QUANTUM COMPUTATIONAL STUDIES ON PORPHYCENE - Theoretical and Computational Research in the 21st Century - page 164

response, and as such the most reactive sites within a molecule. In chem-
istry, often chemical reactivity and molecular properties, in general, are
preferably interpreted in terms of the atoms composing molecular struc-
ture. It is then logical to introduce the so-called atom-condensed Fukui
functions. This means that some way of calculating the change in the total
atomic electron density of an atom α with respect to N is needed. As the
nuclear change of an atom is a constant, one of the easiest ways is to use
the concept of atomic charges, which introduces the following expression
for atom-condensed Fukui function:

f α = −(∂ q α /∂ N ) v (13)

Yang and Mortier [27] were the first to use such atom-condensed Fukui
functions, and used Mulliken charges to obtain values for the above-de-
fined atom-condensed Fukui functions.
Now, let us discuss the operational defi nitions of Fukui functions.
Modern chemical reactivity theory is based on the concept of Frontier
orbitals .The lowest unoccupied molecular orbital (LUMO) and the high-
est occupied molecular orbital (HOMO) or the frontier orbitals are of great
theoretical interest. Parr and Young [26, 28] demonstrated that the most
of the frontier electronic theory of chemical reactivity can be rationalized
from the DFT of electronic structure. In spite of the great success in chem-
istry, frontier orbital theoretical background was not simple to determine
because the molecular orbitals are not quantum mechanical observable.
Parr and Yang [26, 28]considered a species S with N electrons, having
ground-state electronic energy E [ N , v ] and chemical potential μ [ N , v ].
The energy as a function of N has a discontinuity of slope at each integral
N , and so there are three distinct chemical potentials for each integral N ,
μ - = (d E /d N ) v - . μ + = (d E /d N ) v + and μ 0 = (d E /d N ) v 0 = ½(μ + + μ − ) .
Fundamental equation for changes in energy and chemical potential are

d E = μd N +∫ρ( r )d v ( r )d r (14)

and

dμ =2ηd N + ∫ f ( r )d v ( r )d r (15)

The function f ( r ) is defined by