Information Technology Reference
In-Depth Information
be neglected. Then
H
(1) coincides with the corresponding expression for the
hydrogen atom with index
a
, and the wave function
ˆ
1
(1) should coincide with
the wave function of the hydrogen atom at the point “
a
.” Similarly, if
r
b
1
Re
, the
wave function
ˆ
1
(1) should coincide with the wave function of the hydrogen atom
at the point “
b
”:
2
4
3
5
2
2
2
2
r
a
1
R
e
,
H ðÞ!
ℏ
Ze
2
r
a
1
∂
x
1
þ
∂
y
1
þ
∂
,
ˆ
1
ðÞ!ˆ
a
ðÞ
;
r
b
1
2
m
z
1
∂
∂
∂
2
4
3
5
2
2
2
2
Ze
2
r
b
1
R
e
,
H ðÞ!
ℏ
∂
x
1
þ
∂
y
1
þ
∂
,
ˆ
1
ðÞ!ˆ
b
ðÞ:
2
m
z
1
∂
∂
∂
Thus one can choose the wave function
ˆ
1
(1) as
ˆ
1
(1) ¼ const
(1)
ˆ
a
(1)
+ const
(2)
ˆ
b
(1).
Note that the hydrogen molecule possesses a rather high degree of symmetry,
with the rotation axis of infinite order passing through the two hydrogen atoms.
Furthermore, among the symmetry elements there exists a plane that passes through
the middle of the distance between the nuclei of atoms and perpendicular to
it. Therefore, the wave function
ˆ
1
(1) can be written as
ð
1
ðÞ
¼
N
s
ˆ
a
ðÞþˆ
b
ðÞ
ˆ
½
,
aðÞ
1
ˆ
ðÞ
¼
N
as
ˆ
a
ðÞˆ
b
ðÞ
½
:
Upon reflection in the symmetry plane the atoms exchange positions (Fig.
3.6
).
However the square modulus of the wave function (i.e., the electron density
distribution) remains unchanged. As shown in the figure the function
s
1
(1) must
correspond to the bond between the two hydrogen atoms, since the probability
density of finding electrons in the space between the nuclei increases. According to
this criterion the function
ˆ
as
ˆ
1
(1) does not contribute to that bond. Therefore it is
a
1
(1) bonding and antibonding orbitals. Numerical
solution confirms these qualitative considerations. The function
1
(1) and
conventional to call
ˆ
ˆ
1
(1) leads to the
potential energy of the nuclei of the molecule with a minimum corresponding to the
bonding of two hydrogen atoms (figure), whereas the function
ˆ
as
ˆ
1
(1) leads to their
repulsion.
The simple model we have considered, which qualitatively describes the order
and relative positions of electronic levels in the potential well, is often applied to
more complex molecular fragments. It is especially useful when analyzing electron
transitions in a system consisting of fragments which can be regarded as
independent.
