Newns-Anderson Hamiltonian

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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... 

In a series of publications, Voth and coworkers [21,24,25,26,27] explored various aspects of electron-transfer reactions. Their calculations are based on a version of the Anderson-Newns Hamiltonian presented in Sect. 1.2.4.2, which allows the direct computation of adiabatic free energy surfaces. For a Fe +/Fe " couple at a fixed distance to a Pt(lll) surface, they compared a classical and a quantized water model [24]. The quantization... 

The solvent dynamics of a one-dimensional system was explored by Kuznetsov and Schmickler [30]. They singled out one important solvent coordinate and represented all the others by a heat bath. This is a similar concept to analytical theories such as Kramer s [20]. Adiabatic potential-energy surfaces were calculated from the Anderson-Newns Hamiltonian (see Fig. 3). The result is a symmetric double well, the barrier being the lower and the less sharp, the higher the electronic coupling A. M D-simulations of a motion on this surface coupled to the heat bath... 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

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... 

Persson and Baratoff [6] have been the first to estimate the inelastic contribution to the current by using a scattering-like approach. They show that fundamental aspects of the transport problem can be taken into account only if the propagation of the electron is treated on the same level as the vibration excitation. Similar approaches are those by Gata and Antoniewicz [20], and Spataru and Budau [21]. All of these approaches start by writing a Newns-Anderson type Hamiltonian ... 

These ideas can be cast into an Hamiltonian based on the Anderson-Newns model and extended to the electrochemical situation by one of us.64 This Hamiltonian consists of several terms. We start with the terms for the adsorbate orbital, which we label by a. This can take up two electrons with opposite spins a, which interact through a Coulomb repulsion of magnitude U ... 

For quantitative calculations, DFT and theories complement each other well. DFT can provide the electronic parameters for particular reactions, and can compensate the well-known shortcomings of Anderson-Newns " like Hamiltonians. The first applications of the combination of DFT with theory to the hydrogen reaction, which we have presented above, are encouraging They explain very well the different catalytic activities of the various metals, give the correct trend and order of magnitude for the rate constants. 

In our group, we have developed a theoretical framework that can be applied to both kinds of reactions. It is based on a model Hamiltonian incorporating concepts from theories of outer sphere electron transfer [49-51], Anderson-Newns theory [52, 53], and our own ideas. The model as was developed in the 1990s [54, 55] and at that time, it was applied to various processes such as metal deposition/dissolution, anion adsorption, and outer-sphere electron transfer. However, this was at a time when DFT was not widely available, and several important system parameters had to be estimated, so that the applications had a qualitative character nevertheless, they provided a basic understanding of these processes at the molecular level. 

This Hamiltonian, which was introduced by Schmickier [12], is equivalent to earher formulations by Levich and Dogo-nadze in terms of wave mechanics [5] it is also related to the spin-boson model for homogeneous electron exchange [13] and to the Anderson—Newns model for specific adsorption [14]. 

The concepts outlined in the previous section can be based on a model Hamiltonian [7,13], which combines ideas from the Marcus-Hush [1,2] theory of electron transfer with the Anderson-Newns model [8,9], While the resulting theory explains the principles of electrocatalysis well, it suffers from the well-known defects of the Anderson-Newns-type models it does not account for many-body effects and is therefore not good enough for quantitative calculations. Therefore, we have developed a method to combine our theory with DFT calculations. In the following, we present the main ideas the mathematical details are given in the appendix. 

Resonant processes dynamic solution of the Newns-Anderson Hamiltonian... 

In the next step, to analyze the resonant processes associated with charge exchange between He+ and He , we consider a spin-less Newns-Anderson Hamiltonian [18] where the level and the hopping terms Tis /a that have been neglected to calculate ElHe" "] and [He , are intioduced. This Hamiltonian reads... 

RESONANT PROCESSES DYNAMIC SOLUTION OF THE NEWNS-ANDERSON HAMILTONIAN... 

This case is analyzed using the H -levels and the H-Al interactions given in Fig. 6 and Table 1, respectively. As the H -level is always above the metal Fermi level, we can use semiclassical master equations for solving the Newns-Anderson Hamiltonian of this problem [21,22]. 

In the effective medium theory, the electronic interaction between an atom and the solid is replaced by that of the atom and a homogeneous electron gas. The approach is based upon density functional theory, but since the solid is not really behaving as a homogeneous electron gas, gradient corrections to the theory have been devised [210]. Simple model hamiltonians have also been used to study chemisorption of atoms and molecules at metals. Here we mention the Anderson-Grimley-Newns model hamiltonian of the type... 

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