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η =½[(δ
2
E
/δ
N
2
)
V
]
(3)
The softness (
S
) is the reciprocal of the hardness;
S
=1/η
(4)
Invoking fi nite difference approximation, Parr and Pearson [22] gave
approximate and operational formulae for electronegativity and hardness
as under
χ =½ (
I
+
A
)
(5)
η =½(
I
−
A
)
(6)
where
I
and
A
are the first ionization potential and electron affinity of the
chemical species.
According to Koopmans' theorem, the orbital energies of the frontier
orbitals can be written as
−
ε
HOMO
=
I
(7)
and
−
ε
LUMO
=
A
(8)
In 1986, within the limitations of Koopmans' theorem, Pearson [23] sub-
stitututed electronegativity and hardness into a MO framework as follows:
χ
= (∂
E
/∂
N
)
v
= (
I
+
A
)/2
or
χ =−(
ε
LUMO
+
ε
HOMO
)/2
(9)
and
η
= (∂
2
E
/∂
N
2
)
v
= (
I
−
A
)/2
or,
η = (
ε
LUMO
−
ε
HOMO
)/2
(10)
He again pointed out that a hard species has a large HOMO-LUMO gap
and a soft species has a small HOMO-LUMO gap [23].
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