Biomedical Engineering Reference
In-Depth Information
As mentioned above, DD allows one to simultaneously obtain the
preferable sites for nucleophilic attacks (
f
(2)
(
r
)
>
0) and the preferable
sites for electrophilic attacks (
f
(2)
(
r
)
<
0) into the system at point
r
. Dual
descriptor has been demonstrated to be a robust tool to predict specii c
sites of nucleophilic and electrophilic attacks in a much more ei cient
way than the Fukui function by itself because it is able to distinguish
those sites of true nucleophilic and electrophilic behavior, and as a con-
sequence some works have been published with the aim of remarking
the powerfulness of
f
(2)
(
r
) and all those LRDs depending on DD [9, 13,
15, 25-28].
h e general working equation to obtain DD is given by the dif erence
between nucleophilic and electrophilic Fukui function [15]. A well-known
i rst level of approximation implies the use of i nite dif erence method
whereby the sum of electronic densities of the system with one more elec-
tron and one less electron is subtracted by the double of the total electronic
density of the original system. Since this level of approximation implies a
time-consuming computation, a second level of approximation has been
used for some years where the densities of FMOs provide an easier-to-
compute working equation:
(2)
()
f
r
f
()
r
f
()
r
()
r
()
r
L
H
where densities of LUMO and HOMO are represented by
ρ
L
(
r
) and
ρ
H
(
r
),
respectively.
Molecular symmetry can have an inl uence upon the local reactivity,
and on the other hand, it has been demonstrated that the Fukui function
must conserve the symmetry [31]. In addition, as the degeneration that
may arise in frontier molecular orbitals is related with the molecular sym-
metry, within the framework of the second level of approximation, this
phenomenon has been taken into account, thus providing a mathematical
expression to be applied on closed-shell molecular systems [32].
Hence, when an interaction between two species is well described
through the use of this LRD, it is said the reaction is controlled by frontier
molecular orbitals (or frontier-controlled) under the assumption that the
remaining molecular orbitals do not participate during the reaction.
h e dual descriptor can also be condensed through an appropriate inte-
gration within the kth-atomic domain Ω
k
:
(2)
(2)
f
()
rr
d
f
(7.9)
k
k
