Biomedical Engineering Reference
In-Depth Information
one atomic orbital per heavy atom. Hydrogen atoms do not enter into π -electron models.
The basis functions are taken to be orthonormal, so we have
1f
i
=
j
χ
i
(
r
)χ
j
(
r
) dτ
=
0
otherwise
ZDO extends this idea to the two-electron integrals; if we have a two-electron integral of
the type
e
2
4πε
0
...
1
r
12
χ
i
(
r
1
)χ
j
(
r
1
) dτ
1
then the integral is set to zero unless
i
=
j
. So all two-electron integrals
e
2
4πε
0
1
r
12
χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
χ
i
(
r
1
)χ
j
(
r
1
)
(18.3)
are zero unless
i
l
.
At first sight the basis functions are STO 2p
π
, and the remaining integrals can actually
be calculated exactly. When Pariser and Parr first tried to calculate the ionization energies
and spectra of simple conjugated hydrocarbons such as benzene with exact two-electron
integrals, they got very poor agreement with experiment. They therefore proposed that the
two-electron integrals should be treated as parameters to be calibrated against spectroscopic
data. Once again, you have to remember that computers were still in their infancy and that
any simple formula was good news. We had electromechanical calculators and log tables
in those days, but nothing much else. One such popular formula was
=
j
and
k
=
1
r
12
χ
k
(
r
2
) χ
k
(
r
2
) dτ
1
dτ
2
=
1
R
ik
+
χ
i
(
r
1
)χ
i
(
r
1
)
(18.4)
α
ik
where
R
ik
is the separation between the orbital centres. All other two-electron integrals
were taken as zero and the α
ik
term has to be fixed by appeal to spectroscopic experiment.
The expression is what you would get for the mutual potential energy of a pair of electrons
separated by distance
R
ik
+
α
ik
, and these repulsion integrals are traditionally given a
symbol γ
ik
.
Next we have to address the one-electron terms, and the treatment depends on whether
the term is diagonal (
i
=
j
) or off-diagonal (
i
=
j
). In standard HF-LCAO theory the
one-electron terms are given by
χ
j
dτ
N
h
2
8π
2
m
e
∇
e
2
4πε
0
Z
I
R
I
−
2
−
χ
i
(18.5)
I
=
1
where the first term represents the kinetic energy of the electron and the second term gives
the mutual potential energy between the electron and each nucleus.
In Pariser-Parr-Pople theory, the off-diagonal elements are taken as zero unless the atom
pairs are directly bonded. If the atom pairs are directly bonded the matrix element is given
a constant value β for each particular atom pair, which is certainly not the same value
as the β value in ordinary Hückel theory. The value of either β is found by calibration
against experiment. The one-electron diagonal terms are written so as to separate out the
contribution from the nuclear centre
I
on which the atomic orbital is based (usually taken
as minus the valence state ionization energy ω
I
) and the other nuclear centres
J
(usually
