Biomedical Engineering Reference
In-Depth Information
which is arguably unhelpful when we want to describe the bonding in methane. We imagine
a two-step process where one electron is promoted from the 2s to a 2p orbital giving
C:(1s)
2
(2s)
1
(2p)
3
The four valence orbitals are then mixed to form sp
3
hybrids, giving four equivalent orbitals
C:(1s)
2
sp
3
4
which is said to form the valence state. Sadly, it does not correspond to a known spectro-
scopic state of the carbon atom and so you will anticipate that many learned discussions
took place to try and assign an energy value to the diagonal matrix elements of extended
Hückel theory. Things are far, far worse for transition metal ions with variable valencies.
Wolfsberg and Helmholtz went to great pains to try and get reasonable values for the diag-
onal terms by comparison with electronic spectra. Modern thinking is that the parameters
are simply parameters that should be chosen to give a good fit of some measurable physical
property against experiment, and that they should take the same value for a given element
irrespective of its chemical environment.
Wolfsberg and Helmholtz did their calculations in terms of 'group'orbitals. Such orbitals
are simply linear combinations of atomic orbitals that mirror the symmetry of the molecule.
In those days, it was essential to reduce the size of any matrix eigenvalue problem to one
that could be done by hand. We have moved on and do not generally bother with group
orbitals any more, although an understanding of molecular symmetry is vital in order to
understand the spectroscopic transitions.
Once the diagonal elements have been assigned, the off-diagonal elements are calculated
by the key formula
h
ii
+
h
jj
h
ij
=
kS
ij
(18.1)
2
where
S
ij
is the overlap integral between the two basis functions χ
i
and χ
j
. Parameter
k
is
a 'constant' that can be adjusted to give agreement with experiment; common experience
suggests that 1.75 is a reasonable choice.
As a variation on the theme, some early authors chose to write
k
S
ij
h
ii
×
h
ij
=
h
jj
(18.2)
To use the chlorate ion example cited by Wolfsberg and Helmholtz, we would these days
use a basis set comprising 2s and 2p STOs on oxygen with 3s, 3p and 3d orbitals on chlorine
giving a total of 25 basis functions. I have added d orbitals on Cl because it is hypervalent
in the molecule under study and people argue that d orbitals are necessary in such cir-
cumstances. That gives 16 doubly occupied valence shell molecular orbitals (made up as
follows: 6 valence electrons from each oxygen, 7 from chlorine and 1 from the excess neg-
ative charge). A standard modern extended Hückel calculation with a tetrahedral geometry
and bond length of 147 pm gave me the orbital energies listed in Table 18.3. You might get
slightly different results depending on your choice of Hückel parameters; I used the standard
default values in Gaussian 03. Notice how close together the highest occupied orbitals are;
the order obtained is a little different from that reported by Wolfsberg and Helmholtz, but
that is not surprising. Unlike advanced theories where electron repulsion is taken explicitly
into account, Hückel excitation energies are given as differences in orbital energies.
