On the theory of tunnelling in electron and proton transfer reactions

The concept of tunnelling in the theory of electron and proton transfer reactions has recently been questioned on the ground that the situation is a non-stationary one. It has been suggested that time-dependent perturbation theory should be applied to obtain the quantum mechanical transition probability. We have done this for a square barrier. The result for most reactions is the same as obtained by the WKB approximation.     

On the quantum field theory of photoionisation and electron scattering reactions on atoms

Brillouin-Wigner perturbation theory, formulated in the Schrödinger picture of quantum field theory, is employed to derive a perturbative scheme for the scattering matrix for photoionisation and electron scattering reactions on atoms. It is important to note that the intermediate states appearing in this series are physical, i.e. fully correlated, eigenstates of the total Hamiltonian. The scheme is amenable to numerical analysis: The key point is the use of an “energy-optimised”g-Hartree basis which yields an efficient treatment of atomic correlations and makes Brillouin-Wigner and Rayleigh-Schrödinger perturbation theories coincide.     

On the relationship between the one‐electron and Bohm's quantum potential

The one‐electron potential, derived from the electron density, is a three‐dimensional function, whereas Bohm's quantum potential depends on the spatial coordinates of all involved electrons. To analyze the relationship between the two potentials in the many‐electron case, first the dimensionality of the quantum potential needs to be reduced to match that of the one‐electron potential. A possible approach for such dimensionality reduction by calculating the expectation value of the quantum potential over all but one electron, using the factorization of the real part of the wave function into a marginal and conditional function is presented, and a relation between such three‐dimensional local quantum potential and the one‐electron potential is given. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

The theory of electron transfer

This article provides an overview of the theory of electron transfer. Emphasis is placed on the history of key ideas and on the definition of difficult terms. Among the topics considered are the quantum formulation of electron transfer, the role of thermal fluctuations, the structures of transition states, and the physical models of rate constants. The special case of electron transfer from a metal electrode to a molecule in solution is described in detail.     

Nonadiabatic theory of electron transfer

Tunneling between potential wells is considered as affected by random nonadiabatic change in the resonance condition. A general expression is derived for the transition probability on the assumption that the process involves only two states of the discrete spectrum. Detailed consideration is given to the case of an electron localized by the polarized layer it produces.     

An inner-sphere reorganization model of hetero-exchange electron transfer reaction of diatomic reactants

An accurate theoretical scheme for obtaining directly the inner-sphere reorganization energies of the hetero-exchange electron transfer reactions from ionization potentials and electron affinities is first reported in this paper. Ionization potentials and electron affinities are alternatively obtained from the Rydberg spectroscopic data via a numerical procedure for some diatomic molecules. The inner-sphere reorganization energy values are calculated for the hetero-exchange electron transfer reactions (AB + CD⁺ → AB⁺ + CD) of diatomic molecules and are compared with those from other approximate methods.     

The new theory of electron transfer. Thermodynamic potential profiles in the inverted and superverted regions

In a previous paper (S Fletcher, J Solid State Electrochem 11:965, 2007) a non-Marcus theory of electron transfer was developed, with results applicable to the normal region of thermodynamic driving forces. In the present paper, the theory is extended to highly exergonic reactions (the inverted region) and to highly endergonic reactions (the superverted region). The results are presented mathematically and in the form of Gibbs energy profiles plotted against a charge fluctuation reaction coordinate. The new theory utilizes the concept of donor and acceptor “supermolecules,” which consist of conventional donor and acceptor species plus their associated ionic atmospheres. The key findings are as follows. (1) In the inverted region, donor supermolecules are positively charged both before and after the electron transfer event. (2) In the normal region, donor supermolecules change polarity from negative to positive during the electron transfer event. (3) In the superverted region, the donor supermolecule is negatively charged both before and after the electron transfer event. This overall pattern of events makes it possible for polar solvents to catalyse electron transfer in the inverted and superverted regions. Because this new effect is predicted only by the present theory and not by the Marcus theory, it provides a clear means of distinguishing between them.     

Integral encounter theory of strong electron transfer

The integral encounter theory (IET) has been extended to the reactions limited by diffusion along the reaction coordinate to the level crossing points where either thermal or hot electron transfer occurs. IET describes the bimolecular ionization of the instantaneously excited electron donor D* followed by the hot geminate backward transfer which precedes the ion pair equilibration and its subsequent thermal recombination. We demonstrate that the fraction of ion pairs which avoids the hot recombination is much smaller than their initial number when the electron tunneling is strong.     

On the construction of diabatic and adiabatic potential energy surfaces based on ab initio valence bond theory

A theoretical model is presented for deriving effective diabatic states based on ab initio valence bond self-consistent field (VBSCF) theory by reducing the multiconfigurational VB Hamiltonian into an effective two-state model. We describe two computational approaches for the optimization of the effective diabatic configurations, resulting in two ways of interpreting such effective diabatic states. In the variational diabatic configuration (VDC) method, the energies of the diabatic states are variationally minimized. In the consistent diabatic configuration (CDC) method, both the configuration coefficients and orbital coefficients are simultaneously optimized to minimize the adiabatic ground-state energy in VBSCF calculations. In addition, we describe a mixed molecular orbital and valence bond (MOVB) approach to construct the CDC diabatic and adiabatic states for a chemical reaction. Note that the VDC-MOVB method has been described previously. Employing the symmetric S(N)2 reaction between NH(3) and CH(3)NH(3)(+) as a test system, we found that the results from ab initio VBSCF and from ab initio MOVB calculations using the same basis set are in good agreement, suggesting that the computationally efficient MOVB method is a reasonable model for VB simulations of condensed phase reactions. The results indicate that CDC and VDC diabatic states converge, respectively, to covalent and ionic states as the molecular geometries are distorted from the minimum of the respective diabatic state along the reaction coordinate. Furthermore, the resonance energy that stabilizes the energy of crossing between the two diabatic states, resulting in the transition state of the adiabatic ground-state reaction, has a strong dependence on the overlap integral between the two diabatic states and is a function of both the exchange integral and the total diabatic ground-state energy.     

Light-harvesting energy transfer and subsequent electron transfer of cationic porphyrin complexes on clay surfaces

A novel energy-transfer system involving nonaggregated cationic porphyrins adsorbed on an anionic-type clay surface and the electron-transfer reaction that occurs after light harvesting are described. In the clay-porphyrin complexes, photochemical energy transfer from excited singlet zinc porphyrins to free-base porphyrins proceeds. The photochemical electron-transfer reaction from an electron donor in solution (hydroquinone) to the adsorbed porphyrin in the excited singlet state was also examined. Because the electron-transfer rate from the hydroquinone to the excited singlet free-base porphyrin is larger than that to the excited singlet zinc porphyrin, we conclude that the energy transfer accelerates the overall electron-transfer reaction.     

Non-Hermitian quantum mechanics: wave packet propagation on autoionizing potential energy surfaces

The correspondence between the time-dependent and time-independent molecular dynamic formalisms is shown for autoionizing processes. We demonstrate that the definition of the inner product in non-Hermitian quantum mechanics plays a key role in the proof. When the final state of the process is dissociative, it is technically favorable to introduce a complex absorbing potential into the calculations. The conditions which this potential should fulfill are briefly discussed. An illustrative numerical example is presented involving three potential energy surfaces.     

Raman studies on the interaction of the reactants with the platinum nanoparticle surface during the nanocatalyzed electron transfer reaction

Raman studies are conducted to understand the specific interactions between the individual reactants and the platinum nanoparticle surface during the nanocatalyzed electron transfer reaction between hexacyanoferrate (III) ions and thiosulfate ions. When Pt nanoparticles are added to the thiosulfate ion solution, a shift in the symmetric SS stretching mode is observed compared to the frequency observed for the free thiosulfate ions in solution, suggesting that binding to the Pt nanoparticle surface occurs via the S- ion. It is also observed that there are no shifts in the symmetric and asymmetric OSO bending or SO stretching frequencies. This suggests that the thiosulfate ions do not bind to the nanoparticle surface via the O- ion. When platinum nanoparticles are added to the hexacyanoferrate(III) ion solution, evidence is found for both adsorbed hexacyanoferrate(III) ions and a platinum cyanide complex. For adsorbed hexacyanoferrate(III) ions, the CN stretching frequency is observed at 2101 cm(-1) and the Fe-C stretching frequency is found at 368 cm(-1). The observed CN stretching frequencies located at 2147 and 2167 cm(-1) provide strong evidence that there is a Pt(CN)4(2-) platinum cyanide complex formed. In addition, the Pt-CN band is also observed at 2054 cm(-1). These observed bands provide spectroscopic evidence that the hexacyanoferrate(III) ions dissolve by forming a complex with the surface platinum atoms of the nanoparticles. Raman spectra of the product mixtures are obtained after the completion of the reaction when carried out with higher reactant concentrations to observe the Raman spectra, but with a similar 10:1 ratio of thiosulfate to hexacyanoferrate(III) ions as used previously, with and without PVP-Pt nanoparticles at a correspondingly higher concentration. It is observed that there are no shifts in the characteristic Raman bands associated with hexacyanoferrate(II) ions and no evidence for the formation of adsorbed hexacyanoferrate(II) species or platinum cyanide complexes in the presence of the platinum nanoparticles. In addition, there is evidence for the shifted symmetric SS stretching mode, suggesting that some of the unreacted thiosulfate (present in large excess) is bound to the Pt nanoparticle surface. Thus, under the actual reaction conditions, the hexacyanoferrate(III) ions preferentially react with adsorbed thiosulfate ions to form the reaction products, and this supports the surface catalytic mechanism we proposed previously.     

Ab initio quantum mechanical/molecular mechanical simulation of electron transfer process: fractional electron approach

Electron transfer (ET) reactions are one of the most important processes in chemistry and biology. Because of the quantum nature of the processes and the complicated roles of the solvent, theoretical study of ET processes is challenging. To simulate ET processes at the electronic level, we have developed an efficient density functional theory (DFT) quantum mechanical (QM)/molecular mechanical (MM) approach that uses the fractional number of electrons as the order parameter to calculate the redox free energy of ET reactions in solution. We applied this method to study the ET reactions of the aqueous metal complexes Fe(H(2)O)(6)(2+/3+) and Ru(H(2)O)(6)(2+/3+). The calculated oxidation potentials, 5.82 eV for Fe(II/III) and 5.14 eV for Ru(II/III), agree well with the experimental data, 5.50 and 4.96 eV, for iron and ruthenium, respectively. Furthermore, we have constructed the diabatic free energy surfaces from histogram analysis based on the molecular dynamics trajectories. The resulting reorganization energy and the diabatic activation energy also show good agreement with experimental data. Our calculations show that using the fractional number of electrons (FNE) as the order parameter in the thermodynamic integration process leads to efficient sampling and validate the ab initio QM/MM approach in the calculation of redox free energies.     

Structural dynamics of surfaces by ultrafast electron crystallography: experimental and multiple scattering theory

Recent studies in ultrafast electron crystallography (UEC) using a reflection diffraction geometry have enabled the investigation of a wide range of phenomena on the femtosecond and picosecond time scales. In all these studies, the analysis of the diffraction patterns and their temporal change after excitation was performed within the kinematical scattering theory. In this contribution, we address the question, to what extent dynamical scattering effects have to be included in order to obtain quantitative information about structural dynamics. We discuss different scattering regimes and provide diffraction maps that describe all essential features of scatterings and observables. The effects are quantified by dynamical scattering simulations and examined by direct comparison to the results of ultrafast electron diffraction experiments on an in situ prepared Ni(100) surface, for which structural dynamics can be well described by a two-temperature model. We also report calculations for graphite surfaces. The theoretical framework provided here allows for further UEC studies of surfaces especially at larger penetration depths and for those of heavy-atom materials.     

On the origins of intersecting potential energy surfaces

Techniques developed for investigating nonadiabatic processes in molecular systems are adapted to study the structure and properties of holomorphic and meromorphic functions of a complex variable, \(f(z)=\mathfrak {R}(f)+i\,\mathfrak {I}(f)\). The connection is that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) are correlated two-dimensional scalar functions, interrelated by the Cauchy–Riemann equations. Exploiting this fact, it is demonstrated that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) of f can be envisaged in Euclidean \({\mathbb {R}}^{3}\) space as a two-state set of constrained, intersecting two-dimensional potential energy surfaces (PESs), called the graph of f. Importantly, the analytic and algebraic properties of f dictate the geometric structure evinced in the graph of f. This parallels multi-state sets of higher-dimensional, constrained, intersecting PESs linked with correlated electronic eigenstates of the parameterized molecular Hamiltonian operator. In view of this association, the language and mathematical infrastructure devised by chemists for discussing and analyzing intersections in higher-dimensional PESs are suitably modified for f. Notably, an algorithm capable of optimizing roots and poles of f through analysis of the real, two-dimensional \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) functions is derived, which is based on intersection-adapted coordinate and constrained Lagrangian methodologies. As constrained, intersecting PESs are indispensible for conceptualizing and characterizing the physics governing nonadiabatic phenomena, f represents a foundational bridge to these more abstract constructions.     

The new theory of electron transfer: application to the photosynthetic reaction centre

Based on recent developments in the theory of electron transfer, we prove that a non-polar environment is needed to maintain the high efficiency and chemical integrity of the photosynthetic reaction centre. We also determine the Gibbs energy diagram for the primary act of charge separation in photosynthesis, and propose an equivalent circuit that captures the principal features of the entire acceptor side of the electron transport chain in photosystem II.     

Nonlinear coupling in the quantum statistical theory of proton transfer dynamics

The possible role of the nonlinear coupling on the character of the dynamics of particle transfer process is investigated. The analysis and solutions of the kinetic equation indicate that nonlinear coupling causes symmetry breaking of particle transfer potential and determines possible equilibrium structure of the system. Dissipative coupling characterizes the rate of the system to reach thermodynamic equilibrium and along with nonlinear coupling and parameters of the system determines in a unique way the resulting equilibrium structure of the system.