▪ Abstract Proton-coupled electron transfer (PCET) is an important mechanism for charge transfer in a wide variety of systems including biology- and materials-oriented venues. We review several are...

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Proton-Coupled Electron Transfer

The latest release of the Crystal program for solid-state quantum-mechanical ab initio simulations is presented. The program adopts atom-centered Gaussian-type functions as a basis set, which makes it possible to perform all-electron as well as pseudopotential calculations. Systems of any periodicity can be treated at the same level of accuracy (from 0D molecules, clusters and nanocrystals, to 1D polymers, helices, nanorods, and nanotubes, to 2D monolayers and slab models for surfaces, to actual 3D bulk crystals), without any artificial repetition along nonperiodic directions for 0–2D systems. Density functional theory calculations can be performed with a variety of functionals belonging to several classes: local-density (LDA), generalized-gradient (GGA), meta-GGA, global hybrid, range-separated hybrid, and self-consistent system-specific hybrid. In particular, hybrid functionals can be used at a modest computational cost, comparable to that of pure LDA and GGA formulations, because of the efficient implementation of exact nonlocal Fock exchange. Both translational and point-symmetry features are fully exploited at all steps of the calculation, thus drastically reducing the corresponding computational cost. The various properties computed encompass electronic structure (including magnetic spin-polarized open-shell systems, electron density analysis), geometry (including full or constrained optimization, transition-state search), vibrational properties (frequencies, infrared and Raman intensities, phonon density of states), thermal properties (quasi-harmonic approximation), linear and nonlinear optical properties (static and dynamic [hyper]polarizabilities), strain properties (elasticity, piezoelectricity, photoelasticity), electron transport properties (Boltzmann, transport across nanojunctions), as well as X-ray and inelastic neutron spectra. The program is distributed in serial, parallel, and massively parallel versions. In this paper, the original developments that have been devised and implemented in the last 4 years (since the distribution of the previous public version, Crystal14, occurred in December 2013) are described.

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Quantum-mechanical condensed matter simulations with CRYSTAL

Coupled electron and proton transfer reactions play a key role in the mechanisms of biological energy transduction.1–3 Such reactions are also fundamental for artificial energy-related systems such as fuel cells, chemical sensors, and other electrochemical devices. Biological examples include, among others, cytochrome c oxidase,4,5 bc1 complex,6,7 and photosynthetic reaction centers.8,9 In such systems, electrons tunnel between redox cofactors of an enzyme, while the coupled protons are transferred either across a single hydrogen bond or between protonatable groups along special proton-conducting channels.

In this paper general theories and models of coupled electron transfer/proton transfer (ET/PT) reactions are discussed. Pure electron transfer reactions in proteins have been thoroughly studied in the past, both experimentally10–17 and theoretically.18–25 The coupled reactions are relatively new and currently are gaining attention in the field.6,8,26–43

Two types of coupled reactions can be distinguished. In concerted electron and proton transfer reactions (denoted PCET in Refs. 29,30,43–45, although this term is also used more generally), both the ET and PT transitions occur in one step. Such concerted processes occur in reactions in which proton transfer is typically limited to one hydrogen bond; however, examples with multiple hydrogen bond rearrangements are also known.46 In sequential reactions, the transitions occur in two steps: ET/PT or PT/ET. Typically each individual step is uphill in energy, while the coupled reaction is downhill.

A sequential reaction can proceed along two parallel channels: ET then PT (EP) or PT then ET (PE). In each channel the reaction involves two sequential steps: uphill activation, and then downhill reaction to the final product state. The lifetime of the activated complex is limited by the back reaction. The general formula for the rate of such reactions can be easily developed. In the context of bioenergetics issues, however, it is interesting to analyze all of the possible cases separately because each corresponds to a different mechanism: for example, an electron can go first and pull out a proton; alternatively, a proton can go first and pull out an electron; or an electron can jump back and forth between donor and acceptor and gradually pull out a proton. In enzymes involving coupled proton and electron transport, the exact mechanism of the reaction is of prime interest.

First we will consider a simple four-state model of reactions where the proton moves across a single hydrogen bond; both concerted and sequential reactions will be treated. Then we will consider models for long-distance proton transfer, also denoted proton transport or proton translocation. Typically, electron transfer coupled to proton translocation in proteins involves an electron tunneling over a long distance between two redox cofactors, coupled to a proton moving along a proton conducting channel in a classical, diffusion-like random walk fashion. Again, separately the electron and proton transfer reactions are typically uphill, while the coupled reaction is downhill in energy. The schematics of this process is shown in Fig. 1. The kinetics of such reactions can be much different from those involving proton transfer across a single hydrogen bond. In this paper, we will discuss the specifics of such long-distance proton-coupled reactions.



Fig. 1

Schematics of the electron transfer reaction coupled to proton translocation. In the reaction, an electron is tunneling over a long distance between two redox cofactors, O and R, and a coupled proton is transferred over a proton conducting channel. The ...



Following the review of theoretical concepts, a few applications will be discussed. First the phenoxyl/phenol and benzyl/toluene self-exchange reactions will be examined. The phenoxyl/phenol reaction involves electronically nonadiabatic proton transfer and corresponds to a proton-coupled electron transfer (PCET) mechanism, whereas the benzyl/toluene reaction involves electronically adiabatic proton transfer and corresponds to a hydrogen atom transfer (HAT) mechanism. Comparison of these two systems provides insight into fundamental aspects of electron-proton interactions in these types of systems. Next a series of theoretical calculations on experimentally studied PCET reactions in solution and enzymes will be summarized, along with general predictions concerning the dependence of rates and kinetic isotope effects (the ratio of the rate constants for hydrogen and deuterium transfer) on system properties such as temperature and driving force. The final application that will be discussed is cytochrome c oxidase (CcO). CcO is the terminal component of the electron transport chain of the respiratory system in mitochondria and is one of the key enzymes responsible for energy generation in cells. The intricate correlation between the electron and proton transport via electrostatic interactions, as well as the kinetics of the coupled transitions, appear to be the basis of the pumping mechanism in this enzyme.

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Theory of Coupled Electron and Proton Transfer Reactions

Electrochemistry of redox-active self-assembled monolayers

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn–Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.

https://www.pnas.org/content/pnas/114/11/2801.full.pdf

Understanding band gaps of solids in generalized Kohn–Sham theory

A self-consistent scheme for determining the optimal fraction of exact exchange for full-range hybrid functionals is presented and applied to the calculation of band gaps and dielectric constants of solids. The exchange-correlation functional is defined in a similar manner to the PBE0 functional, but the mixing parameter is set equal to the inverse macroscopic dielectric function and it is determined self-consistently by computing the optimal dielectric screening. We found excellent agreement with experiments for the properties of a broad class of systems, with band gaps ranging between 0.7 and 21.7 eV and dielectric constants within 1.23 and 15.9. We propose that the eigenvalues and eigenfunctions obtained with the present self-consistent hybrid scheme may be excellent inputs for ${G}_{0}{W}_{0}$ calculations.

Self-consistent hybrid functional for condensed systems

Dielectric-dependent hybrid (DDH) functionals were recently shown to yield accurate energy gaps and dielectric constants for a wide variety of solids, at a computational cost considerably less than that of $\mathit{GW}$ calculations. The fraction of exact exchange included in the definition of DDH functionals depends (self-consistently) on the dielectric constant of the material. Here we introduce a range-separated (RS) version of DDH functionals where short- and long-range components are matched using system-dependent, nonempirical parameters. We show that RS-DDHs yield accurate electronic properties of inorganic and organic solids, including energy gaps and absolute ionization potentials. Furthermore we show that these functionals may be generalized to finite systems.

Nonempirical range-separated hybrid functionals for solids and molecules

The vibronic couplings for the phenoxyl/phenol and the benzyl/toluene self-exchange reactions are calculated with a semiclassical approach, in which all electrons and the transferring hydrogen nucleus are treated quantum mechanically. In this formulation, the vibronic coupling is the Hamiltonian matrix element between the reactant and product mixed electronic−proton vibrational wavefunctions. The magnitude of the vibronic coupling and its dependence on the proton donor−acceptor distance can significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of proton-coupled electrontransfer reactions. Both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. For the phenoxyl/phenol system, the electrons are unable to rearrange fast enough to follow the prot...

Calculation of vibronic couplings for phenoxyl/phenol and benzyl/toluene self-exchange reactions: implications for proton-coupled electron transfer mechanisms.

Toward the accurate calculation of pKa values in water and acetonitrile.

The Lewis acidity of metal–organic frameworks (MOFs) has attracted much research interest in recent years. We report here the development of two quantitative methods for determining the Lewis acidity of MOFs—based on electron paramagnetic resonance (EPR) spectroscopy of MOF-bound superoxide (O2•–) and fluorescence spectroscopy of MOF-bound N-methylacridone (NMA)—and a simple strategy that significantly enhances MOF Lewis acidity through ligand perfluorination. Two new perfluorinated MOFs, Zr6-fBDC and Zr6-fBPDC, where H2fBDC is 2,3,5,6-tetrafluoro-1,4-benzenedicarboxylic acid and H2fBPDC is 2,2′,3,3′,5,5′,6,6′-octafluoro-4,4′-biphenyldicarboxylic acid, were shown to be significantly more Lewis acidic than nonsubstituted UiO-66 and UiO-67 as well as the nitrated MOFs Zr6-BDC-NO2 and Zr6-BPDC-(NO2)2. Zr6-fBDC was shown to be a highly active single-site solid Lewis acid catalyst for Diels–Alder and arene C–H iodination reactions. Thus, this work establishes the important role of ligand perfluorination in enh...

Jonathan H. Skone

Papers

Self-consistent hybrid functional for condensed systems

Nonempirical range-separated hybrid functionals for solids and molecules

Calculation of vibronic couplings for phenoxyl/phenol and benzyl/toluene self-exchange reactions: implications for proton-coupled electron transfer mechanisms.

Toward the accurate calculation of pKa values in water and acetonitrile.

Tuning Lewis Acidity of Metal–Organic Frameworks via Perfluorination of Bridging Ligands: Spectroscopic, Theoretical, and Catalytic Studies