Biomedical Engineering Reference
In-Depth Information
-0.8
-0.9
-1
-1.1
-1.2
0.5
1
1.5
2
2.5
3
3.5
4
Distance/atomic unit
Figure 15.6
Simple dihydrogen calculations
15.6
James and Coolidge Treatment
Just as for H
2
+
, it is again found that an accurate value of the binding energy can be obtained
by writing the wavefunction in terms of elliptic coordinates. James and Coolidge (1933)
wrote such a wavefunction that included the interelectron distance
r
12
explicitly:
μ
2
))
klmnp
c
klmnp
μ
1
μ
l
2
ν
1
ν
2
u
p
μ
l
1
μ
2
ν
1
ν
2
u
p
ψ
=
exp (
−
δ (μ
1
+
+
(15.10)
r
A,1
+
r
B,1
r
A,1
−
r
B,1
2
r
12
R
AB
μ
1
=
ν
1
=
u
=
R
AB
R
AB
where
k
,
l
,
m
,
n
and
p
are integers and the form of the function is that it is symmetric to
interchange of electron names. Parameter δ is the orbital exponent. In order to make the
wavefunction symmetric in the nuclear coordinates, the authors included only those terms
having (
m
n
) as an even integer. They found that a 13-term function gave essentially
complete agreement with experiment.
+
15.7 Population Analysis
Our simple treatment of the hydrogen molecule ion was based on the bonding orbital
1
√
2 (1
ψ
+
=
S
)
(1s
A
+
1s
B
)
+
2
.We normally use the symbol ρ
for volume charge densities and to emphasize that it depends on positions in space we write
which corresponds to a charge distribution of density
−
e
ψ
+
e
(ψ
+
(
r
))
2
ρ (
r
)
=−
