Biomedical Engineering Reference
In-Depth Information
Electron
G
A
r
G
B
C
r
C
r
A
r
B
Origin
Figure 16.5
Overlap integral between two GTOs
N
exp
exp
dτ
2
2
α
A
|
r
−
r
A
|
α
B
|
r
−
r
B
|
S
AB
=
−
−
a
0
a
0
(16.28)
N
exp
dτ
2
α
C
|
r
−
r
C
|
=
−
a
0
The product GTO
G
C
has exponent α
C
and centre
r
C
given by
α
C
=
α
A
+
α
B
1
α
A
+
r
C
=
α
B
(α
A
r
A
+
α
B
r
B
)
The remaining integral is a product of three standard integrals
exp
dτ
α
C
|
r
−
r
C
|
2
−
a
0
=
exp
d
x
exp
d
y
exp
d
z
α
C
(
x
−
x
C
)
2
a
0
α
C
(
y
−
y
C
)
2
a
0
α
C
(
z
−
z
C
)
2
a
0
−
−
−
One major problem that concerned early workers is that GTOs do not give terribly good
energies. If we try a variational calculation on a hydrogen atom with a single s-type GTO
α
π
a
0
3/4
exp
α
r
2
a
0
G
(α)
=
−
and calculate the optimal Gaussian exponent, we find α
opt
=
0.283 with a variational
energy of
0.5
E
h
and the error is some 15%. A
second problem is that the shapes of GTOs and STOs are quite different. Figure 16.6 shows
the dependences of the STO (with exponent ζ
−
0.424
E
h
. The experimental energy is
−
0.283)
on distance for a hydrogen atom. The plot is of wavefunction/
a
−
3/
0
versus distance from
the nucleus,
r
/
a
0
. The full curve is the STO, the dashed curve the best GTO.
=
1) and the GTO (with exponent α
=
