Biomedical Engineering Reference
In-Depth Information
Similar considerations apply to the one-electron terms
−
χ
j
dτ
N
h
2
8π
2
m
e
∇
1
4πε
0
Z
α
R
α
2
χ
i
−
α
=
1
When
i
=
j
and the basis function χ
i
is centred on nucleus A we write the integral
χ
i
χ
i
dτ
h
2
8π
2
m
e
∇
1
4πε
0
Z
A
R
A
2
U
ii
=
−
−
and determine
U
from atomic spectral data. The remaining terms in the sum are called
penetration integrals
and written
V
AB
.
If
i
=
j
and the basis functions are on different atomic centres then all three-centre
contributions are ignored. The remaining two-centre terms involving atoms A and B are
written β
AB
S
ij
, where
S
is the overlap integral and β a 'bonding' parameter. The bonding
parameter is nonzero only for bonded pairs.
Collecting up terms and simplifying we find that a CNDO HF-LCAO Hamiltonian has
elements
P
AA
−
2
P
ii
γ
AA
+
1
h
ii
=
U
ii
+
(
P
BB
γ
AB
−
V
AB
)
B
=
A
1
2
P
ij
γ
AB
h
ij
=
β
AB
S
ij
−
(18.9)
A and B label atoms,
i
and
j
label the basis functions and
P
AA
is the sum of the diagonal
charge density matrix for those basis functions centred on atomA.
The original parameter scheme was called CNDO/1. Electron repulsion integrals were
calculated exactly, on the assumption that the basis functions were s-type STOs, and all
overlap integrals were calculated exactly. The bonding parameters β
AB
were chosen by
comparison with (crude)
ab initio
calculations on relevant small molecules, and a simple
additivity scheme was employed:
β
AB
=
β
A
+
β
B
18.8 CNDO/2
It turned out that CNDO/1 calculations gave poor predictions of molecular geometries, and
this failing was analysed as due to the approximations made for
U
ii
and the penetration
term
V
AB
. These problems were corrected in CNDO/2;
V
AB
is no longer calculated exactly;
rather it is taken as -
Z
B
γ
AB
. The atomic terms become
Z
A
−
γ
AA
1
2
(
I
i
+
1
2
U
ii
=−
E
i
)
−
(18.10)
