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η(
r
) = 1/N ∫η(
r
,
r
′)ρ(
r
′)
dr
′
(62)
These two quantities are also local in nature, i.e.,
2 ∫
s
(
r
) η(
r
)
dr
= 1
(63)
They defi ned the term
local hardness
, ή as the change in chemical po-
tential with changing electron density in different parts of the molecule.
Thus
ή
= ½(δμ/δρ)
v
(64)
s
(
r
) = (δρ/δμ)
v
= S
f
(65)
Thus, the local softness depends on the one-electronic density function of
the frontier orbital,
f.
If the global hardness and the frontier orbitals are
known, the local softness can be calculated at each atom. In a chemical
reaction, the most efficient interaction will be between the softest part of
each molecule. This allows the maximum transfer of electrons for a given
difference in chemical potentials.
By using the concept of electron density ρ(r), one can defi ne a descrip-
tor of local softness [60].
The local softness at an orbital
r
is defi ned using the following equa-
tion:
s
(
r
) =
f(r)
S
(66)
where
f
(
r
) is known as the Fukui function
The condensed softness, or the softness of an atom in a molecule, may
be derived through the chemical potential equalization principle that leads
to defi ne the Fukui function of the
k
th atom in a molecule A with
N
A
elec-
trons, in the following form:
S
+
k
=
f
+
k
S for nucleophilic attack
(67)
and
S
-
k
=
f
-
k
S for electrophilic attack
(68)
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