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following statement. If the electron cloud is strongly held by the nucleus,
the chemical species is “hard,” but if the electron cloud is loosely held by
the nucleus, the system is “soft” [9, 24].
Parr and Pearson [46] invoked the calculus of fi nite difference approxi-
mation to suggest an approximate and operational formula of hardness and
electronegativity as under:
The energy (
E
) of a chemical system having
N
number of the valence
electrons can be written in the form of the quadratic approximate equation
as follows:
E
=
aN
+
bN
2
(23)
where '
a
' is a constant, and it is a combination of core integral and a va-
lence shell electron pair repulsion integral and '
b
' is half of the average
valence shell electron-electron repulsion integral.
Now differentiating Eq. (23) with respect to
N
at constant external po-
tential,
v
we obtain
(∂E/∂N)
v
= −
a
− 2
bN
= (
I
+
A
)/2
(24)
where
I
and
A
are the first ionization potential and electron affinity of the
chemical species.
The right-hand side of the equation is the Mulliken [47] electronegativ-
ity (χ
M
).
(∂E/∂N)
v
= (
I
+
A
)/2 = χ
M
(25)
It is interesting to note that Putz [48], using finite difference approxima-
tion, showed that the Mulliken electronegativity definition can be recov-
ered from the Parr electronegativity (χ
P
) definition [43].
χ
M
= (
I
+
A
)/2 = {(
E
N
−
E
N + 1
)+(
E
N − 1
−
E
N
)}/2 = (-
E
N + 1
+
E
N − 1
)/2
(26)
χ
P
= -μ
= -(δ
E
/δ
N
)
v
=(-
E
N + 1
+
E
N − 1
)/2 = χ
M
(27)
Now, second derivative of Eq. (24) is
½(∂
2
E
/∂
N
2
)
v
=
b
= (
I
−
A
)/2
(28)
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