Biomedical Engineering Reference
In-Depth Information
Electron densities around the two hydrogen atoms have to be the same by symmetry, and
the nuclear labels A and B can be used interchangeably, which means that
c
A
=
c
B
c
A
=±
c
B
This gives two possibilities, which I will label ψ
+
and ψ
−
, and we usually write them in
normalized form as
1
√
2 (1
ψ
+
=
S
)
(1s
A
+
1s
B
)
+
1
√
2 (1
ψ
−
=
S
)
(1s
A
−
1s
B
)
(15.5)
−
where
S
=
1s
A
1s
B
dτ
exp
−
1
R
AB
a
0
R
AB
a
0
+
R
AB
3
a
0
=
+
We can test these approximate wavefunctions by calculating the variational energies,
which can be written
e
2
4πε
0
R
AB
−
ε
AA
±
ε
AB
ε
±
=
+
ε
H
(1s)
1
±
S
1
s
A
r
A
dτ
e
2
4πε
0
ε
AA
=
1
1
exp
e
2
4πε
0
a
0
a
0
R
AB
R
AB
a
0
2
R
AB
a
0
=
−
+
−
(15.6)
e
2
4πε
0
1s
A
r
B
dτ
ε
AB
=
1
exp
e
2
4πε
0
a
0
R
AB
a
0
R
AB
a
0
=
+
−
The analytical solution of these integrals together with the overlap integral
S
(in Equa-
tion (15.5)) is covered by the classic texts such as Eyring
et al.
(1944); I have just quoted the
results. This gives us the two potential energy curves shown in many elementary quantum
chemistry textbooks (Figure 15.3). There is a subtle point: they are often called molecular
potential energy curves because the nuclei experience just this potential.
The ground-state energy of a hydrogen atom is
1
/
2
E
h
and the curves tend asymptot-
ically to the correct limit. The upper curve describes an excited state whilst the lower
curve describes the ground state. The calculated binding energy is in poor agreement with
experiment (Table 15.1), whilst the equilibrium bond length is in modest agreement with
experiment.
A hydrogenic 1s orbital has the form
−
ζ
3
π
a
0
exp
ζ
r
a
0
1s
=
−
(15.7)
