Revisiting electronic couplings and incoherent hopping models for electron transport in crystalline C60 at ambient temperatures

Non-negligible Outer-Shell Reorganization Energy for Charge Transfer in Nonpolar Systems

Many charge-transporting molecular systems function as ordered or disordered arrays of solid state materials composed of nonpolar (or weakly polar) molecules. Due to low dielectric constants for nonpolar systems, it is common to ignore the effects of outer-shell reorganization energy (λout). However, ignoring λout has not been properly supported and it can severely impact predictions and insights derived. Here, we estimate λout by two means: from experimental ultraviolet photoelectron spectra, in which vibronic progression in these spectra can be fitted with the widths of peaks determining the low-frequency component in reorganization energy, regarded to be closely associated with λout, and from molecular dynamic (MD) simulation of nonpolar molecules, in which disorder or fluctuation statistics for energies of charged molecules are calculated. An upper bound for λout was obtained as 505 and 549 meV for crystalline anthracene (140 K) and pentacene (50 K), respectively, by fitting of experimental data, and 212 and 170 meV, respectively, from MD simulations. These values are comparable to the inner-sphere reorganization energy (λin) arising from intramolecular vibration. With corresponding spectral density functions calculated, we found that λout is influenced both by low- and high-frequency dynamics, in which the former arises from constrained translational and rotational motions of surrounding molecules. In an amorphous state, about half of the λout's obtained are from high-frequency components, which is quite different from the conventional polar solvation. Moreover, crystalline systems exhibit super-Ohmic spectral density, whereas amorphous systems are sub-Ohmic.

Electron Localization and Mobility in Monolayer Fullerene Networks

The novel 2D quasi-hexagonal phase of covalently bonded fullerene molecules (qHP C60), the so-called graphullerene, has displayed far superior electron mobilities, if compared to the parent van der Waals three-dimensional crystal (vdW C60). Herein, we present a comparative study of the electronic properties of vdW and qHP C60 using state-of-the-art electronic-structure calculations and a full quantum-mechanical treatment of electron transfer. We show that both materials entail polaronic localization of electrons with similar binding energies (≈0.1 eV) and, therefore, they share the same charge transport via polaron hopping. In fact, we quantitatively reproduce the sizable increment of the electron mobility measured for qHP C60 and identify its origin in the increased electronic coupling between C60 units.

Charge Transport in Organic Semiconductors: The Perspective from Nonadiabatic Molecular Dynamics

Organic semiconductors (OSs) are an exciting class of materials that have enabled disruptive technologies in this century including large-area electronics, flexible displays, and inexpensive solar cells. All of these technologies rely on the motion of electrical charges within the material and the diffusivity of these charges critically determines their performance. In this respect, it is remarkable that the nature of the charge transport in these materials has puzzled the community for so many years, even for apparently simple systems such as molecular single crystals: some experiments would better fit an interpretation in terms of a localized particle picture, akin to molecular or biological electron transfer, while others are in better agreement with a wave-like interpretation, more akin to band transport in metals.

Exciting recent progress in the theory and simulation of charge carrier transport in OSs has now led to a unified understanding of these disparate findings, and this Account will review one of these tools developed in our laboratory in some detail: direct charge carrier propagation by quantum-classical nonadiabatic molecular dynamics. One finds that even in defect-free crystals the charge carrier can either localize on a single molecule or substantially delocalize over a large number of molecules depending on the relative strength of electronic couplings between the molecules, reorganization, or charge trapping energy of the molecule and thermal fluctuations of electronic couplings and site energies, also known as electron–phonon couplings.

Our simulations predict that in molecular OSs exhibiting some of the highest measured charge mobilities to date, the charge carrier forms “flickering” polarons, objects that are delocalized over 10–20 molecules on average and that constantly change their shape and extension under the influence of thermal disorder. The flickering polarons propagate through the OS by short (≈10 fs long) bursts of the wave function that lead to an expansion of the polaron to about twice its size, resulting in spatial displacement, carrier diffusion, charge mobility, and electrical conductivity. Arguably best termed “transient delocalization”, this mechanistic scenario is very similar to the one assumed in transient localization theory and supports its assertions. We also review recent applications of our methodology to charge transport in disordered and nanocrystalline samples, which allows us to understand the influence of defects and grain boundaries on the charge propagation.

Unfortunately, the energetically favorable packing structures of typical OSs, whether molecular or polymeric, places fundamental constraints on charge mobilities/electronic conductivity compared to inorganic semiconductors, which limits their range of applications. In this Account, we review the design rules that could pave the way for new very high-mobility OS materials and we argue that 2D covalent organic frameworks are one of the most promising candidates to satisfy them.

We conclude that our nonadiabatic dynamics method is a powerful approach for predicting charge carrier transport in crystalline and disordered materials. We close with a brief outlook on extensions of the method to exciton transport, dissociation, and recombination. This will bring us a step closer to an understanding of the birth, survival, and annihiliation of charges at interfaces of optoelectronic devices.

Comparing the Expense and Accuracy of Methods to Simulate Atomic Vibrations in Rubrene

Atomic vibrations can inform about materials properties from hole transport in organic semiconductors to correlated disorder in metal-organic frameworks. Currently, there are several methods for predicting these vibrations using simulations, but the accuracy-efficiency tradeoffs have not been examined in depth. In this study, rubrene is used as a model system to predict atomic vibrational properties using six different simulation methods: density functional theory, density functional tight binding, density functional tight binding with a Chebyshev polynomial-based correction, a trained machine learning model, a pretrained machine learning model called ANI-1, and a classical forcefield model. The accuracy of each method is evaluated by comparison to the experimental inelastic neutron scattering spectrum. All methods discussed here show some accuracy across a wide energy region, though the Chebyshev-corrected tight-binding method showed the optimal combination of high accuracy with low expense. We then offer broad simulation guidelines to yield efficient, accurate results for inelastic neutron scattering spectrum prediction.

Free Charge Carriers in Homo-Sorted π-Stacks of Donor-Acceptor Conjugates

Inspired by the high photoconversion efficiency observed in natural light-harvesting systems, the hierarchical organization of molecular building blocks has gained impetus in the past few decades. Particularly, the molecular arrangement and packing in the active layer of organic solar cells (OSCs) have garnered significant attention due to the decisive role of the nature of donor/acceptor (D/A) heterojunctions in charge carrier generation and ultimately the power conversion efficiency. This review focuses on the recent developments in emergent optoelectronic properties exhibited by self-sorted donor-on-donor/acceptor-on-acceptor arrangement of covalently linked D-A systems, highlighting the ultrafast excited state dynamics of charge transfer and transport. Segregated organization of donors and acceptors promotes the delocalization of photoinduced charges among the stacks, engendering an enhanced charge separation lifetime and percolation pathways with ambipolar conductivity and charge carrier yield. Covalently linking donors and acceptors ensure a sufficient D-A interface and interchromophoric electronic coupling as required for faster charge separation while providing better control over their supramolecular assemblies. The design strategies to attain D-A conjugate assemblies with optimal charge carrier generation efficiency, the scope of their application compared to state-of-the-art OSCs, current challenges, and future opportunities are discussed in the review. An integrated overview of rational design approaches derived from the comprehension of underlying photoinduced processes can pave the way toward superior optoelectronic devices and bring in new possibilities to the avenue of functional supramolecular architectures.

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.

Delocalised kinetic Monte Carlo for simulating delocalisation-enhanced charge and exciton transport in disordered materials

Charge transport is well understood in both highly ordered materials (band conduction) or highly disordered ones (hopping conduction). In moderately disordered materials-including many organic semiconductors-the approximations valid in either extreme break down, making it difficult to accurately model the conduction. In particular, describing wavefunction delocalisation requires a quantum treatment, which is difficult in disordered materials that lack periodicity. Here, we present the first three-dimensional model of partially delocalised charge and exciton transport in materials in the intermediate disorder regime. Our approach is based on polaron-transformed Redfield theory, but overcomes several computational roadblocks by mapping the quantum-mechanical techniques onto kinetic Monte Carlo. Our theory, delocalised kinetic Monte Carlo (dKMC), shows that the fundamental physics of transport in moderately disordered materials is that of charges hopping between partially delocalised electronic states. Our results reveal why standard kinetic Monte Carlo can dramatically underestimate mobilities even in disordered organic semiconductors, where even a little delocalisation can substantially enhance mobilities, as well as showing that three-dimensional calculations capture important delocalisation effects neglected in lower-dimensional approximations.

Crystal structure and metallization mechanism of the π-radical metal

Radical electrons tend to localize on individual molecules, resulting in an insulating (Mott-Hubbard) bandgap in the solid state. Herein, we report the crystal structure and intrinsic electronic properties of the first single crystal of a π-radical metal, tetrathiafulvalene-extended dicarboxylate (TED). The electrical conductivity is up to 30 000 S cm-1 at 2 K and 2300 S cm-1 at room temperature. Temperature dependence of resistivity obeys a T 3 power-law above T > 100 K, indicating a new type of metal. X-ray crystallographic analysis clarifies the planar TED molecule, with a symmetric intramolecular hydrogen bond, is stacked along longitudinal (the a-axis) and transverse (the b-axis) directions. The π-orbitals are distributed to avoid strong local interactions. First-principles electronic calculations reveal the origin of the metallization giving rise to a wide bandwidth exceeding 1 eV near the Fermi level. TED demonstrates the effect of two-dimensional stacking of π-orbitals on electron delocalization, where a high carrier mobility of 31.6 cm2 V-1 s-1 (113 K) is achieved.

PyCDFT: A Python package for constrained density functional theory

We present PyCDFT, a Python package to compute diabatic states using constrained density functional theory (CDFT). PyCDFT provides an object-oriented, customizable implementation of CDFT, and allows for both single-point self-consistent-field calculations and geometry optimizations. PyCDFT is designed to interface with existing density functional theory (DFT) codes to perform CDFT calculations where constraint potentials are added to the Kohn-Sham Hamiltonian. Here, we demonstrate the use of PyCDFT by performing calculations with a massively parallel first-principles molecular dynamics code, Qbox, and we benchmark its accuracy by computing the electronic coupling between diabatic states for a set of organic molecules. We show that PyCDFT yields results in agreement with existing implementations and is a robust and flexible package for performing CDFT calculations. The program is available at https://dx.doi.org/10.5281/zenodo.3821097.

Evolutionary chemical space exploration for functional materials: computational organic semiconductor discovery

Computational methods, including crystal structure and property prediction, have the potential to accelerate the materials discovery process by enabling structure prediction and screening of possible molecular building blocks prior to their synthesis. However, the discovery of new functional molecular materials is still limited by the need to identify promising molecules from a vast chemical space. We describe an evolutionary method which explores a user specified region of chemical space to identify promising molecules, which are subsequently evaluated using crystal structure prediction. We demonstrate the methods for the exploration of aza-substituted pentacenes with the aim of finding small molecule organic semiconductors with high charge carrier mobilities, where the space of possible substitution patterns is too large to exhaustively search using a high throughput approach. The method efficiently explores this large space, typically requiring calculations on only ∼1% of molecules during a search. The results reveal two promising structural motifs: aza-substituted naphtho[1,2-a]anthracenes with reorganisation energies as low as pentacene and a series of pyridazine-based molecules having both low reorganisation energies and high electron affinities.

Quantum localization and delocalization of charge carriers in organic semiconducting crystals

Charge carrier transport in organic semiconductors is at the heart of many revolutionary technologies ranging from organic transistors, light-emitting diodes, flexible displays and photovoltaic cells. Yet, the nature of charge carriers and their transport mechanism in these materials is still unclear. Here we show that by solving the time-dependent electronic Schrödinger equation coupled to nuclear motion for eight organic molecular crystals, the excess charge carrier forms a polaron delocalized over up to 10-20 molecules in the most conductive crystals. The polaron propagates through the crystal by diffusive jumps over several lattice spacings at a time during which it expands more than twice its size. Computed values for polaron size and charge mobility are in excellent agreement with experimental estimates and correlate very well with the recently proposed transient localization theory.

Solid-State Order and Charge Mobility in [5]- to [12]Cycloparaphenylenes

We report a computational study of mesoscale morphology and charge-transport properties of radially π-conjugated cycloparaphenylenes ([ n]CPPs) of various ring sizes ( n = 5-12, where n is the number of repeating phenyl units). These molecules are considered structural constituents of fullerenes and carbon nanotubes. [ n]CPP molecules are nested in a unique fashion in the solid state. Molecular dynamics simulations show that while intramolecular structural stability (order) increases with system size, intermolecular structural stability decreases. Density functional calculations reveal that reorganization energy, an important parameter in charge transfer, decreases as n is increased. Intermolecular charge-transfer electronic couplings in the solid state are relatively weak (due to curved π-conjugation and loose intermolecular contacts) and are on the same order of magnitude (∼10 meV) for each system. Intrinsic charge-carrier mobilities were simulated from kinetic Monte Carlo simulations; hole mobilities increased with system size and scaled as ∼ n⁴. We predict that disordered [ n]CPPs exhibit hole mobilities as high as 2 cm²/(V·s). Our computations show a strong correlation between reorganization energy and hole mobility (μ ∼ λ^-4). Quantum mechanical calculations were performed on cofacially stacked molecular pairs for varying phenyl units and reveal that orbital delocalization is responsible for both decreasing reorganization energies and electronic couplings as n is increased.

Resolving orbital pathways for intermolecular electron transfer

Over 60 years have passed since Taube deduced an orbital-mediated electron transfer mechanism between distinct metal complexes. This concept of an orbital pathway has been thoroughly explored for donor-acceptor pairs bridged by covalently bonded chemical residues, but an analogous pathway has not yet been conclusively demonstrated for formally outer-sphere systems that lack an intervening bridge. In our present study, we experimentally resolve at an atomic level the orbital interactions necessary for electron transfer through an explicit intermolecular bond. This finding was achieved using a homologous series of surface-immobilized ruthenium catalysts that bear different terminal substituents poised for reaction with redox active species in solution. This arrangement enabled the discovery that intermolecular chalcogen⋯iodide interactions can mediate electron transfer only when these interactions bring the donor and acceptor orbitals into direct contact. This result offers the most direct observation to date of an intermolecular orbital pathway for electron transfer.

Polaron States in Fullerene Adducts Modeled by Coarse-Grained Molecular Dynamics and Tight Binding

Strong electron-phonon coupling leads to polaron localization in molecular semiconductor materials and influences charge transport, but it is expensive to calculate atomistically. Here, we propose a simple and efficient model to determine the energy and spatial extent of polaron states within a coarse-grained representation of a disordered molecular film. We calculate the electronic structure of the molecular assembly using a tight-binding Hamiltonian and determine the polaron state self-consistently by perturbing the site energies by the dielectric response of the surrounding medium to the charge. When applied to fullerene derivatives, the method shows that polarons extend over multiple molecules in C60 but localize on single molecules in higher adducts of phenyl-C61-butyric-acid-methyl-ester (PCBM) because of packing disorder and the polar side chains. In PCBM, polarons localize on single molecules only when energetic disorder is included or when the fullerene is dispersed in a blend. The method helps to establish the conditions under which a hopping transport model is justified.

Probing charge transfer dynamics in a single iron tetraphenylporphyrin dyad adsorbed on an insulating surface

Although the dynamics of charge transfer (CT) processes can be probed with ultimate lifetime resolution, the inability to control CT at the nanoscale is one of the most important roadblocks to revealing some of its deep fundamental aspects. In this work, we present an investigation of CT dynamics in a single iron tetraphenylporphyrin (Fe-TPP) donor/acceptor dyad adsorbed on a CaF2/Si(100) insulating surface. The tip of a scanning tunneling microscope (STM) is used to create local ionic states in one fragment of the dyad. The CT process is monitored by imaging subsequent changes in the neighbor acceptor molecule and its efficiency is mapped, revealing the influence of the initial excited state in the donor molecule. In the validation of the experiments, simulations based on density functional theory show that holes have a higher donor-acceptor CT rate compared to electrons and highlight a noticeable initial state dependence on the CT process. We leverage the unprecedented spatial resolution achieved in our experiments to show that the CT process in the dyad is governed via molecule-molecule coherent tunneling with negligible surface-mediated character.

Nonlocal Electron-Phonon Coupling in Prototypical Molecular Semiconductors from First Principles

A substantial amount of evidence indicates a relevant role played by the nonlocal electron-phonon couplings in the mechanism of charge transport in organic semiconductors. In this work, we compute the nonlocal electron-phonon coupling for the prototypical molecular semiconductors rubrene and tetracene using the phonon modes obtained from ab initio methods. We do not make the rigid molecular approximation allowing a mixing of intra- and intermolecular modes, and we use a supercell approach to sample the momentum space. Indeed, we find that some low-frequency intramolecular modes are mixed with the rigid-molecule translations and rotations in the modes with the strongest electron-phonon coupling. To rationalize the results we propose a convenient decomposition of the delocalized lattice modes into molecular-based modes.

Detailed balance, internal consistency, and energy conservation in fragment orbital-based surface hopping

We have recently introduced an efficient semi-empirical non-adiabatic molecular dynamics method for the simulation of charge transfer/transport in molecules and molecular materials, denoted fragment orbital-based surface hopping (FOB-SH) [J. Spencer et al., J. Chem. Phys. 145, 064102 (2016)]. In this method, the charge carrier wavefunction is expanded in a set of charge localized, diabatic electronic states and propagated in the time-dependent potential due to classical nuclear motion. Here we derive and implement an exact expression for the non-adiabatic coupling vectors between the adiabatic electronic states in terms of nuclear gradients of the diabatic electronic states. With the non-adiabatic coupling vectors (NACVs) available, we investigate how different flavours of fewest switches surface hopping affect detailed balance, internal consistency, and total energy conservation for electron hole transfer in a molecular dimer with two electronic states. We find that FOB-SH satisfies detailed balance across a wide range of diabatic electronic coupling strengths provided that the velocities are adjusted along the direction of the NACV to satisfy total energy conservation upon a surface hop. This criterion produces the right fraction of energy-forbidden (frustrated) hops, which is essential for correct population of excited states, especially when diabatic couplings are on the order of the thermal energy or larger, as in organic semiconductors and DNA. Furthermore, we find that FOB-SH is internally consistent, that is, the electronic surface population matches the average quantum amplitudes, but only in the limit of small diabatic couplings. For large diabatic couplings, inconsistencies are observed as the decrease in excited state population due to frustrated hops is not matched by a corresponding decrease in quantum amplitudes. The derivation provided here for the NACV should be generally applicable to any electronic structure approach where the electronic Hamiltonian is constructed in a diabatic electronic state basis.

Crossover from Hopping to Band-Like Charge Transport in an Organic Semiconductor Model: Atomistic Nonadiabatic Molecular Dynamics Simulation

The mechanism of charge transport (CT) in a 1D atomistic model of an organic semiconductor is investigated using surface hopping nonadiabatic molecular dynamics. The simulations benefit from a newly implemented state tracking algorithm that accounts for trivial surface crossings and from a projection algorithm that removes decoherence correction-induced artificial long-range charge transfer. The CT mechanism changes from slow hopping of a fully localized charge to fast diffusion of a polaron delocalized over several molecules as electronic coupling between the molecules exceeds the critical threshold V ≥ λ/2 (λ is the reorganization energy). With increasing temperature, the polaron becomes more localized and the mobility exhibits a "band-like" power law decay due to increased site energy and electronic coupling fluctuations (local and nonlocal electron-phonon coupling). Thus, reducing both types of electron-phonon coupling while retaining high mean electronic couplings should be part of the strategy toward discovery of new organics with high room-temperature mobilities.

Fragment-orbital tunneling currents and electronic couplings for analysis of molecular charge-transfer systems

In theoretical charge-transfer research, calculation of the electronic coupling element is crucial for examining the degree of the electronic donor-acceptor interaction. The tunneling current (TC), representing the magnitudes and directions of electron flow, provides a way of evaluating electronic couplings, along with the ability of visualizing how electrons flow in systems. Here, we applied the TC theory to π-conjugated organic dimer systems, in the form of our fragment-orbital tunneling current (FOTC) method, which uses the frontier molecular-orbitals of system fragments as diabatic states. For a comprehensive test of FOTC, we assessed how reasonable the computed electronic couplings and the corresponding TC densities are for the hole- and electron-transfer databases HAB11 and HAB7. FOTC gave 12.5% mean relative unsigned error with regard to the high-level ab initio reference. The shown performance is comparable with that of fragment-orbital density functional theory, which gave the same error by 20.6% or 13.9% depending on the formulation. In the test of a set of nucleobase π stacks, we showed that the original TC expression is also applicable to nondegenerate cases under the condition that the overlap between the charge distributions of diabatic states is small enough to offset the energy difference. Lastly, we carried out visual analysis on the FOTC densities of thiophene dimers with different intermolecular alignments. The result depicts an intimate topological connection between the system geometry and electron flow. Our work provides quantitative and qualitative grounds for FOTC, showing it to be a versatile tool in characterization of molecular charge-transfer systems.

Density Functional Physicality in Electronic Coupling Estimation: Benchmarks and Error Analysis

Electronic coupling estimates from constrained density functional theory configuration interaction (CDFT-CI) depend critically on choice of density functional. In this Letter, the orbital multielectron self-interaction error (OMSIE), vertical electron affinity (VEA), and vertical ionization potential (VIP) are shown to be the key indicators inherited from the density functional that determine the accuracy of electronic coupling estimates. An error metric η is derived to connect the three properties, based on the linear proportionality between electronic coupling and overlap integral, and the hypothesis that the slope of this line is a function of VEA/VIP, η = (1/N_testset)Σ_i^testset|-VE^Ref × OMSIE + ΔVE - ΔVE × OMSIE|_i. Based on η, BH&HLYP and LRC-ωPBEh are suggested as the best functionals for electron and hole transfer, respectively. Error metric η is therefore a useful predictor of errors in CDFT-CI electronic coupling, showing that the physical correctness of the density functional has a direct effect on the accuracy of the electronic coupling.

Charge Transport in Molecular Materials: An Assessment of Computational Methods

The booming field of molecular electronics has fostered a surge of computational research on electronic properties of organic molecular solids. In particular, with respect to a microscopic understanding of transport and loss mechanisms, theoretical studies assume an ever-increasing role. Owing to the tremendous diversity of organic molecular materials, a great number of computational methods have been put forward to suit every possible charge transport regime, material, and need for accuracy. With this review article we aim at providing a compendium of the available methods, their theoretical foundations, and their ranges of validity. We illustrate these through applications found in the literature. The focus is on methods available for organic molecular crystals, but mention is made wherever techniques are suitable for use in other related materials such as disordered or polymeric systems.

Critical analysis of fragment-orbital DFT schemes for the calculation of electronic coupling values

We present a critical analysis of the popular fragment-orbital density-functional theory (FO-DFT) scheme for the calculation of electronic coupling values. We discuss the characteristics of different possible formulations or "flavors" of the scheme which differ by the number of electrons in the calculation of the fragments and the construction of the Hamiltonian. In addition to two previously described variants based on neutral fragments, we present a third version taking a different route to the approximate diabatic state by explicitly considering charged fragments. In applying these FO-DFT flavors to the two molecular test sets HAB7 (electron transfer) and HAB11 (hole transfer), we find that our new scheme gives improved electronic couplings for HAB7 (-6.2% decrease in mean relative signed error) and greatly improved electronic couplings for HAB11 (-15.3% decrease in mean relative signed error). A systematic investigation of the influence of exact exchange on the electronic coupling values shows that the use of hybrid functionals in FO-DFT calculations improves the electronic couplings, giving values close to or even better than more sophisticated constrained DFT calculations. Comparing the accuracy and computational cost of each variant, we devise simple rules to choose the best possible flavor depending on the task. For accuracy, our new scheme with charged-fragment calculations performs best, while numerically more efficient at reasonable accuracy is the variant with neutral fragments.

Confronting surface hopping molecular dynamics with Marcus theory for a molecular donor–acceptor system

We investigate the performance of fewest switches surface hopping (SH) in describing electron transfer (ET) for a molecular donor–acceptor system. Computer simulations are carried out for a wide range of reorganisation energy (λ), electronic coupling strength (H_ab) and driving force using our recently developed fragment orbital-based SH approach augmented with a simple decoherence correction. This methodology allows us to compute SH ET rates over more than four orders of magnitude, from the sub-picosecond to the nanosecond time regime. We find good agreement with semi-classical ET theory in the non-adiabatic ET regime. The correct scaling of the SH ET rate with electronic coupling strength is obtained and the Marcus inverted regime is reproduced, in line with previously reported results for a spin-boson model. Yet, we find that the SH ET rate falls below the semi-classical ET rate in the adiabatic regime, where the free energy barrier is in the order of k_BT in our simulations. We explain this by first signatures of non-exponential population decay of the initial charge state. For even larger electronic couplings (H_ab = λ/2), the free energy barrier vanishes and ET rates are no longer defined. At this point we observe a crossover from ET on the vibronic time scale to charge relaxation on the femtosecond time scale that is well described by thermally averaged Rabi oscillations. The extension of the analysis from the non-adiabatic limit to large electronic couplings and small or even vanishing activation barriers is relevant for our understanding of charge transport in organic semiconductors.

Electronic couplings for molecular charge transfer: benchmarking CDFT, FODFT and FODFTB against high-level ab initio calculations. II

A new database (HAB7-) of electronic coupling matrix elements (Hab) for electron transfer in seven medium-sized negatively charged π-conjugated organic dimers is introduced. Reference data are obtained with spin-component scaled approximate coupled cluster method (SCS-CC2) and large basis sets. Assessed DFT-based approaches include constrained density functional theory (CDFT), fragment-orbital DFT (FODFT), self-consistent charge density functional tight-binding (FODFTB) and the recently described analytic overlap method (AOM). This complements the previously reported HAB11 database where only cationic dimers were considered. The CDFT method in combination with a functional based on PBE and including 50% of exact exchange (HFX) was found to provide best estimates, with a mean relative unsigned error (MRUE) of 8.2%. CDFT couplings systematically increase with decreasing fraction of HFX as a consequence of increasing delocalisation of the SOMO orbital. The FODFT method is found to be very robust underestimating electronic couplings by 28%. The FODFTB and AOM methods, although orders of magnitude more efficient in terms of computational effort than the DFT approaches, perform well with reasonably small errors of 54% and 29%, respectively, translating in errors in the non-adiabatic electron transfer rate of a factor of 2.4 and 1.7, respectively. We discuss carefully various sources of errors and the scope and limitations of all assessed methods taking into account the results obtained for both HAB7- and HAB11 databases.

Multi-haem cytochromes in Shewanella oneidensis MR-1: structures, functions and opportunities

Multi-haem cytochromes are employed by a range of microorganisms to transport electrons over distances of up to tens of nanometres. Perhaps the most spectacular utilization of these proteins is in the reduction of extracellular solid substrates, including electrodes and insoluble mineral oxides of Fe(III) and Mn(III/IV), by species of Shewanella and Geobacter. However, multi-haem cytochromes are found in numerous and phylogenetically diverse prokaryotes where they participate in electron transfer and redox catalysis that contributes to biogeochemical cycling of N, S and Fe on the global scale. These properties of multi-haem cytochromes have attracted much interest and contributed to advances in bioenergy applications and bioremediation of contaminated soils. Looking forward, there are opportunities to engage multi-haem cytochromes for biological photovoltaic cells, microbial electrosynthesis and developing bespoke molecular devices. As a consequence, it is timely to review our present understanding of these proteins and we do this here with a focus on the multitude of functionally diverse multi-haem cytochromes in Shewanella oneidensis MR-1. We draw on findings from experimental and computational approaches which ideally complement each other in the study of these systems: computational methods can interpret experimentally determined properties in terms of molecular structure to cast light on the relation between structure and function. We show how this synergy has contributed to our understanding of multi-haem cytochromes and can be expected to continue to do so for greater insight into natural processes and their informed exploitation in biotechnologies.

QM/MM calculations with deMon2k

The density functional code deMon2k employs a fitted density throughout (Auxiliary Density Functional Theory), which offers a great speed advantage without sacrificing necessary accuracy. Powerful Quantum Mechanical/Molecular Mechanical (QM/MM) approaches are reviewed. Following an overview of the basic features of deMon2k that make it efficient while retaining accuracy, three QM/MM implementations are compared and contrasted. In the first, deMon2k is interfaced with the CHARMM MM code (CHARMM-deMon2k); in the second MM is coded directly within the deMon2k software; and in the third the Chemistry in Ruby (Cuby) wrapper is used to drive the calculations. Cuby is also used in the context of constrained-DFT/MM calculations. Each of these implementations is described briefly; pros and cons are discussed and a few recent applications are described briefly. Applications include solvated ions and biomolecules, polyglutamine peptides important in polyQ neurodegenerative diseases, copper monooxygenases and ultra-rapid electron transfer in cryptochromes.

A faux hawk fullerene with PCBM-like properties

Reaction of C60, C6F5CF2I, and SnH(n-Bu)3 produced, among other unidentified fullerene derivatives, the two new compounds 1,9-C60(CF2C6F5)H (1) and 1,9-C60(cyclo-CF2(2-C6F4)) (2). The highest isolated yield of 1 was 35% based on C60. Depending on the reaction conditions, the relative amounts of 1 and 2 generated in situ were as high as 85% and 71%, respectively, based on HPLC peak integration and summing over all fullerene species present other than unreacted C60. Compound 1 is thermally stable in 1,2-dichlorobenzene (oDCB) at 160 °C but was rapidly converted to 2 upon addition of Sn2(n-Bu)6 at this temperature. In contrast, complete conversion of 1 to 2 occurred within minutes, or hours, at 25 °C in 90/10 (v/v) PhCN/C6D6 by addition of stoichiometric, or sub-stoichiometric, amounts of proton sponge (PS) or cobaltocene (CoCp2). DFT calculations indicate that when 1 is deprotonated, the anion C60(CF2C6F5)- can undergo facile intramolecular SNAr annulation to form 2 with concomitant loss of F-. To our knowledge this is the first observation of a fullerene-cage carbanion acting as an SNAr nucleophile towards an aromatic C-F bond. The gas-phase electron affinity (EA) of 2 was determined to be 2.805(10) eV by low-temperature PES, higher by 0.12(1) eV than the EA of C60 and higher by 0.18(1) eV than the EA of phenyl-C61-butyric acid methyl ester (PCBM). In contrast, the relative E1/2(0/-) values of 2 and C60, -0.01(1) and 0.00(1) V, respectively, are virtually the same (on this scale, and under the same conditions, the E1/2(0/-) of PCBM is -0.09 V). Time-resolved microwave conductivity charge-carrier yield × mobility values for organic photovoltaic active-layer-type blends of 2 and poly-3-hexylthiophene (P3HT) were comparable to those for equimolar blends of PCBM and P3HT. The structure of solvent-free crystals of 2 was determined by single-crystal X-ray diffraction. The number of nearest-neighbor fullerene-fullerene interactions with centroid···centroid (⊙···⊙) distances of ≤10.34 Å is significantly greater, and the average ⊙···⊙ distance is shorter, for 2 (10 nearest neighbors; ave. ⊙···⊙ distance = 10.09 Å) than for solvent-free crystals of PCBM (7 nearest neighbors; ave. ⊙···⊙ distance = 10.17 Å). Finally, the thermal stability of 2 was found to be far greater than that of PCBM.

Theoretical study of exciton dissociation through hot states at donor–acceptor interface in organic photocell

We theoretically study the dissociation of geminate electron–hole pairs (i.e., excitons) through vibrational hot states at the donor–acceptor interface of organic photocells.

Electronic couplings for molecular charge transfer: benchmarking CDFT, FODFT, and FODFTB against high-level ab initio calculations

We introduce a database (HAB11) of electronic coupling matrix elements (H(ab)) for electron transfer in 11 π-conjugated organic homo-dimer cations. High-level ab inito calculations at the multireference configuration interaction MRCI+Q level of theory, n-electron valence state perturbation theory NEVPT2, and (spin-component scaled) approximate coupled cluster model (SCS)-CC2 are reported for this database to assess the performance of three DFT methods of decreasing computational cost, including constrained density functional theory (CDFT), fragment-orbital DFT (FODFT), and self-consistent charge density functional tight-binding (FODFTB). We find that the CDFT approach in combination with a modified PBE functional containing 50% Hartree-Fock exchange gives best results for absolute H(ab) values (mean relative unsigned error = 5.3%) and exponential distance decay constants β (4.3%). CDFT in combination with pure PBE overestimates couplings by 38.7% due to a too diffuse excess charge distribution, whereas the economic FODFT and highly cost-effective FODFTB methods underestimate couplings by 37.6% and 42.4%, respectively, due to neglect of interaction between donor and acceptor. The errors are systematic, however, and can be significantly reduced by applying a uniform scaling factor for each method. Applications to dimers outside the database, specifically rotated thiophene dimers and larger acenes up to pentacene, suggests that the same scaling procedure significantly improves the FODFT and FODFTB results for larger π-conjugated systems relevant to organic semiconductors and DNA.

An Effective Two-Orbital Quantum Chemical Model for Organic Photovoltaic Materials

We present a coarse-grained quantum chemical model of organic photovoltaic materials, which is based on the classic idea that the main physical processes involve the electrons occupying the frontier orbitals (HOMO and LUMO) of each molecule or "site". This translates into an effective electronic Hamiltonian with two electrons and two orbitals per site. The on-site parameters (one- and two-electron integrals) can be rigorously related to the ionization energy, electron affinity, and singlet and triplet first excitation energies of that site. The intersite Hamiltonian parameters are introduced in a way that is consistent with classical electrostatics, and for the one-electron part, we use a simple approximation that could be refined using information from atomistic quantum chemical calculations. The model has been implemented within the GAMESS-US package. This allows the exploration of the physics of these materials using state-of-the art quantum chemical methods on relatively large systems (hundreds of electron-donor and electron-acceptor sites). To illustrate this point, we present ground- and excited-state calculations on dimers and two-dimensional arrays of sites using the Hartree-Fock, configuration interaction, and coupled-cluster methods. The calculations provide evidence for the possibility of low-energy, long-range electron transfer in donor-acceptor heterojunctions characterized by a moderate degree of disorder.

Electron flow in multiheme bacterial cytochromes is a balancing act between heme electronic interaction and redox potentials

The naturally widespread process of electron transfer from metal reducing bacteria to extracellular solid metal oxides entails unique biomolecular machinery optimized for long-range electron transport. To perform this function efficiently, microorganisms have adapted multiheme c-type cytochromes to arrange heme cofactors into wires that cooperatively span the cellular envelope, transmitting electrons along distances greater than 100 Å. Implications and opportunities for bionanotechnological device design are self-evident. However, at the molecular level, how these proteins shuttle electrons along their heme wires, navigating intraprotein intersections and interprotein interfaces efficiently, remains a mystery thus far inaccessible to experiment. To shed light on this critical topic, we carried out extensive quantum mechanics/molecular mechanics simulations to calculate stepwise heme-to-heme electron transfer rates in the recently crystallized outer membrane deca-heme cytochrome MtrF. By solving a master equation for electron hopping, we estimate an intrinsic, maximum possible electron flux through solvated MtrF of 10(4)-10(5) s(-1), consistent with recently measured rates for the related multiheme protein complex MtrCAB. Intriguingly, our calculations show that the rapid electron transport through MtrF is the result of a clear correlation between heme redox potential and the strength of electronic coupling along the wire: thermodynamically uphill steps occur only between electronically well-connected stacked heme pairs. This observation suggests that the protein evolved to harbor low-potential hemes without slowing down electron flow. These findings are particularly profound in light of the apparently well-conserved staggered cross-heme wire structural motif in functionally related outer membrane proteins.

A hybrid approach to simulation of electron transfer in complex molecular systems

Electron transfer (ET) reactions in biomolecular systems represent an important class of processes at the interface of physics, chemistry and biology. The theoretical description of these reactions constitutes a huge challenge because extensive systems require a quantum-mechanical treatment and a broad range of time scales are involved. Thus, only small model systems may be investigated with the modern density functional theory techniques combined with non-adiabatic dynamics algorithms. On the other hand, model calculations based on Marcus's seminal theory describe the ET involving several assumptions that may not always be met. We review a multi-scale method that combines a non-adiabatic propagation scheme and a linear scaling quantum-chemical method with a molecular mechanics force field in such a way that an unbiased description of the dynamics of excess electron is achieved and the number of degrees of freedom is reduced effectively at the same time. ET reactions taking nanoseconds in systems with hundreds of quantum atoms can be simulated, bridging the gap between non-adiabatic ab initio simulations and model approaches such as the Marcus theory. A major recent application is hole transfer in DNA, which represents an archetypal ET reaction in a polarizable medium. Ongoing work focuses on hole transfer in proteins, peptides and organic semi-conductors.

On the Inapplicability of Electron-Hopping Models for the Organic Semiconductor Phenyl-C61-butyric Acid Methyl Ester (PCBM)

Phenyl-C61-butyric acid methyl ester (PCBM) is one of the most popular semiconductors in organic photovoltaic cells, but the electron-transport mechanism in the microcrystalline domains of this material as well as its preferred packing structure remain unclear. Here we use density functional theory to calculate electronic-coupling matrix elements, reorganization energies, and activation energies for available experimental and model crystal structures. We find that the picture of an excess electron hopping from one fullerene to another does not apply for any of the crystalline phases, rendering traditional rate equations inappropriate. We also find that the cohesive energy increases in the order body-centered-cubic < hexagonal < simple cubic < monoclinic < triclinic, independently of the type of dispersion correction used. Our results indicate that the coupled electron-ion dynamics needs to be solved explicitly to obtain a realistic description of charge transfer in this material.

Efficient algorithms for the simulation of non-adiabatic electron transfer in complex molecular systems: application to DNA

In this work, a fragment-orbital density functional theory-based method is combined with two different non-adiabatic schemes for the propagation of the electronic degrees of freedom. This allows us to perform unbiased simulations of electron transfer processes in complex media, and the computational scheme is applied to the transfer of a hole in solvated DNA. It turns out that the mean-field approach, where the wave function of the hole is driven into a superposition of adiabatic states, leads to over-delocalization of the hole charge. This problem is avoided using a surface hopping scheme, resulting in a smaller rate of hole transfer. The method is highly efficient due to the on-the-fly computation of the coarse-grained DFT Hamiltonian for the nucleobases, which is coupled to the environment using a QM/MM approach. The computational efficiency and partial parallel character of the methodology make it possible to simulate electron transfer in systems of relevant biochemical size on a nanosecond time scale. Since standard non-polarizable force fields are applied in the molecular-mechanics part of the calculation, a simple scaling scheme was introduced into the electrostatic potential in order to simulate the effect of electronic polarization. It is shown that electronic polarization has an important effect on the features of charge transfer. The methodology is applied to two kinds of DNA sequences, illustrating the features of transfer along a flat energy landscape as well as over an energy barrier. The performance and relative merit of the mean-field scheme and the surface hopping for this application are discussed.