Biomedical Engineering Reference
In-Depth Information
taken as -
Z
J
γ
IJ
, where
Z
is the formal number of electrons that an atom contributes to the
π system). There is no problem with hydrocarbons in that all carbon atoms are assigned
the same valence state ionization energies, irrespective of their chemical environment, but
many papers were written to justify one particular choice of ω
I
against another for nitrogen.
For the record, the PPP HF-LCAO Hamiltonian
h
F
for hydrocarbons and other first-row
systems has dimension equal to the number of conjugated atoms and can be written
P
jj
−
Z
j
γ
ij
1
2
P
ii
γ
ii
+
h
ii
=
ω
i
+
j
=
i
(18.6)
1
2
P
ij
γ
ij
h
ij
=
β
ij
−
Here
P
is the usual matrix of charges and bond orders. The HF-LCAO equations are solved
by the usual techniques.
In those early days, no one dreamed of geometry optimization; benzene rings were
assumed to be regular hexagons with equal C-C bond length of 140 pm. PPP calculations
were state of the art in the early 1970s and consumed many hours of early computer time.
Once again there was an argument as to the role of d orbitals on sulfur, and a number of
long-forgotten schemes and methodologies were advanced.
18.5 Which Basis Functions Are They?
I seem to have made two contradictory statements about the basis functions used in semiem-
pirical work. On the one hand, I have said that they are orthonormal and so their overlap
matrix is the unit matrix; on the other hand, I have used overlap integrals to calculate certain
integrals.
Think of a Hückel π -electron treatment of ethene, and call the carbon 2p
π
orbitals χ
1
and χ
2
. The matrix of overlap integrals is
1
p
=
S
p
1
where
p
is the overlap integral of the two atomic (STO) orbitals in question. The eigenvalues
of this matrix are 1
±
p
and the normalized eigenvectors are
1
2
1
2
1
1
1
−
v
1
=
and
v
2
=
1
A little matrix algebra will show that
p
)
v
1
v
1
+
p
)
v
2
v
2
S
=
(1
+
(1
−
Mathematicians have a rather grand expression for this: they talk about the
spectral decom-
position
of a matrix. We can make use of the expression to calculate powers of matrices,
such as the negative square root
