First-principles density-functional theory calculations of electron-transfer rates in azurin dimers

Controlling the charge-transfer dynamics of two-level systems around avoided crossings

Two-level quantum systems are fundamental physical models that continue to attract growing interest due to their crucial role as a building block of quantum technologies. The exact analytical solution of the dynamics of these systems is central to control theory and its applications, such as that to quantum computing. In this study, we reconsider the two-state charge transfer problem by extending and using a methodology developed to study (pseudo)spin systems in quantum electrodynamics contexts. This approach allows us to build a time evolution operator for the charge transfer system and to show new opportunities for the coherent control of the system dynamics, with a particular emphasis on the critical dynamic region around the transition state coordinate, where the avoided crossing of the energy levels occurs. We identify and propose possible experimental implementations of a class of rotations of the charge donor (or acceptor) that endow the electronic coupling matrix element with a time-dependent phase that can be employed to realize controllable coherent dynamics of the system across the avoided level crossing. The analogy of these rotations to reference frame rotations in generalized semiclassical Rabi models is discussed. We also show that the physical rotations in the charge-transfer systems can be performed so as to implement quantum gates relevant to quantum computing. From an exquisitely physical-mathematical viewpoint, our approach brings to light situations in which the time-dependent state of the system can be obtained without resorting to the special functions appearing in the Landau-Zener approach.

HAB79: A new molecular dataset for benchmarking DFT and DFTB electronic couplings against high-level ab initio calculations

A new molecular dataset called HAB79 is introduced to provide ab initio reference values for electronic couplings (transfer integrals) and to benchmark density functional theory (DFT) and density functional tight-binding (DFTB) calculations. The HAB79 dataset is composed of 79 planar heterocyclic polyaromatic hydrocarbon molecules frequently encountered in organic (opto)electronics, arranged to 921 structurally diverse dimer configurations. We show that CASSCF/NEVPT2 with a minimal active space provides a robust reference method that can be applied to the relatively large molecules of the dataset. Electronic couplings are largest for cofacial dimers, in particular, sulfur-containing polyaromatic hydrocarbons, with values in excess of 0.5 eV, followed by parallel displaced cofacial dimers. V-shaped dimer motifs, often encountered in the herringbone layers of organic crystals, exhibit medium-sized couplings, whereas T-shaped dimers have the lowest couplings. DFT values obtained from the projector operator-based diabatization (POD) method are initially benchmarked against the smaller databases HAB11 (HAB7-) and found to systematically improve when climbing Jacob's ladder, giving mean relative unsigned errors (MRUEs) of 27.7% (26.3%) for the generalized gradient approximation (GGA) functional BLYP, 20.7% (15.8%) for hybrid functional B3LYP, and 5.2% (7.5%) for the long-range corrected hybrid functional omega-B97X. Cost-effective POD in combination with a GGA functional and very efficient DFTB calculations on the dimers of the HAB79 database give a good linear correlation with the CASSCF/NEVPT2 reference data, which, after scaling with a multiplicative constant, gives reasonably small MRUEs of 17.9% and 40.1%, respectively, bearing in mind that couplings in HAB79 vary over 4 orders of magnitude. The ab initio reference data reported here are expected to be useful for benchmarking other DFT or semi-empirical approaches for electronic coupling calculations.

Fluctuating hydrogen-bond networks govern anomalous electron transfer kinetics in a blue copper protein

We combine experimental and computational methods to address the anomalous kinetics of long-range electron transfer (ET) in mutants of Pseudomonas aeruginosa azurin. ET rates and driving forces for wild type (WT) and three N47X mutants (X = L, S, and D) of Ru(2,2'-bipyridine)₂ (imidazole)(His83) azurin are reported. An enhanced ET rate for the N47L mutant suggests either an increase of the donor-acceptor (DA) electronic coupling or a decrease in the reorganization energy for the reaction. The underlying atomistic features are investigated using a recently developed nonadiabatic molecular dynamics method to simulate ET in each of the azurin mutants, revealing unexpected aspects of DA electronic coupling. In particular, WT azurin and all studied mutants exhibit more DA compression during ET (>2 Å) than previously recognized. Moreover, it is found that DA compression involves an extended network of hydrogen bonds, the fluctuations of which gate the ET reaction, such that DA compression is facilitated by transiently rupturing hydrogen bonds. It is found that the N47L mutant intrinsically disrupts this hydrogen-bond network, enabling particularly facile DA compression. This work, which reveals the surprisingly fluctional nature of ET in azurin, suggests that hydrogen-bond networks can modulate the efficiency of long-range biological ET.

Diabatic Population Matrix Formalism for Performing Molecular Mechanics Style Simulations with Multiple Electronic States

An accurate description of nonbonded interactions is important in investigating dynamics of molecular systems. In many situations, fixed point charge models are successfully applied to explaining various chemical phenomena. However, these models with conventional formulations will not be appropriate in elucidating the detailed dynamics during nonadiabatic events. This is mainly because the chemical properties of any molecule, especially its electronic populations, significantly change with respect to molecular distortions in the vicinity of the surface crossing. To overcome this issue in molecular simulations yet within the framework of the fixed point charge model, we define a diabatic electronic population matrix and substitute it for the conventional adiabatic partial charges. We show that this matrix can be readily utilized toward attaining more reliable descriptions of Coulombic interactions, in combination with the interpolation formalism for obtaining the intramolecular interaction potential. We demonstrate how the mixed formalism with the diabatic charges and the interpolation can be applied to molecular simulations by conducting adiabatic and nonadiabatic molecular dynamics trajectory calculations of the green fluorescent protein chromophore anion in aqueous environment.

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.

Progress and challenges in simulating and understanding electron transfer in proteins

This Review presents an overview of the most common numerical simulation approaches for the investigation of electron transfer (ET) in proteins. We try to highlight the merits of the different approaches but also the current limitations and challenges. The article is organized into three sections. Section 2 deals with direct simulation algorithms of charge migration in proteins. Section 3 summarizes the methods for testing the applicability of the Marcus theory for ET in proteins and for evaluating key thermodynamic quantities entering the reaction rates (reorganization energies and driving force). Recent studies interrogating the validity of the theory due to the presence of non-ergodic effects or of non-linear responses are also described. Section 4 focuses on the tunneling aspects of electron transfer. How can the electronic coupling between charge transfer states be evaluated by quantum chemistry approaches and rationalized? What interesting physics regarding the impact of protein dynamics on tunneling can be addressed? We will illustrate the different sections with examples taken from the literature to show what types of system are currently manageable with current methodologies. We also take care to recall what has been learned on the biophysics of ET within proteins thanks to the advent of atomistic simulations.

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.

Dynamical treatment of charge transfer through duplex nucleic acids containing modified adenines

We address the issue of whether chemical alterations of nucleobases are an effective tool to modulate charge transfer through DNA molecules. Our investigation uses a multilevel computational approach based on classical molecular dynamics and quantum chemistry. We find yet another piece of evidence that structural fluctuations are a key factor to determine the electronic structure of double-stranded DNA. We argue that the electronic structure and charge transfer ability of flexible polymers is the result of a complex intertwining of various structural, dynamical and chemical factors. Chemical intuition may be used to design molecular wires, but this is not the sole component in the complex charge transfer mechanism through DNA.

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

We assess the validity of incoherent hopping models that have previously been used to describe electron transport in crystalline C(60) at room temperature. To this end we present new density functional theory based calculations of the electron transfer parameter defining these models. Specifically, we report electronic coupling matrix elements for several ten thousand configurations that are thermally accessible to the C(60) molecules through rotational diffusion around their lattice sites. We find that the root-mean-square fluctuations of the electronic coupling matrix element (11 meV) are almost as large as the average value (14 meV) and that the distribution is well approximated by a Gaussian. Importantly, due to the small reorganisation energy of the C(60) dimer (≈0.1 eV), the ET is almost activationless for the majority of configurations. Yet, for a small but significant fraction of orientations the coupling is so strong compared to reorganisation energy that no charge-localised state exists, a situation that is aggravated if zero-point motion of the nuclei is taken into account. The present calculations indicate that standard hopping models do not provide a sound description of electron transport in C(60), which might be the case for many other organics as well, and that approaches are needed that solve the electron dynamics directly.

Modeling and computations of the intramolecular electron transfer process in the two-heme protein cytochrome c(4)

The di-heme protein Pseudomonas stutzeri cytochrome c(4) (cyt c(4)) has emerged as a useful model for studying long-range protein electron transfer (ET). Recent experimental observations have shown a dramatically different pattern of intramolecular ET between the two heme groups in different local environments. Intramolecular ET in homogeneous solution is too slow (>10 s) to be detected but fast (ms-μs) intramolecular ET in an electrochemical environment has recently been achieved by controlling the molecular orientation of the protein assembled on a gold electrode surface. In this work we have performed computational modeling of the intramolecular ET process by a combination of density functional theory (DFT) and quantum mechanical charge transfer theory to disclose reasons for this difference. We first address the electronic structures of the model heme core with histidine and methionine axial ligands in both low- and high-spin states by structure-optimized DFT. The computations enable estimating the intramolecular reorganization energy of the ET process for different combinations of low- and high-spin heme couples. Environmental reorganization free energies, work terms ("gating") and driving force were determined using dielectric continuum models. We then calculated the electronic transmission coefficient of the intramolecular ET rate using perturbation theory combined with the electronic wave functions determined by the DFT calculations for different heme group orientations and Fe-Fe separations. The reactivity of low- and high-spin heme groups was notably different. The ET rate is exceedingly low for the crystallographic equilibrium orientation but increases by several orders of magnitude for thermally accessible non-equilibrium configurations. Deprotonation of the propionate carboxyl group was also found to enhance the ET rate significantly. The results are discussed in relation to the observed surface immobilization effect and support the notion of conformationally gated ET.

Nonorthogonality Problem and Effective Electronic Coupling Calculation: Application to Charge Transfer in π-Stacks Relevant to Biochemistry and Molecular Electronics

A recently proposed method for the calculation of the effective electronic coupling (or charge-transfer integral) in a two-state system is discussed and related to other methods in the literature. The theoretical expression of the coupling is exact within the two-state model and applies to the general case where the charge transfer (CT) process involves nonorthogonal initial and final diabatic (localized) states. In this work, it is shown how this effective electronic coupling is also the one to be used in a suitable extension of Rabi's formula to the nonorthogonal representation of two-state dynamical problems. The formula for the transfer integral is inspected in the regime of long-range CT and applied to CT reactions in redox molecular systems of interest to biochemistry and/or to molecular electronics: the guanine-thymine stack from regular B-DNA, the polyaromatic perylenediimide stack, and the quinol-semiquinone couple. The calculations are performed within the framework of the Density Functional Theory (DFT), using hybrid exchange-correlation (XC) density functionals, which also allowed investigation of the appropriateness of such hybrid-DFT methods for computing electronic couplings. The use of the recently developed M06-2X and M06-HF density functionals in appropriate ways is supported by the results of this work.

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.

Full-electron calculation of effective electronic couplings and excitation energies of charge transfer states: Application to hole transfer in DNA pi-stacks

In this work I develop and apply a theoretical method for calculating effective electronic couplings (or transfer integrals) between redox sites involved in hole or electron transfer reactions. The resulting methodology is a refinement and a generalization of a recently developed approach for transfer integral evaluation. In fact, it holds for any overlap between the charge-localized states used to represent charge transfer (CT) processes in the two-state model. The presented theoretical and computational analyses show that the prototype approach is recovered for sufficiently small overlaps. The method does not involve any empirical parameter. It allows a complete multielectron description, therefore including electronic relaxation effects. Furthermore, its theoretical formulation holds at any value of the given reaction coordinate and yields a formula for the evaluation of the vertical excitation energy (i.e., the energy difference between the adiabatic ground and first-excited electronic states) that rests on the same physical quantities used in transfer integral calculation. In this paper the theoretical approach is applied to CT in B-DNA base dimers within the framework of Density Functional Theory (DFT), although it can be implemented in other computational schemes. The results of this work, as compared with previous Hartree-Fock (HF) and post-HF evaluations, support the applicability of the current implementation of the method to larger pi-stacked arrays, where post-HF approaches are computationally unfeasible.

First principles effective electronic couplings for hole transfer in natural and size-expanded DNA

Hole transfer processes between base pairs in natural DNA and size-expanded DNA (xDNA) are studied and compared, by means of an accurate first principles evaluation of the effective electronic couplings (also known as transfer integrals), in order to assess the effect of the base augmentation on the efficiency of charge transport through double-stranded DNA. According to our results, the size expansion increases the average electronic coupling, and thus the CT rate, with potential implications in molecular biology and in the implementation of molecular nanoelectronics. Our analysis shows that the effect of the nucleobase expansion on the charge-transfer (CT) rate is sensitive to the sequence of base pairs. Furthermore, we find that conformational variability is an important factor for the modulation of the CT rate. From a theoretical point of view, this work offers a contribution to the CT chemistry in pi-stacked arrays. Indeed, we compare our methodology against other standard computational frameworks that have been adopted to tackle the problem of CT in DNA, and unravel basic principles that should be accounted for in selecting an appropriate theoretical level.

Mapping protein electron transfer pathways with QM/MM methods

Mixed quantum mechanics/molecular mechanics (QM/MM) methods offer a valuable computational tool for understanding the electron transfer pathway in protein-substrate interactions and protein-protein complexes. These hybrid methods are capable of solving the Schrödinger equation on a small subset of the protein, the quantum region, describing its electronic structure under the polarization effects of the remainder of the protein. By selectively turning on and off different residues in the quantum region, we are able to obtain the electron pathway for short- and large-range interactions. Here, we summarize recent studies involving the protein-substrate interaction in cytochrome P450 camphor, ascorbate peroxidase and cytochrome c peroxidase, and propose a novel approach for the long-range protein-protein electron transfer. The results on ascorbate peroxidase and cytochrome c peroxidase reveal the importance of the propionate groups in the electron transfer pathway. The long-range protein-protein electron transfer has been studied on the cytochrome c peroxidase-cytochrome c complex. The results indicate the importance of Phe82 and Cys81 on cytochrome c, and of Asn196, Ala194, Ala176 and His175 on cytochrome c peroxidase.

Long Distance Electron-Transfer Mechanism in Peptidylglycine α-Hydroxylating Monooxygenase: A Perfect Fitting for a Water Bridge

The active sites of copper enzymes have been the subject of many theoretical and experimental investigations from a number of years. Such studies have embraced topics devoted to the modeling of the first coordination sphere at the metallic cations up to the development of biomimetic, or bioinspired, catalytic systems. At least from the theoretical viewpoint, fewer efforts have been dedicated to elucidate how the two copper cations act concertedly in noncoupled dicopper enzymes such as peptidylglycine alpha-hydroxylating monooxygenase (PHM) and dopamine beta-monooxygenase (DbetaM). In these metalloenzymes, an electronic transfer is assumed between the two distant copper cations (11 A). Recent experimental results suggest that this transfer occurs through water molecules, a phenomenon which has been theoretically evidenced to be of high efficiency in the case of cytochrome b5 (Science, 2005, 310, 1311). In the present contribution dedicated to PHM, we overpass the common theoretical approaches dedicated to the electronic and geometrical structures of sites CuM or CuH restricted to their first coordination spheres and aim at directly comparing theoretical results to the experimentally measured activity of the PHM enzyme. To achieve this goal, molecular dynamics simulations were performed on wild-type and various mutants of PHM. More precisely, we provide an estimate of the electron-transfer efficiency between the CuM and CuH sites by means of such molecular dynamics simulations coupled to Marcus theory joined to the Beratan model to approximate the required coupling matrix elements. The theoretical results are compared to the kinetics measurements performed on wild and mutated PHM. The present work, the dynamic aspects of which are essential, accounts for the experimental results issued from mutagenesis. It supports the conclusion that an electronic transfer can occur between two copper(I) sites along a bridge involving a set of hydrogen and chemical bonds. Residue Gln170 is evidenced to be the keystone of this water-mediated pathway.

Electronic coupling between azurin and gold at different protein/substrate orientations

By means of constrained classical molecular dynamics simulations, we have computed the structure of azurin deposited on a Au(111) surface at different possible orientations and the azimuthal forces acting on the protein at each sampled conformation. We have then evaluated the effect of the angular variation on the speed of electron tunneling between the protein redox site and the metal surface. We find that the azurin/gold electronic coupling has a strong dependence on the molecular orientation and is greatly enhanced by inclining the protein to lie as flat as possible on the surface. We discuss the implications of our results for scanning probe microscopy experiments in which tunneling currents are measured while the protein is subjected to mechanical forces exerted by the tip of the instrument.

Water-Mediated Electron Transfer between Protein Redox Centers

Recent experimental and theoretical investigations show that water molecules between or near redox partners can significantly affect their electron-transfer (ET) properties. Here we study the effects of intervening water molecules on the electron self-exchange reaction of azurin (Az), by performing a conformational sampling on the water medium and by using a newly developed ab initio method to calculate transfer integrals between molecular redox sites. We show that the insertion of water molecules at the interface between the copper active sites of Az dimers slightly increases the overall ET rate, while some favorable water conformations can considerably enhance the ET kinetics. These features are traced back to the interplay of two competing factors: the electrostatic interaction between the water and protein subsystems (mainly opposing the ET process for the water arrangements drawn from MD simulations) and the effectiveness of water in mediating ET coupling pathways. Such an interplay provides a physical basis for the found absence of correlation between the electronic couplings derived through ab initio electronic structure calculations and the related quantities obtained through the Empirical Pathways (EP) method. In fact, the latter does not account for electrostatic effects on the transfer integrals. Thus, we conclude that the water-mediated electron tunneling is not controlled by the geometry of a single physical pathway. We discuss the results in terms of the interplay between different ET pathways controlled by the conformational changes of one of the water molecules via its electrostatic influence. Finally, we examine the dynamical effects of the interfacial water and check the validity of the Condon approximation.

Water Effects on Electron Transfer in Azurin Dimers

Recent experimental and theoretical analyses indicate that water molecules between or near redox partners can significantly affect their electron-transfer (ET) properties. Here, we study the effects of intervening water molecules on the electron self-exchange reaction of azurin (Az) by using a newly developed ab-initio method to calculate transfer integrals between molecular sites. We show that the insertion of water molecules in the gap between the copper active sites of Az dimers slows down the exponential decay of the ET rates with the copper-to-copper distance. Depending on the distance between the redox sites, water can enhance or suppress the electron-transfer kinetics. We show that this behavior can be ascribed to the simultaneous action of two competing effects: the electrostatic interaction of water with the protein subsystem and its ability to mediate ET coupling pathways.

Extracting electron transfer coupling elements from constrained density functional theory

Constrained density functional theory (DFT) is a useful tool for studying electron transfer (ET) reactions. It can straightforwardly construct the charge-localized diabatic states and give a direct measure of the inner-sphere reorganization energy. In this work, a method is presented for calculating the electronic coupling matrix element (Hab) based on constrained DFT. This method completely avoids the use of ground-state DFT energies because they are known to irrationally predict fractional electron transfer in many cases. Instead it makes use of the constrained DFT energies and the Kohn-Sham wave functions for the diabatic states in a careful way. Test calculations on the Zn2+ and the benzene-Cl atom systems show that the new prescription yields reasonable agreement with the standard generalized Mulliken-Hush method. We then proceed to produce the diabatic and adiabatic potential energy curves along the reaction pathway for intervalence ET in the tetrathiafulvalene-diquinone (Q-TTF-Q) anion. While the unconstrained DFT curve has no reaction barrier and gives Hab approximately 17 kcal/mol, which qualitatively disagrees with experimental results, the Hab calculated from constrained DFT is about 3 kcalmol and the generated ground state has a barrier height of 1.70 kcal/mol, successfully predicting (Q-TTF-Q)- to be a class II mixed-valence compound.