Semi-empirical AM-1 Studies on Porphyrin - Modern Trends in Chemistry and Chemical Engineering - page 129

based on the linear combination of atomic orbitals molecular orbital method. The
charge thus arises from the Mulliken population analysis.
Let us consider C μi is the coeffi cients of the basis functions in the molecular orbital
for the μth basis function in the ith molecular orbital. Then the density matrix terms
are given by:
D μν = 2 C * νi (23)
For a closed shell system where each molecular orbital is doubly occupied the
population matrix P has terms:
P μν = D μν S μν (24)
where S is the overlap matrix of the basis functions and the sum of all terms of P μν
is N—the total number of electrons.
The Mulliken population analysis aims fi rst to divide N among all the basis func-
tions. This is done by taking the diagonal element of P μν and then dividing the off-
diagonal elements equally between the two appropriate basis functions. Since the off-
diagonal terms include P μν and P μν , this simplifi es to just the sum of a row. This defi nes
the gross orbital population (GOP) as

(25)
The (GOP)μ terms sum to N and thus divide the total number of electrons between
the basis functions. It remains to sum these terms over all basis functions on a given
atom A to give the gross atom population (GAP). The sum of (GAP) A terms is also N.
The charge, Q A , is then defi ned as the difference between the number of electrons on
the isolated free atom, which is the atomic number Z A , and the gross atom population:
Q A =Z A - (GAP) A (26)
The problem with this approach is the equal division of the off-diagonal terms
between the two basis functions. This leads to charge separations in molecules that
are exaggerated. In a modifi ed [36] Mulliken population analysis, this problem can
be reduced by dividing the overlap populations P μν between the corresponding orbital
populations P μμ and P νν in the ratio between the latter. This choice, although still arbi-
trary, relates the partitioning in some way to the electronegativity difference between
the corresponding atoms. Numerous approximations were being tried out for getting
around the problem of computing the more diffi cult integrals. In the zero differential
overlap (ZDO) approximation in which the product of two different atomic orbitals is
set to zero. The integral which survived the ZDO approximation were partly computed
using the uniformly charged sphere approximation and the rest parameterized. The
result procedure was a quantitative theory, which went well beyond Hückel theory by
explicitly taking into account electron repulsions. Pariser and Parr [37] used the meth-
od for the prediction of the spectral procedure of conjugate systems. Pople [38] inde-
pendently used the ZDO approximation to work out the same computational strategy.
Now, let us have a look on the zero differential overlap approximation [21] to
pound over the subject.
It is well known in quantum chemistry that for two different atomic functions χ μ
and χ v the overlap integral is:

(GOP) μ = ∑ ν P μν

S μv = ∫χ μ χ v dτ (μ≠v) (27)