Environmental Engineering Reference
In-Depth Information
the previous section. This entails a reduction of the electronic energy. We will
discuss this effect in greater detail for the catalysis of bond breaking, where it is
much more pronounced.
2.8 ELECTROCATALYSIS OF BOND BREAKING
In electrocatalysis, the reactants are in contact with the electrode, and electronic inter-
actions are strong. Therefore, the one-electron approximation is no longer justified: at
least two spin states on a valence orbital must be considered. Further, the form of the
bond Hamiltonian (2.12) is not satisfactory, since it simply switches between two elec-
tronic states. This approach becomes impractical with two spin states in one orbital;
also, it has an ad hoc nature, which is not satisfactory.
In order to obtain a better model for the molecular bond, [Santos et al., 2006]
employed the extended H ¨ ckel, or tight binding, theory. For the breaking of the
bond in a diatomic molecule according to the schemes
A
B
þ
2e
!
A
þ
B
or A
B
!
A
þ
þ
B
þ
þ
2e
(2
:
18)
we consider one atomic orbital for each atom, which can take up two electrons with
opposite spin. We shall later specialize to the case of a homonuclear molecule, but
it is convenient to start with the general case. In the tight binding theory, the
Hamiltonian of the isolated molecule is
H
mol
¼
X
s
1
a
n
a,s
þ
1
b
n
b,s
þ
V
ab
c
as
c
bs
þ
V
ab
c
bs
c
as
þ
U
a
n
a,s
n
a,
s
þ
U
b
n
b,s
n
b,
s
(2
:
19)
Here the indices a and b stand for the valence orbitals on the two atoms; as before, n is
a number operator, c
þ
and c are creation and annihilation operators, and sis the spin
index. The third and fourth terms in the parentheses effect electron exchange and are
responsible for the bonding between the two atoms, while the last two terms stand for
the Coulomb repulsion between electrons of opposite spin on the same orbital. As is
common in tight binding theory, we assume that the two orbitals a and b are orthog-
onal; we shall correct for this neglect of overlap later. The coupling V
ab
can be taken as
real; we set V
ab
¼
b
0.
The Hamiltonian for the metal is the same as before, but the metal states now couple
to two atomic orbitals:
H
met
¼
X
k,s
[1
k
n
k,s
þ
(V
a,k
c
k,s
c
a,s
þ
V
b,k
c
k,s
c
b,s
þ
h.c.)]
(2
:
20)
where h.c. stands for the hermitian conjugate. The phonon bath interacts with
the charges on both ions. The region of interest is immediately after the bond
breaking, when the ions are close; therefore, we can assume that the interaction is
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