Biomedical Engineering Reference
In-Depth Information
An approximate form of the molecular orbital theory of unsaturated hydrocarbon molecules
in their ground states is developed. The molecular orbital equations rigorously derived from
the correct many-electron Hamiltonian are simplified by a series of systematic approximations
and reduce to equations comparable with those used in the semi-empirical method based on
an incompletely defined one-electron Hamiltonian. The two sets of equations differ, however,
in that those of this paper include certain important terms representing electronic interactions.
The theory is used to discuss the resonance energies, ionization potentials, charge densities,
bond orders and bond lengths of some simple hydrocarbons. The electron interaction terms
introduced in the theory are shown to play an important part in determining the ionization
potentials...
You should have picked upmany of the key phrases. He started from the HF-LCAO equa-
tions and made what is now known as the σ
π separation approximation; the π -electrons
are treated separately and the effect of the remaining σ -electrons is absorbed into the
HF-LCAO Hamiltonian. The HF-LCAO equations have to be solved iteratively in order
to get the HF-LCAO π -electron molecular orbitals, and many of the two-electron integ-
rals (the 'electronic interaction terms') are retained. In order to take account of the effect
of σ
−
π separation, most integrals are calibrated by appeal to experiment. The 'charges
and bond orders' are simply the Mulliken populations calculated with an overlap matrix
equal to the unit matrix, and ionization energies are calculated according to Koopmans'
theorem.
The second keynote paper by Pariser and Parr (1953) also gives a snapshot of the times,
when there was a great deal of interest in the electronic spectra of conjugated molecules.
They wrote:
−
A semi-empirical theory is outlined which is designed for the correlation and prediction of the
wavelengths and intensities of the first main visible or ultraviolet bands and other properties of
complex unsaturated molecules, and preliminary application of the theory is made to ethylene
and benzene.
The theory is formulated in the language of the purely theoretical method of the antisym-
metrized products of molecular orbitals (in LCAO approximation) including configuration
interaction, but departs from this theory in several essential respects. First, atomic orbital
integrals involving the core Hamiltonian are expressed in terms of quantities which may
be regarded as semi-empirical. Second, an approximation of zero differential overlap is
employed and an optionally uniformly charged sphere representation of atomic π -orbitals
is introduced, which greatly simplify the evaluation of electronic repulsion integrals and
make applications to complex molecules containing heteroatoms relatively simple. Finally,
although the theory starts from the π -electron approximation, in which the unsaturated
electrons are treated apart from the rest, provision is included for the adjustment of
the σ -electrons to the π -electron distribution in a way which does not complicate the
mathematics.
Once again you should have picked up many of the key phrases. We often speak of
the
PPP method
in honour of its three originators.
18.4 Zero Differential Overlap
Akey phrase that I have yet to explain is
zero differential overlap (ZDO)
; the atomic orbital
basis set is not rigorously defined but we can imagine it to comprise the relevant STO with
