spin echo envelope modulation pulse electron paramagnetic resonance experiment would be able to detect this effect in oriented radical pairs.

Electronic Couplings for Photoinduced Charge Transfer and Excitation Energy Transfer Based on Fragment Particle-Hole Densities

A new scheme is proposed to calculate the electronic couplings for photoinduced charge transfer and excitation energy transfer for both singlet and triplet states. In this scheme, the locally excited and charge-transfer states are constructed from the adiabatic ones by maximally localizing the particle (i.e., electron) and hole densities in terms of predefined molecular fragments. The construction process, after which the electronic couplings are directly obtained, is highly efficient and can be combined with various kinds of preliminary electronic structure calculations as long as the adiabatic excitation energies and transition densities are available. The method also applies to the systems with multiple charge or excitation centers. Its validity is demonstrated by the applications to the 6,13-dichloropentacene dimer and tetramer and the C₆₀-Zn porphyrin dyad. The results reveal that the environment has a strong impact on the electronic couplings and can even enlarge those for long-range charge transfer by several orders of magnitude.

Unconventional singlet fission materials

Singlet fission (SF) is a photophysical downconversion pathway, in which a singlet excitation transforms into two triplet excited states. As such, it constitutes an exciton multiplication generation process, which is currently at the focal point for future integration into solar energy conversion devices. Beyond this, various other exciting applications were proposed, including quantum cryptography or organic light emitting diodes. Also, the mechanistic understanding evolved rapidly during the last year. Unfortunately, the number of suitable SF-chromophores is still limited. This is per se problematic, considering the wide range of envisaged applicability. With that in mind, we emphasize uncommon SF-scaffolds and outline requirements as well as strategies to expand the chromophore pool of SF-materials.

Theoretical Description of the Primary Proton-Coupled Electron Transfer Reaction in the Cytochrome bc1 Complex

The cytochrome bc₁ complex is a transmembrane enzymatic protein complex that plays a central role in cellular energy production and is present in both photosynthetic and respiratory chain organelles. Its reaction mechanism is initiated by the binding of a quinol molecule to an active site, followed by a series of charge transfer reactions between the quinol and protein subunits. Previous work hypothesized that the primary reaction was a concerted proton-coupled electron transfer (PCET) reaction because of the apparent absence of intermediate states associated with single proton or electron transfer reactions. In the present study, the kinetics of the primary bc₁ complex PCET reaction is investigated with a vibronically nonadiabatic PCET theory in conjunction with all-atom molecular dynamics simulations and electronic structure calculations. The computed rate constants and relatively high kinetic isotope effects are consistent with experimental measurements on related biomimetic systems. The analysis implicates a concerted PCET mechanism with significant hydrogen tunneling and nonadiabatic effects in the bc₁ complex. Moreover, the employed theoretical framework is shown to serve as a general strategy for describing PCET reactions in bioenergetic systems.

Multiscale modelling of charge transport in P3HT:DIPBI bulk heterojunction organic solar cells

Charge transport properties of a P3HT:DIPBI bulk heterojunction solar cell are modelled by kinetic Monte Carlo simulations based on a morphology obtained from coarse-grained molecular dynamics. Different methods for calculating the hopping integrals entering the charge transfer rates are compared and calibrated for hole transport in amorphous P3HT. The influence of intermolecular and intramolecular charge transfer on the total charge carrier mobility and hence the power conversion efficiency is investigated in detail. An analysis of the most probable pathways with low resistance for hole transport is performed, establishing a connection between charge mobility and morphology.

Block-Localized Excitation for Excimer Complex and Diabatic Coupling

We describe a block-localized excitation (BLE) method to carry out constrained optimization of block-localized orbitals for constructing valence bond-like, diabatic excited configurations using multistate density functional theory (MSDFT). The method is an extension of the previous block-localized wave function method through a fragment-based ΔSCF approach to optimize excited determinants within a molecular complex. In BLE, both the number of electrons and the electronic spin of different fragments in a whole system can be constrained, whereas electrostatic, exchange, and polarization interactions among different blocks can be fully taken into account of. To avoid optimization collapse to unwanted states, a ΔSCF projection scheme and a maximum overlap of wave function approach have been presented. The method is illustrated by the excimer complex of two naphthalene molecules. With a minimum of eight spin-adapted configurational state functions, it was found that the inversion of L_a^- and L_b^- states near the optimal structure of the excimer complex is correctly produced, which is in quantitative agreement with DMRG-CASPT2 calculations and experiments. Trends in the computed transfer integrals associated with excited-state energy transfer both in the singlet and triplet states are discussed. The results suggest that MSDFT may be used as an efficient approach to treat intermolecular interactions in excited states with a minimal active space (MAS) for interpretation of the results and for dynamic simulations, although the selection of a small active space is often system dependent.

Excited state diabatization on the cheap using DFT: Photoinduced electron and hole transfer

Excited state electron and hole transfer underpin fundamental steps in processes such as exciton dissociation at photovoltaic heterojunctions, photoinduced charge transfer at electrodes, and electron transfer in photosynthetic reaction centers. Diabatic states corresponding to charge or excitation localized species, such as locally excited and charge transfer states, provide a physically intuitive framework to simulate and understand these processes. However, obtaining accurate diabatic states and their couplings from adiabatic electronic states generally leads to inaccurate results when combined with low-tier electronic structure methods, such as time-dependent density functional theory, and exorbitant computational cost when combined with high-level wavefunction-based methods. Here, we introduce a density functional theory (DFT)-based diabatization scheme that directly constructs the diabatic states using absolutely localized molecular orbitals (ALMOs), which we denote as Δ-ALMO(MSDFT2). We demonstrate that our method, which combines ALMO calculations with the ΔSCF technique to construct electronically excited diabatic states and obtains their couplings with charge-transfer states using our MSDFT2 scheme, gives accurate results for excited state electron and hole transfer in both charged and uncharged systems that underlie DNA repair, charge separation in donor-acceptor dyads, chromophore-to-solvent electron transfer, and singlet fission. This framework for the accurate and efficient construction of excited state diabats and evaluation of their couplings directly from DFT thus offers a route to simulate and elucidate photoinduced electron and hole transfer in large disordered systems, such as those encountered in the condensed phase.

Electronic couplings for photo-induced processes from subsystem time-dependent density-functional theory: The role of the diabatization

Subsystem time-dependent density-functional theory (sTDDFT) making use of approximate non-additive kinetic energy (NAKE) functionals is known to be capable of describing excitation energy transfer processes in a variety of applications. Here, we show that sTDDFT, especially when combined with projection-based embedding (PbE), can be employed for the entire range of photo-induced electronic couplings essential for modeling photophysical properties of complex chemical and biological systems and therefore represents a complete toolbox for this class of problems. This means that it is capable of capturing the interaction/coupling associated with local- and charge-transfer (CT) excitons. However, this requires the choice of a reasonable diabatic basis. We therefore propose different diabatization strategies of the virtual orbital space in PbE-sTDDFT and show how CT excitations can be included in sTDDFT using NAKE functionals via a phenomenological approach. Finally, these electronic couplings are compared to couplings from a multistate fragment excitation difference (FED)-fragment charge difference (FCD) diabatization procedure. We show that both procedures, multistate FED-FCD and sTDDFT (with the right diabatization procedure chosen), lead to an overall good agreement for the electronic couplings, despite differences in their general diabatization strategy. We conclude that the entire range of photo-induced electronic couplings can be obtained using sTDDFT (with the right diabatization procedure chosen) in a black-box manner.