Free energy functionals for polarization fluctuations: Pekar factor revisited

Redox potentials in ionic liquids: Anomalous behavior?

Redox potentials depend on the nature of the solvent/electrolyte through the solvation energies of the ionic solute species. For concentrated electrolytes, ion solvation may deviate significantly from the Born model predictions due to ion pairing and correlation effects. Recently, Ghorai and Matyushov [J. Phys. Chem. B 124, 3754-3769 (2020)] predicted, on the basis of linear response theory, an anomalous trend in the solvation energies of room temperature ionic liquids, with deviations of hundreds of kJ/mol from the Born model for certain size solutes/ions. In this work, we computationally evaluate ionic solvation energies in the prototypical ionic liquid, 1-butyl-3-methylimidazolium tetrafluoroborate (BMIM/BF4), to further explore this behavior and benchmark several of the approximations utilized in the solvation energy predictions. For comparison, we additionally compute solvation energies within acetonitrile and molten NaCl salt to illustrate the limiting behavior of purely dipolar and ionic solvents. We find that the overscreening effect, which results from the inherent charge oscillations of the ionic liquid, is substantially reduced in magnitude due to screening from the dipoles of the molecular ions. Therefore, for the molten NaCl salt, for which the ions do not have permanent dipoles, modulation of ionic solvation energies from the overscreening effect is most significant. The conclusion is that ionic liquids do indeed exhibit unique solvation behavior due to peak(s) in the electrical susceptibility caused by the ion shell structure; redox potential shifts for BMIM/BF4 are of more modest order ∼0.1 V, but may be larger for other ionic liquids that approach molten salt behavior.

Electrical noise in electrolytes: a theoretical perspective

Seemingly unrelated experiments such as electrolyte transport through nanotubes, nano-scale electrochemistry, NMR relaxometry and surface force balance measurements, all probe electrical fluctuations: of the electric current, the charge and polarization, the field gradient (for quadrupolar nuclei) and the coupled mass/charge densities. The fluctuations of such various observables arise from the same underlying microscopic dynamics of the ions and solvent molecules. In principle, the relevant length and time scales of these dynamics are encoded in the dynamic structure factors. However, modelling the latter for frequencies and wavevectors spanning many orders of magnitude remains a great challenge to interpret the experiments in terms of physical processes such as solvation dynamics, diffusion, electrostatic and hydrodynamic interactions between ions, interactions with solid surfaces, etc. Here, we highlight the central role of the charge-charge dynamic structure factor in the fluctuations of electrical observables in electrolytes and offer a unifying perspective over a variety of complementary experiments. We further analyze this quantity in the special case of an aqueous NaCl electrolyte, using simulations with explicit ions and an explicit or implicit solvent. We discuss the ability of the standard Poisson-Nernst-Planck theory to capture the simulation results, and how the predictions can be improved. We finally discuss the contributions of ions and water to the total charge fluctuations. This work illustrates an ongoing effort towards a comprehensive understanding of electrical fluctuations in bulk and confined electrolytes, in order to enable experimentalists to decipher the microscopic properties encoded in the measured electrical noise.

Electron Tunneling in Biology: When Does it Matter?

Electrons can tunnel between cofactor molecules positioned along biological electron transport chains up to a distance of ≃ 20 Å on the millisecond time scale of enzymatic turnover. This tunneling range determines the design of biological energy chains facilitating the cross-membrane transport of electrons. Tunneling distance and cofactors' redox potentials become the main physical parameters affecting the rate of electron transport. In addition, universal charge-transport properties are assigned to all proteins, making protein identity, flexibility, and dynamics insignificant. This paradigm is challenged by dynamical models of electron transfer, showing that the electron hopping rate is constant within the crossover distance R* ≃ 12 Å, followed with an exponential falloff at longer distances. If this hypothesis is fully confirmed, natural and man-made energy chains for electron transport should be best designed by placing redox cofactors near the crossover distance R*. Protein flexibility and dynamics affect the magnitude of the maximum hopping rate within the crossover distance. Changes in protein flexibility between forward and backward transitions contribute to vectorial charge transport. For biological energy chains, charge transport through proteins is not defined by universal parameters, and protein identity matters.

Reorganization energy of electron transfer

The theory of electron transfer reactions establishes the conceptual foundation for redox solution chemistry, electrochemistry, and bioenergetics. Electron and proton transfer across the cellular membrane provide all energy of life gained through natural photosynthesis and mitochondrial respiration. Rates of biological charge transfer set kinetic bottlenecks for biological energy storage. The main system-specific parameter determining the activation barrier for a single electron-transfer hop is the reorganization energy of the medium. Both harvesting of light energy in natural and artificial photosynthesis and efficient electron transport in biological energy chains require reduction of the reorganization energy to allow fast transitions. This review article discusses mechanisms by which small values of the reorganization energy are achieved in protein electron transfer and how similar mechanisms can operate in other media, such as nonpolar and ionic liquids. One of the major mechanisms of reorganization energy reduction is through non-Gibbsian (nonergodic) sampling of the medium configurations on the reaction time. A number of alternative mechanisms, such as electrowetting of active sites of proteins, give rise to non-parabolic free energy surfaces of electron transfer. These mechanisms, and nonequilibrium population of donor-acceptor vibrations, lead to a universal phenomenology of separation between the Stokes shift and variance reorganization energies of electron transfer.

Reorganization Energy of Electron Transfer in Ionic Liquids

Bandshape analysis of charge-transfer optical bands in room-temperature ionic liquids (ILs) was performed to extract the reorganization energy of electron transfer. Remarkably, the reorganization energies in ILs are close to those in cyclohexane. This result runs against common wisdom in the field since conducting ILs, which are characterized by an infinite static dielectric constant, and nonpolar cyclohexane fall to the opposite ends of the polarity scale based on their dielectric constants. Theoretical calculations employing structure factors of ILs from molecular dynamics simulations support the low values of the reorganization energy. Standard dielectric arguments do not apply to solvation in ILs, and nonergodic reorganization energies are required for a quantitative analysis.

Molecules, the Ultimate Nanomotor: Linking Chemical Reaction Intermediates to their Molecular Diffusivity

The intellectual community focused on nanomotors has recently become interested in extending these concepts to individual molecules. Here, we study a chemical reaction according to whose mechanism some intermediate species should speed up while others slow down in predictable ways, if the nanomotor hypothesis of boosted diffusion holds. Accordingly, we scrutinize the absolute diffusion coefficient (D) during intermediate steps of the catalytic cycle for the CuAAC reaction (copper-catalyzed azide-alkyne cycloaddition click reaction), using proton pulsed field-gradient nuclear magnetic resonance to discriminate between the diffusion of various reaction intermediates. We observe time-dependent diffusion that is enhanced for some intermediate molecular species and depressed for those whose size increases owing to complex formation. These findings point to the failure of the conventional Stokes-Einstein equation to fully explain diffusivity during chemical reaction. Without attempting a firm explanation, this paper highlights aspects of the physics of chemical reactions that are imperfectly understood and presents systematic data that can be used to assess hypotheses.

Ewald sum corrections in simulations of ion and dipole solvation and electron transfer

Periodic boundary conditions and Ewald sums used in standard simulation protocols require finite-size corrections when the total charge of the simulated system is nonzero. Corrections for ion solvation were introduced by Hummer, Pratt, and García, [J. Chem. Phys. 107, 9275 (1997)]. The latter approach is extended here to derive finite-size correction for the Stokes-shift and reorganization energy applied to electron-transfer reactions. The same correction term, scaling inversely with the box size, adds to the reorganization energy from the energy-gap variance but is subtracted from the reorganization energy calculated from the Stokes shift. Finite-size corrections thus widen the gap between these two quantities, which were recently found to diverge for protein electron transfer. Corrections to the free energy of dipole solvation and the variance of the electric field scale as m²/L³ with the solute dipole m and the box size L.

Electron transfer in nonpolar media

Electron transfer in nonpolar media violates the temperature scaling predicted by the fluctuation–dissipation theorem.

Ergodicity-Breaking in Thermal Biological Electron Transfer? Cytochrome C Revisited

It was recently suggested that certain redox proteins operate in an ergodicity-breaking regime to facilitate biological electron transfer (ET). A signature for this is a large variance reorganization free energy (several electronvolts) but a significantly smaller Stokes reorganization free energy due to incomplete protein relaxation on the time scale of the ET event. Here we investigate whether this picture holds for oxidation of cytochrome c in aqueous solution, at various levels of theory including classical molecular dynamics with two additive and one electronically polarizable force field, and QM/MM calculations with the QM region treated by full electrostatic DFT embedding and by the perturbed matrix method. Sampling the protein and energy gap dynamics over more than 250 ns, we find no evidence for ergodicity-breaking effects. In particular, the inclusion of electronic polarizability of the heme group at QM/MM levels did not induce nonergodic effects, contrary to previous reports by Matyushov et al. The well-known problem of overestimation of reorganization free energies with additive force fields is cured when the protein and solvent are treated as electronically polarizable. Ergodicity-breaking effects may occur in other redox proteins, and our results suggest that long simulations, ideally on the ET time scale, with electronically polarizable force fields are required to obtain strong numerical evidence for them.

Screening of Coulomb interactions in liquid dielectrics

The interaction of charges in dielectric materials is screened by the dielectric constant of the bulk dielectric. In dielectric theories, screening is assigned to surface charges appearing from preferential orientations of dipoles along the local field in the interface. For liquid dielectrics, such interfacial orientations are affected by the interfacial structure characterized by a separate interfacial dipolar susceptibility. We argue that dielectric properties of polar liquids should be characterized by two distinct susceptibilities responsible for local response (solvation) and long-range response (dielectric screening). A microscopic model for dipolar screening in polar liquids is developed here. It shows that the standard bulk dielectric constant is responsible for screening at large distances. The potential of mean force between ions in polar liquids becomes oscillatory at short distances. Oscillations arise from the coupling of collective longitudinal excitations (dipolarons) of the polar liquid with its interfacial density profile. Interfacial electrostatics demonstrates a crossover beyond the solute radius of  ∼1 nm from the scaling laws roughly consistent with continuum expectations for small solutes to new scaling trends characterizing the much softer nano-scale solute-solvent interface. This crossover also marks the transition to a continuum-type electrostatic screening between ions, when short-distance oscillation of the potential of mean force become strongly damped. Screening oscillations are enhanced for more structured interfaces.

Wetting of the Protein Active Site Leads to Non-Marcusian Reaction Kinetics

Enzymes exist in continuously fluctuating water bath dramatically affecting their function. Water not only forms the solvation shell but also penetrates into the protein interior. Changing the wetting pattern of the protein's active site in response to altering redox state initiates a highly nonlinear structural change and non-Gaussian electrostatic fluctuations at the active site. The free-energy surfaces of electron transfer are highly nonparabolic (non-Marcusian), as shown by atomistic molecular dynamics simulations of hydrated ferredoxin protein and by an analytical model in agreement with simulations. The reorganization energy of electron transfer passes through a spike marking equal probabilities of the wet and dry states of the active site. The activation thermodynamics affected by wetting leads to a non-Arrhenius, passing through a maximum, plot for the reaction rate vs the inverse temperature.

Q-model of electrode reactions: altering force constants of intramolecular vibrations

A theory of redox reactions involving electron transfer between a metal electrode and a molecule in solution is formulated in terms of two types of nuclear coordinates of the thermal bath: electrostatic polarization of the medium and local low-frequency vibrations. The polarization fluctuations follow Gaussian statistics. In contrast, the vibrational coordinate is allowed to change its force constant between two oxidation states of the reactant, which is projected onto non-Gaussian fluctuations of the reactant's electronic states. A closed-form analytical theory for the electrode redox reactions is formulated in terms of three reorganization energies: the reorganization energy for the electrostatic polarization of the medium and two internal (vibrational) reorganization energies for the reduced and oxidized states of the reactant. The theory predicts asymmetry between the cathodic and anodic branches of the electrode current driven by the corresponding difference in the vibrational force constants.

Simulating biological charge transfer: Continuum dielectric theory or molecular dynamics?

We study the thermodynamic parameters of Marcus's theory of charge transfer, the driving forces and the reorganization energies, using two widely applied approaches to bioenergetic problems that seem to be radically different: continuum dielectric theory via a numerical solution of Poisson's equation, and the thermodynamic integration approach based upon classical Newtonian molecular dynamics, as perfomed by Na et al., PCCP 19, 18,938 (2017). With application to a nitrite reductase NrfHA protein heterodimer, we obtain an excellent agreement between the respective driving forces with an r.m.s. deviation of 1.7 kcal/mol, and a lower limit to the reorganization energies. The computational methods turn out to be mutually supportive: molecular dynamics can be used to determine the parameters of a dielectric theory computation, which on the other hand can be used to properly rescale the reorganization energies and partition them into aqueous and protein contributions. In addition, we use the electrostatic approach to study the influence of Ca²⁺ ions on the free energy landscape of charge transfer.

Theory and Electrochemistry of Cytochrome c

Extensive simulations of cytochrome c in solution are performed to address the apparent contradiction between large reorganization energies of protein electron transfer typically reported by atomistic simulations and much smaller values produced by protein electrochemistry. The two sets of data are reconciled by deriving the activation barrier for electrochemical reaction in terms of an effective reorganization energy composed of half the Stokes shift (characterizing the medium polarization in response to electron transfer) and the variance reorganization energy (characterizing the breadth of electrostatic fluctuations). This effective reorganization energy is much smaller than each of the two components contributing to it and is fully consistent with electrochemical measurements. Calculations in the range of temperatures between 280 and 360 K combine long, classical molecular dynamics simulations with quantum calculations of the protein active site. The results agree with the Arrhenius plots for the reaction rates and with cyclic voltammetry of cytochrome c immobilized on self-assembled monolayers. Small effective reorganization energy, and the resulting small activation barrier, is a general phenomenology of protein electron transfer allowing fast electron transport within biological energy chains.