Biomedical Engineering Reference
In-Depth Information
E
()
r
(7.1)
()
N
r
But when chemical reactions are governed by interactions mainly
of covalent nature, in such a case a second order LRD called Fukui
function [18] is used instead of electronic density. Fukui function is
dei ned in terms of the derivative of
ρ
(
r
) with respect to
N
; through
a Maxwell relation, the same descriptor is interpreted as the varia-
tion of
μ
with respect to
υ
(
r
) [18]:
()
r
f
()
r
(7.2)
N
()
N
r
()
r
h e function
f
(
r
) rel ects the ability of a molecular site to accept or
donate electrons. High values of
f
(
r
) are related to a high reactivity at
point
r
[18].
Since the number of electrons
N
is a discrete variable [20], right and let
derivatives of
ρ
(
r
) with respect to
N
have emerged. By applying a i nite dif-
ference approximation to Equation 7.2, two dei nitions of Fukui functions
depending on total electronic densities are obtained:
()
r
f
()
r
()
r
()
r
(7.3)
N
1
N
N
()
r
()
r
(7.4)
f
()
r
()
r
()
r
N
N
1
N
()
r
where
ρ
N
+1
(
r
),
ρ
N
(
r
) and
ρ
N −
1
(
r
) are the electronic densities at point
r
for
the system with
N
+ 1,
N
and
N −
1 electrons, respectively. h e i rst one,
f
+
(
r
), has been associated to reactivity for a nucleophilic attack so that it
measures the intramolecular reactivity at the site
r
toward a nucleophilic
reagent. h e second one,
f
−
(
r
), has been associated to reactivity for an elec-
trophilic attack so that this function measures the intramolecular reactivity
at the site
r
toward an electrophilic reagent [21].
h e densities of frontier molecular orbitals (FMOs),
ρ
L
(
r
) (LUMO den-
sity) and
ρ
H
(
r
) (HOMO density), come into the picture since it has been
shown [21, 22] that when the frozen orbital approximation (FOA) is used
