Biomedical Engineering Reference
In-Depth Information
40
-0.003
-0.003
0.006
-0.022
30
0.006
0.025
0.025
-0.012
-0.012
0.016
0.072
0.016
0.072
0.082
-0.041
0.063
0.054
0.1010.006
0.016
0.016
0.025
0.035
0.054
0.063
-0.031
-0.031
-0.022
0.00
0.025
0.035
0.044
0.044
0.006
0.006
20
-0.003
-0.012
-0.003
-0.003
10
0
0
10
20
30
40
Figure 15.5
Density difference for excited state
15.4 Heitler-London Treatment of Dihydrogen
The simplest molecule of any true chemical importance is dihydrogen. The methods dis-
cussed above for the hydrogen molecule ion are not applicable to dihydrogen, because
of the extra electron and the electron-electron repulsion. The first successful treatment of
dihydrogen was that of Heitler and London (1927). They argued as follows. Consider a
'reaction'
H
2
where the left-hand-side hydrogen atoms labelledAand B are initially at infinite separation.
Each hydrogen atom is in its lowest electronic state and there are two spin orbitals per atom
giving four in total: 1s
A
α,1s
A
β,1s
B
α and 1s
B
β. As we bring the atoms closer and closer
together, eventually their orbitals will overlap significantly and we have to take account
of the Pauli principle. We need to make sure that the total wavefunction is antisymmetric
to exchange of electron names. There are (4
H
A
+
H
B
→
12 possible ways of distributing two
electrons amongst four atomic spin orbitals but not all such distributions satisfy the Pauli
principle. Analysis along the lines given for helium in Chapter 14 gives the following
unnormalized building blocks (Slater determinants):
!
/2
!
)
=
1
s
A
(
r
1
) α (
s
1
)
1
s
B
(
r
1
) β (
s
1
)
1
s
A
(
r
1
) β (
s
1
)
1
s
B
(
r
1
) α (
s
1
)
D
1
=
D
2
=
1
s
A
(
r
2
) α (
s
2
)
1
s
B
(
r
2
) β (
s
2
)
1
s
A
(
r
2
) β (
s
2
)
1
s
B
(
r
2
) α (
s
2
)
1
s
A
(
r
1
) α (
s
1
)
1
s
B
(
r
1
) α (
s
1
)
1
s
A
(
r
1
) β (
s
1
)
1
s
B
(
r
1
) β (
s
1
)
D
3
=
D
4
=
1
s
A
(
r
2
) α (
s
2
)
1
s
B
(
r
2
) α (
s
2
)
1
s
A
(
r
2
) β (
s
2
)
1
s
B
(
r
2
) β (
s
2
)
