Electrochemical Electron Transfer: From Marcus Theory to Electrocatalysis - Fuel Cell Catalysis: A Surface Science Approach - page 49

proportional to the charge on both ions. Using normalized coordinates, the corre-
sponding terms are

where n ¼ X
s

,

H sol ¼ l q 2 þ p 2 þ 2(Z n)q

(n a,s þ n b,s ) (2 : 21)

Z is the charge number of the molecule when the orbitals a and b are empty.
By using the Hartree - Fock approximation, the Coulomb repulsion terms in the
molecular Hamiltonian H mol can be reduced to one-electron terms, and the density
of states can be calculated. The isolated molecule shows the expected splitting into
a bonding and an antibonding orbital. In the homonuclear case, which we consider
from here on, the energy separation between the two orbitals is determined by the
interaction b and by the Coulomb repulsion U ¼ U a ¼ U b . The interaction b
depends on the separation r between the two atoms, usually exponentially:
b ¼ A exp( r = l), A . 0. If we re-introduce the neglected orbital overlap and use
the Wolfsberg - Helmholtz approximation, the potential energy of the isolated mol-
ecule assumes a Morse shape, similar to Sav´ant's model. The details are given in
[Santos et al., 2006, 2008].
From the given Hamiltonian, adiabatic potential energy surfaces for the reaction can
be calculated numerically [Santos and Schmickler 2007a, b, c; Santos and Schmickler
2006]; they depend on the solvent coordinate q and the bond distance r, measured with
respect to its equilibrium value. A typical example is shown in Fig. 2.16a (Plate 2.4) it
refers to a reduction reaction at the equilibrium potential in the absence of a d-band
(D d ¼ 0). The stable molecule correspond to the valley centered at q ¼ 0, r ¼ 0, and
the two separated ions correspond to the trough seen for larger r and centered at
q ¼ 2. The two regions are separated by an activation barrier, which the system has
to overcome.
In order to understand the nature of electrocatalysis, it is instructive to consider the
evolution of the density of states of the reactant (Fig. 2.17). In the initial state, it shows
a filled bonding orbital below the Fermi level and an empty antibonding orbital above.
In the final state, the bond has been broken, the difference between bonding and anti-
bonding orbitals has vanished, and the system has just one doubly degenerate orbital
below the Fermi level, filled with four electrons. The crucial stage in the reaction
occurs when, owing to a thermal fluctuation, the antibonding orbital passes through
the Fermi level. For a reaction at equilibrium, the saddle point corresponds to the situ-
ation where the antibonding orbital is about half filled. In the absence of d-band
catalysis, the antibonding orbital is somewhat broadened by the interactions with the
sp-band. This broadening reduces the energy of activation somewhat, because the elec-
tronic energy of the system is obtained by integrating 1r(1) up to the Fermi level.
Figure 2.16b shows a potential energy surface in the presence of a band situated
near the Fermi level and coupling strongly to the reactant. The energy of activation
is significantly reduced. This catalysis can be readily understood by examining the
density of states (Fig. 2.18). At the saddle point, the antibonding orbital of the mol-
ecule is split into two parts: a bonding and an antibonding part with respect to the
metal d-band. This is the splitting mechanism for strong interactions that we discussed