Biomedical Engineering Reference
In-Depth Information
p
)
−
1/2
v
2
v
2
To cut a long story short, we regard the basis functions used in semiempirical calculations
as related to ordinary STOs χ
1
, χ
2
,...χ
n
by the matrix transformation
χ
1
S
−
1/2
p
)
−
1/2
v
1
v
1
+
=
(1
+
(1
−
χ
n
S
−
1/2
χ
2
...
(18.7)
These have the property that they are orthonormal, yet resemble the ordinary STOs as
closely as possible, in a least-squares sense.
18.6 All Valence Electron ZDO Models
The early π -electron semiempirical models proved a great success, and they are still some-
times encountered in current publications. Attempts to extend them to the σ systems or
to inorganic molecules met with mixed fortune for a variety of reasons, especially the
following three.
•
If we draw a molecule and then arbitrarily choose an axis system, physical properties
such as the energy should not depend on the choice of axis system.We speak of
rotational
invariance
; the answers should be the same if we rotate a local molecular axis system.
•
Also, despite what one reads in elementary organic texts, calculated physical properties
ought to be the same whether one works with ordinary atomic orbitals or the mixtures
we call hybrids.
•
Finally, we should get the same answers if we workwith symmetry-adapted combinations
of atomic orbitals rather than the 'raw' atomic orbitals. Whatever the outcome, we will
not get a different energy.
18.7 CNDO
Pople and Segal (1965) seem to be the first authors to have addressed these problems.
The most elementary theory that retains all the main features is the complete neglect of
differential overlap (CNDO) model. The first paper dealt mainly with hydrocarbons, and
only the valence electrons were treated. The inner shells contribute to the core that modifies
the one-electron terms in the HF-LCAO Hamiltonian. The ZDO approximation is applied
to all two-electron integrals so that
χ
i
(
r
1
)χ
j
(
r
1
)
1
r
12
χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
(18.8)
is zero unless
i
l
. Suppose now that atoms A and B are both carbon, and
so we take 2s, 2p
x
,2p
y
and 2p
z
basis functions for either atom. In addition to the ZDO
approximation, the CNDO model requires that all remaining two-electron integrals of type
(18.8) involving basis functions on A and B are equal to a common value denoted γ
AB
.
Parameter γ
AB
depends on the nature of the atoms A and B but not on the details of the
atomic orbitals. This simplification guarantees rotational invariance.
=
j
and
k
=
