Biomedical Engineering Reference
In-Depth Information
because most of their interesting molecules did not contain benzene rings and inorganic
chemists were particularly aggrieved because there was no mention of inorganic molecules
in Hückel's theory.
Hückel's calculations referred to the so-called π -electron system. What few calculations
had been done on the core had used a method described as the 'linear combination of bond
orbitals', where Hückel-type schemes were used to try and combine together (for example)
sp
3
hybrid orbitals.
18.2.1 Wolfsberg and Helmholtz
The pioneering work of Max Wolfsberg and Lindsay Helmholtz on the electronic structure
of the ions CrO
4
, MnO
4
and CrO
2
4
is usually cited as the first serious attempt to apply
the Hückel model to inorganic complexes. The synopsis is given below (Wolfsberg and
Helmholtz 1952).
We have made use of a semi empirical treatment to calculate the energies of the molecular
orbitals for the ground state and the first few excited states of permanganate, chromate and
perchlorate ions. The calculations of the excitation energies is in agreement with the qual-
itative features of the observed spectra, i.e. absorption in the far ultraviolet for ClO
4
with
two strong maxima in the visible or near ultraviolet for MnO
4
and CrO
2
4
with the chro-
mate spectrum displaced towards higher energies. An approximate calculation of the relative
f
-values for the first two transitions in CrO
4
2
−
andMnO
4
−
is also in agreement with experiment
The authors point out that the tetrahedral ions PO
4
2
−
,SO
4
2
−
and ClO
4
−
show no visible
or near-ultraviolet absorption whilst the ions XO
4
−
of the fourth row transition elements,
which have the same number of valence electrons, show characteristic visible and near-
ultraviolet absorption, namely two strong maxima with the corresponding peaks displaced
towards shorter wavelengths (higher energies) as the atomic number of the central atom
decreases. The oscillator strength (
f
) relates to the area under an absorption peak and is
also a theoretical measure of the intensity of a given transition.
The extendedHückel model treats all valence electrons, within the spirit of the π -electron
model. Each molecular orbital is written as an LCAO expansion of these valence orbitals,
which can be thought of as being Slater-type orbitals; these orbitals are very similar to
hydrogenic ones except that they do not have radial nodes. Once again we can understand
the model best by considering the HF-LCAO equations.
For H and He, the atomic basis set consists of a single 1s orbital. For Li through Ne, the
inner shell electrons are treated as part of the core and the basis functions used are atomic
2s, 2p
x
,2p
y
and 2p
z
. For Na through Al, the inner shell is treated as part of the core and we
consider only 3s, 3p
x
,3p
y
and 3p
z
orbitals. For Si through Cl we have to decide on whether
or not to include the atomic 3d orbitals in addition, and practice varies. Most authors
included them, especially for hypervalent molecules such as ClO
4
−
. The diagonal elements
of the HF-LCAO matrix are taken to be the negatives of the valence shell ionization energy
(VSIE) for the orbital in question. You have probably met VSIE elsewhere; the orbital
configuration for carbon ground state is written
C:(1s)
2
(2s)
2
(2p)
2
