Biomedical Engineering Reference
In-Depth Information
there is a direct relation between
f
+
/−
(
r
) and the density of the appropri-
ate FMO thus avoiding calculations of the system with
N
+ 1 and
N −
1
electrons:
f
()
r
(),
r
(7.5)
L
f
()
r
().
r
(7.6)
H
On the other hand, the use of Equations 7.5 and 7.6 instead of Equations
7.3 and 7.4 allows one to diminish the computational ef ort without loos-
ing the qualitative picture of the local reactivity, but this approach should
always be checked by comparison with these two pairs of working equa-
tions because the i rst level of approximation based on total electronic
densities will always be more accurate than that of the second level of
approximation based on densities of FMOs.
Condensation to atoms is achieved through integration within the
kth
-
atomic domain Ω
k
[23, 24]:
/
/
f
f
()
rr
(7.7)
k
k
k
f
k
+/−
is now an atomic index that is used to characterize the electrophilic/
nucleophilic power of atom
k
.
Even better, Morell
et al.
[9, 13, 15, 25-28] have proposed a local reac-
tivity descriptor (LRD) which is called the dual descriptor (DD)
f
(2)
(
r
)
Δ
f
(
r
). In spite of having been discovered several years ago, a solid physi-
cal interpretation was not provided in such a moment [29]. h ey used the
notation Δ
f
(
r
), but currently it has been replaced by the modern notation
f
(2)
(
r
) in order to highlight that this is a Fukui function of second order.
Its physical meaning is to reveal nucleophilic and electrophilic sites on a
molecular system at the same time. Mathematically it is dei ned in terms of
the derivative of the Fukui function,
f
(
r
) [18], with respect to the number
of electrons,
N
. h rough a Maxwell relation, this LRD may be interpreted
as the variation of
η
(the molecular hardness which measures the resis-
tance to charge transfer [30]) with respect to
υ
(
r
), the external potential.
h e dei nition of
f
(2)
(
r
) is shown as indicated by Morell
et al.
[9, 13]:
f
()
r
(2)
f
()
r
(7.8)
N
()
N
r
()
r
