Wide band approximation

In outer sphere electron transfer, the reactant is not adsorbed therefore, the interaction with the metal is not as strong as with the catal5d ic reactions discussed below. Hence, the details of the metal band structure are not important, and the couphng A(s) can be taken as constant. This is the so-called wide band approximation, because it corresponds to the interaction with a wide, structureless band on the metal. In this approximation, the function A(s) vanishes, and the reactant s density of states takes the form of a Lorentzian. The simation is illustrated in Fig. 2.3. 

This integral diverges—a consequence of the wide band approximation— however, this poses no problem. The relevant quantities are the differences in energy between various states. It is natural to take the initial state as a reference. This gives [Schmickler, 1986]... 

In the electron transfer theories discussed so far, the metal has been treated as a structureless donor or acceptor of electrons—its electronic structure has not been considered. Mathematically, this view is expressed in the wide band approximation, in which A is considered as independent of the electronic energy e. For the. sp-metals, which near the Fermi level have just a wide, stmctureless band composed of. s- and p-states, this approximation is justified. However, these metals are generally bad catalysts for example, the hydrogen oxidation reaction proceeds very slowly on all. sp-metals, but rapidly on transition metals such as platinum and palladium [Trasatti, 1977]. Therefore, a theory of electrocatalysis must abandon the wide band approximation, and take account of the details of the electronic structure of the metal near the Fermi level [Santos and Schmickler, 2007a, b, c Santos and Schmickler, 2006]. 

As demonstrated in Section 2.2, the energy of activation of simple electron transfer reactions is determined by the energy of reorganization of the solvent, which is typically about 0.5-1 eV. Thus, these reactions are typically much faster than bondbreaking reactions, and do not require catalysis by a J-band. However, before considering the catalysis of bond breaking in detail, it is instructive to apply the ideas of the preceding section to simple electron transfer, and see what effects the abandomnent of the wide band approximation has. 

A full analysis of the SOA suggests that it should be restricted to narrow bands or to high-velocity atoms, but that, within these regimes, it works well. The complementary approximation for broad bands and low velocity atoms is the wide-band approximation (WB A) The essence of the WB A lies in the... 

Here (a, b) is the inner product of two bi-vectors. As shown in Appendix 1, by using the wide band approximation (i.e. by taking the electron density of states in the leads u to be constant) the equation for the bi-vectors ca takes the form... 

In wide-band approximation we suppose v(e) =const, therefore e e (4 -t) 27xv5 t — t) and... 

This Hamiltonian can be treated at various levels of sophistication. In the simplest approximation, the width A, which can be a function of the electronic energy co, is taken as constant (wide-band approximation),67 and the Coulomb interaction is treated at the restricted Hartree-Fock level, so that both spin states have the same occupation probability, na,a) = naro) = ). In this case, the density of states of the adsorbate takes the form of a Lorenzian ... 

In the case of weak interactions with the electronic levels of the electrode (wide band approximation) Eq. (27) has the form of two Lorentz distributions centred at the energies of the bonding and antibonding states (see Fig. 12). The integral of Eq. (27) up to the Fermi level gives the occupation of both, bonding and antibonding orbitals and in the case of the wide band approximation it is an analytical expression ... 

In the case of weak interactions, according to the wide band approximation an analytical expression for the total expectation of the energy is obtained ""... 

The wide band approximation can be applied to describe very well the behaviour of metals with large, stractureless sp bands. However, the more interesting materials showing electrocatalytic properties, such as platinum or rathenium, posess narrow d bands. Then the next step in the development of the model is to abandon the wide band approximation and consider the electronic stmcture of the bands." " Next, we discuss the superposition of a wide sp... 

One of these assumption was that the continuum / extends from —oo to oo. This is often a good approximation to the situation where the edge(s) ofthe continuum is(are) far from the energetic region of interest, in this case the energy Ei. In the solid state physics literature this is sometunes referred to as the wide band approximation. 

Note that in writing Eqs (17.44) we have invoked the wide band approximation (Section 9.1). The bridge Green s function, Eq. (17.23), then satisfies... 

Here G /" denotes the retarted or advanced Green s function of the central region, while r depends on the surface density of states of the leads and the molecule-lead coupling. Both terms are directly amenable to a first principles evaluation using effective single-particle theories like Hartree-Fock or density functional theory (DFT). In this way the actual electronic structure of the device is accounted for, without the need to resort to few level models for the molecule or empirical wide band approximations for the leads. 

Consider now Eq. (9.52). The behavior of S as a function of time depends on the behavior of Cico) about coq. If C(cu) was constant in all range —oodifferent from zero, and approximated by a constant, only in some finite frequency interval about... 

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