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where (∂μ/∂R)
e
is the dipole moment derivative at geometric equilibrium.
Differentiation of the equation (47) with respect to R gives the atomic polar tensor
of B atom:
P
x
B
= (∂μ/∂R)
e
= - (χ
B
- χ
A
). 2R
AB
r
A
r
B
χ
A
χ
B
/C(r
A
χ
A
+ r
B
χ
A
)
2
(48)
BOND STRETCHING FREQUENCY AND FORCE CONSTANT
Several correlations have been shown between infrared stretching frequencies of cer-
tain bonds and the electronegativities of the atoms involved however, this is not sur-
prising as such stretching frequencies depend in part on bond strength, which enters
into the calculation of Pauling (1987) electronegativities. The most commonly en-
countered form of Hooke's law is probably the spring equation, which relates the force
exerted by a spring to the distance it is stretched by a force constant, k, measured in
force per length.
F = -kx
(49)
with,
k = - W
1
/R
3
(50)
Now, let us discuss the algorithm of Ray et al. [17] for the computation of the force
constant on the basis of SBC model. In this model, the vibrational energy function was
defi ned as:
W = W
0
+ (W
1
/R)+(W
2
/R
2
)
(51)
W
1
/R = - (Z
A
+ Z
B
)[{(Z
A
+ δ)/r
1
} + {(Z
B
-δ)/r
2
} - {(Z
A
+δ)(Z
A
+ δ)/R(Z
A
+ Z
B
)}
(52)
and
W
2
/R
2
= h
2
(Z
A
+ Z
B
)/8mR
2
υ
2
AB
(53)
where W is the Born-Oppenheimer potential and R is the internuclear distance
between A and B. The term, W
1
/R and W
2
/R
2
are assigned to describe the electrostatic
energy of the system and the kinetic energy of the bond charge moving freely in a one-
dimension box of length υR along the bond respectively.
Using the equalized molecular electronegativity expression, χ
AB
= (χ
A
R
AA
+ χ
B
R
BB
)/2R
AB
, Ray et al. [17] obtained:
W
1
/R = -{(Z
A
+ Z
B
)
2
/R}[2-{r
1
r
2
/(r
1
+ r
2
)
2
}]
(54)
Ray et al. [17] found that the quantity in bracket is close to 7/4 for most of the
reasonable values of r
1
and r
2
, thus they set it equal to 7/4 to obtain a formula having
maximum simplicity. Thereafter using the force constant formula, k = -W
1
/R
3
, they
defi ned the force constant as:
k = (7/4) χ
AB
2
/(R
AB
C
2
) (55)
where C is the constant which depends on the nature of the bond between A and B.
Badger [45] correlated equilibrium bond distance (R) and the bond stretching force
constant (k) as:
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