Environmental Engineering Reference
In-Depth Information
Wilson, we have
2
2
½ð
h
=
2
mÞr
þUðrÞy ¼ Hy ¼ Ey
ð
3
:
36
Þ
with
r
the electron position. This gives solutions, to repeat, with no interactions,
y¼y
a
(
x
1
) and
y¼y
b
(
x
2
) at energy
E¼E
o
.
The interactions of the electron and the
rst proton with the second proton,
ke
2
(1/
r
a,2
þ
1/
R
) are now considered. The attractive interaction, primarily occur-
ring when the electron is between the two protons, and is attracted to both nuclear
sites, stabilizes H
2
þ
.
We can write the interaction as
H
int
¼ ke
2
½
1
=
R
1
=
r
a
;
2
;
ð
3
:
37
Þ
where the
first term is the repulsion between the two protons spaced by
R
. Following
Paulings treatment, one
nds
EE
o
¼ðke
2
=
Da
o
ÞþðJ þKÞ=ð
1
þDÞ
for
Y
S
ð
3
:
38
Þ
EE
o
¼ðke
2
=
Da
o
ÞþðJKÞ=ð
1
DÞ;
for
Y
A
;
ð
3
:
39
Þ
where
ðð
y
b
ðx
2
Þ½ke
2
r
a
;
2
Þy
a
ðx
1
Þd
3
x
1
d
3
x
2
¼ðke
2
a
o
Þ
e
D
K ¼
ð
1
=
=
ð
1
þDÞ
3
:
40
Þ
ðð
y
a
ðx
1
Þ½ke
2
r
a
;
2
Þy
a
ðx
1
Þd
3
x
1
d
3
x
2
¼ðke
2
a
o
Þ½D
1
þ
e
2
D
ð
1
þD
1
J ¼
ð
1
=
=
Þ
ð
3
:
41
Þ
ðð
y
b
ðx
2
Þy
a
ðx
1
Þd
3
x
1
d
3
x
2
¼
e
D
ð
1
þDþD
2
D ¼
=
3
Þ;
where
D ¼ R
=
a
o
:
ð
3
:
42
Þ
K
is known as the resonance or exchange or hopping integral, and measures the
rate at which an electron on one sitemoves to the nearest-neighbor site. One sees that
its dependence on spacing is essentially e
D
¼
e
R
/
a
, as one would expect for a
tunneling process, and that the basic energy (the prefactor of the exponential term) is
(
ke
2
/
a
o
)
¼
2
E
o
¼
27.2 eV. In these equations,
k
is the Coulomb constant 9
10
9
.
The energy
E
of the symmetric case is shown in (3.38). The major negative term is
K
, and this term changes sign in (3.39), the antisymmetric case. So the difference in
energy between the symmetric and the antisymmetric cases is about 2
K
, which
amounts to about 2
2.65 eV
¼
5.3 for H
2
þ
. The predicted equilibrium spacing is
2.4
a
o
.
The energy can be expanded as a function of
D¼R
/
a
o
that has a minimum at 2.4.
The energy near the minimum can be expressed as
