Electron transfer Hamiltonian

Intermolecular interconversion occurs via the electron-transfer Hamiltonian, f/7. For parallel chains with nearest neighbour electron transfer this is... 

Kurnikov I V and Beratan D N 1996 Ab initio based effective Hamiltonians for long-range electron transfer Hartree-Fock analysis J. Chem. Phys. 105 9561-73... 

In this chapter, we wiU review electrochemical electron transfer theory on metal electrodes, starting from the theories of Marcus [1956] and Hush [1958] and ending with the catalysis of bond-breaking reactions. On this route, we will explore the relation to ion transfer reactions, and also cover the earlier models for noncatalytic bond breaking. Obviously, this will be a tour de force, and many interesting side-issues win be left unexplored. However, we hope that the unifying view that we present, based on a framework of model Hamiltonians, will clarify the various aspects of this most important class of electrochemical reactions. 

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

The third term, Uqt, in Eq. (27) is due to the partial electron transfer between an ion and solvents in its immediate vicinity. The model Hamiltonian approach [33], described in Section V, has shown that Uqt (= AW in Ref. 33) per primary solvent molecule, for an ion such as the polyanion, can also be expressed as a function of E, approximately a quadratic equation ... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

Let us consider an electron transfer system, whose Hamiltonian may be written (r,Q)=H(r,Q)-t-TN,... 

We first recall that the value pertinent in the electron transfer problem is that evaluated for the nuclear configuration Q Q, where the energy of the interseetion surface of (Q) and Hbb (Q) is a minimum. In some systems, it may happen that vl/ and 1]/ are closely related to stationary states of the Hamiltonian H, so that spectroscopic experiments performed on these states may provide useful information about the value of [47, 48]. To clarify this point, we expand the stationary states )/i (i= 1,2,. . . ) of H(r, Q) in the form ... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

Figure 10. Classical adiabatic free energy curve (solid line) forthe Fe /Fe electron transfer at the water/Pt(lll) interface calculated using the Anderson-Newns Hamiltonian and the molecular dynamics umbrella sampling method. Also shown by the dashed line is the parabolic fit of the data. (Reprinted from Ref. 14.)...

Calhoun and Voth also utilized molecular dynamic simulations using the Anderson-Newns Hamiltonian to determine the free energy profile for an adiabatic electron transfer involving an Fe /Fe redox couple at an electrolyte/Pt(lll) metal interface. This treatment expands upon their earlier simulation by including, in particular, the influence of the motion of the redox ions and the counterions at the interface. 

In the early 1990s a few classical semimoleculai and molecular models of electron transfer reactions involving bond breaking appeared in the literature. A quantum mechanical treatment of a unified mr el of electrochemical electron and ion transfer reactions involving bond breaking was put forward by Schmickler using Anderson-Newns Hamiltonian formalism (see Section V.2). 

In this treatment, the Anderson-Newns Hamiltonian was utilized to determine the potential energy surface for both ion transfer, 21" -> I2 and electron transfer, + e at a Pt electrode. Here the solvent part... 

Fe ". In the two-state model, the electron transfer is viewed as a quantum transition between two localized states V, - and Pf. In IF,-, the ion with charge <7/ is at equilibrium with the interfacial water molecules, and the electron is in the metal. In the metal has lost one electron, and the ion with charge q/ is at equilibrium with the interfacial water. The total Hamiltonian of the system H, including all nuclear and electronic degrees of freedom, is not diagonal in the basis ( , , Pf), and so if the system is prepared in the state P, it will evolve in time according to ... 

The calculation of the transmission coefficient for adiabatic electron transfer modeled by the classical Hamiltonian Hajis based on a similar procedure developed for simulations of general chemical reactions in solution. The basic idea is to start the dynamic trajectory from an equilibrium ensemble constrained to the transition state. By following each trajectory until its fate is determined (reactive or nonreactive), it is possible to determine k. A large number of trajectories are needed to sample the ensemble and to provide an accurate value of k. More details... 

The fact that there are an infinite number of electronic degrees of freedom in the metal, and an analysis of experimental results by Schmick-ler, suggest that electron transfer at the solution/metal interface is near the adiabatic limit. A particularly useful approach is based on the Anderson-Newns approach to adsorption. When it is adapted to the electron transfer problem, the total Hamiltonian of the system is given... 

Feuchtwang, T. E. (1979). Tunneling theory without the transfer Hamiltonian formalism V. A theory of inelastic electron tunneling spectroscopy. Phy.s. Rev. B 20, 430-455, and references therein. 

In line with the Franck-Condon principle, the electron transfer occurs at the seam of the crossing between diabatic (localized) states of donor and acceptor. The electronic coupHng is the off-diagonal matrix element of the Hamiltonian defined at the crossing point. 

V. Levich and R. R. Dogonadze, Dokl. Akad. Nauk. SSSR 124 123 (1959). Hamiltonian formulation for electron transfer dielectric polarization approach. Quantum aspects of Weiss-Marcus model developed. 

W. Schmickler, J. Electroanal. Chem. 204 31 (1986). A discussion of the influence of the choice of Hamiltonian on electron-transfer theory. 

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