Biomedical Engineering Reference
In-Depth Information
where
I
i
and
E
i
are valence state ionization energies and electron affinities. The HF-LCAO
matrix elements are
(
P
AA
−
1)
γ
AA
+
1
2
(
I
i
+
1
2
(
P
ii
−
h
ii
=−
E
i
)
+
Z
A
)
−
(
P
BB
γ
AB
−
V
AB
)
B
=
A
(18.11)
1
2
P
ij
γ
AB
h
ij
=
β
AB
S
ij
−
18.9 CNDO/S
One failing of the π -electron models was their inability to treat n
π
∗
electronic trans-
itions. The CNDOmodels gave a great breakthrough because they afforded for the first time
a systematic treatment of both π
→
π
∗
transitions in conjugated molecules.
The treatment essentially followed the lines of CIS discussed in Chapter 17 but the first res-
ults were in poor agreement with experiment in that they gave excitation energies that were
much too large. The most significant improvement to CNDO for use in spectroscopy was
given by the work of Del Bene and Jaffé (1968). Their major contribution was to modify
the bonding integrals; think of a pair of bonded carbon atoms, with one pair of 2p orbitals
pointing along the bond (σ ) and the remaining two pairs perpendicular to the bond (π ).
Del Bene and Jaffé introduced a new parameter κ such that the π -type interaction was
reduced. They essentially replaced the term β
AB
S
ij
with
→
π
∗
and n
→
2
β
A
+
β
B
S
ij
1
β
AB
S
ij
=
(18.12)
2
κ
β
A
+
β
B
S
ij
1
β
AB
S
ij
=
They recommend a value κ
=
0.585, which gives best agreement with experiment for
many conjugated molecules.
18.10
INDO
The CNDO models make an unnecessary and draconian simplification to the two-electron
integrals
χ
i
(
r
1
)χ
j
(
r
1
)
1
r
12
χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
=
δ
ij
δ
kl
γ
AB
That is to say, even if orbitals χ
i
and χ
j
are on the same centre but
i
j
then the entire
integral is taken to be zero. Direct calculation shows that the integrals can be far from zero.
In fact, it is just one-centre integrals of this kind that give the correct ordering between
atomic spectroscopic states. CNDO cannot distinguish between the
3
P,
=
1
D and
1
S states
arising from orbital configuration
C:(1s)
2
(2s)
2
(2p)
2
