QUANTUM COMPUTATIONAL STUDIES ON PORPHYCENE - Theoretical and Computational Research in the 21st Century - page 168

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 [33] 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 arbitrary, relates the partitioning in some
way to the electronegativity difference between the corresponding atoms.

9.2.3.2 ZERO DIFFERENTIAL OVERLAP CHARGE

Numerous approximations were being tried out for getting around the
problem of computing the more difficult integrals. In the zero differential
overlap (ZDO), approximation the product of two different atomic orbit-
als is set to zero. The integral which survived the ZDO approximation
was 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 [34] used the method for the predic-
tion of the spectral procedure of conjugate systems. Pople [35] indepen-
dently used the ZDO approximation to work out the same computational
strategy.
Now, let us have a look on the ZDO approximation [36] 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

S μv = ∫χ μ χ v dτ (μ ≠ v ) (32)

The differential overlap between these two function which is simply the
product χ μ (i) χ v (ii) gives the probability of finding an electron i in a com-
mon volume element to them. It can be expressed as

χ μ (i) χ v (ii) = δ μv (33)

If χ μ and χ v are centered on two atoms distant from each other or their spa-
tial orientations are quite different their differential overlap is nearly zero.
All approximate LCAO-MO-SCF schemes have made use of this feature
by neglecting all or most of the integrals containing the product χ μ (i) χ v
(ii) unless μ is equal to v . If all such integrals are neglected, we come at