Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.8 Practical applications of modeling 489
mnl mnl
( 12) ( ) ( 12)
c c R Φ [8.129]
= ∑ μv μv
mnl
μv
where the coefficients c mnl
μν ( 12) depend only on the distance between the mass centers of the
particles, and the indexes m, n, l,μ, and v are integers satisfying the following inequalities:
m −n ≤l ≤ m + n, −m ≤μ ≤m, −n ≤v ≤n
[8.130]
The dielectric constant can be obtained from the rotational invariant coefficients of the
pair correlation function. For example, following Fries et al., 112 it can be evaluated using the
Kirkwood g factor:
∞
g = 1− ∫R
h () R dR
4π
2 110
ρ 00
[8.131]
3
4π
2
βρme
g
9
=
0
( ε− 1)( 2ε+ 1)
9ε
[8.132]
In the previous relations, ρ is the number density of molecules, β=1/kT, and m e is the
effective dipole of molecules along their symmetry axes.
Such an approach has been used to evaluate the dielectric constant of liquid
acetonitrile, acetone and chloroform by means of the generalized SCMF-HNC (self-consistent
mean field hypernetted chain), and the values obtained are 30.8 for acetonitrile, 19.9 for
acetone, and 5.66 for chloroform. The agreement with the experimental data, 35.9 for
acetonitrile, 20.7 for acetone, and 4.8 for chloroform (under the same conditions), is good.
Another way of evaluating the dielectric constant is that used by Richardi et al. 113 who
applied molecular Ornstein-Zernike theory and eq. [8.131] and [8.132]. The obtained values
for both acetone and chloroform are low with respect to experimental data, but in excellent
agreement with Monte Carlo simulation calculations. In this case, the dielectric
constant is evaluated by using the following expressions:
μμ
32
2
ρμ
=
9kTε
i=
1 j=
1
2
Nμ
( ε− 1)( 2εrf + 1)
( ε + ε)
rf
0
N
N
∑∑
i j
[8.133]
where ε rf is the dielectric constant of the continuum part of the system, i.e., the part outside
the cutoff sphere, N the number of molecules of the system and μ the dipole moment.
A further improvement in the mentioned approach has been achieved by Richardi et
al. 114 by coupling the molecular Ornstein-Zernike theory with a self-consistent mean-field
approximation in order to take the polarizability into account. For the previously-mentioned
solvents (acetone, acetonitrile and chloroform), the calculated values are in excellent agreement
with experimental data, showing the crucial role of taking into account polarizability
contributions for polar polarizable aprotic solvents.