Environmental Engineering Reference
In-Depth Information
attraction. The covalent bond can involve one, two, ormore electrons. The underlying
effects are shown in the simplest cases, the one-electron bond in H
2
þ
, and the
covalent bond inH
2
. The idea of exchange or hopping between sites is brought into
view by consideration of molecular bonding.
3.4.2
Hydrogen Molecule Ion H
2
þ
The physics of the hydrogen molecular ion, the simplest (one-electron) covalent
bond, is inherently quantum mechanical. For the system of one electron and two
protons at large spacing, there are two obvious wavefunctions,
y
a
(
x
1
) and
y
b
(
x
2
),
which represent, respectively, the electron on the left proton and on the right proton.
For large spacing, these states will be long-lived, but for smaller spacing they will be
unstable. An electron starting in
y
a
(x
1
), say, will tunnel to
y
b
(
x
2
), at a frequency
f
.
To
find the ground state, we make use of the idea that in quantum mechanics a
linear combination of allowed solutions is also a solution. A general solution is
Y ¼ Ay
a
ðx
1
ÞþBy
b
ðx
2
Þ;
ð
3
:
33
Þ
where
A
2
þ B
2
¼
1.
The linear combinations that are stable in time are the symmetric and antisym-
metric
Y
S
¼
2
1
=
2
½y
a
ðx
1
Þþy
b
ðx
2
Þ; Y
A
¼
2
1
=
2
½y
a
ðx
1
Þy
b
ðx
2
Þ:
ð
3
:
34
Þ
These states are stable in time because the electron is equally present on right and
left, and the tunneling instability no longer occurs.
It is easy to understand that the symmetric combination
Y
S
has a lower energy
than
A
, because the probability of finding the electron at the midpoint is nonzero,
while that probability is zero in
Y
A
. The midpoint is an energetically favorable
location for the electron because it sees attraction from both protons. The energy
difference between the symmetric and the antisymmetric cases is
Y
D
E ¼ hf
;
ð
3
:
35
Þ
where
f
is the tunneling frequency of an electron started on one side for tunneling to
the other side. The value
E
is about twice the binding energy [39], which is 2.65 eV
for H
2
þ
. So the tunneling rate is 1.28
10
15
/s, corresponding to a residence time
0.778 fs.
The exchange of an electron between the two sites is also referred to as hopping
or resonance and a more detailed treatment will give us the hopping integral that
determines the rate. (A modi
cation of this treatment will be applied below to a
scheme for inducing D
þ
D fusion.)
Consider two protons (at sites a and b, assume they are massive and
D
xed), a
distance
R
apart, with one electron. If
R
is large and we can neglect interaction of the
electron with the second proton, then, following the discussion of Pauling and
