Chemistry Reference
In-Depth Information
Dipole is a descriptor of the asymmetry of charge distribution. Let us reproduce
the quantum mechanical algorithm developed by Ghosh and Bhattacharya [31].
The permanent quantum mechanical electric dipole moment, μ of the molecule
whose electric state is given by Ψ
el
is
μ = ∫ Ψ
*
el
d
op
Ψ
el
dτ
(16)
where d
op
is the quantum mechanical operator of dipole moment. The electric di-
pole moment operator, d
op
, for a molecule includes summation over both the electronic
and nuclear charges.
d
/
(
er
)
/
Z er
=-+
aaa
(17)
where r
α
is the vector from the origin to the nucleus of atomic number Z
α
and r
i
is
the vector to the electron i.
Since the second term in equation (17) is independent of the electronic coordi-
nates, we have
op
i
i
#
#
n} }x }}x
/
(
er
)
d
/
Z er
d
=
"
-
,
+
aaa
*
(18)
el
i
1
el
el
el
#
or,
μ
e
}
el
2
/
Z r
(19)
=-
aaa
We can also rewrite this expression as
#
n
eN
}
2
/
r d
x
e
/
Z r
=-
+
aaa
(20)
el
i
i
where N is the number of electrons in the molecule and r
i
is the position vector of
electron i. or,
μ = - e ∫ ∫ ∫ ρ (x, y, z) r dxdydz +
eZre
/
aaa
(21)
where ρ(x, y, z) is the electronic probability density.
Now expanding ρ in terms of the molecular orbitals and then expanding the mo-
lecular orbitals, in turn, in terms of the atomic orbitals according to the LCAO-MO
SCF scheme and invoking the necessary approximations of Pople's method [32, 33],
the molecular dipole moments are obtained as a sum of two components.
μ
total
= μ
at
+ μ
sp
(22)
The fi rst term arises from the contribution of net atomic charge densities and sec-
ond one is the contribution of atomic dipoles resulting from the mixing (hybridization)
of s and p orbitals on the same atom center. This method has been applied to solve
many puzzles of chemistry. We mention one important application of the method of
dipole correlation of electronic structure is in resolving the mystery—whether the two
lone pairs of electrons of O of water molecule is the same (squirrel ear) or not? [34].
Pauling [1, 2] fi rst assumed that the ionic character may be obtained from the
dipole moment of a compound and he compared the dipole moment with the moment
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