Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 653
where Q is the numerical value of the ionic charge. The next term (l = 0) corresponds to the
total dipole interaction with the corresponding reaction field (Onsager dipolar term)
E Onsager
a
=
2
( 1−ε)
μ
3 ( 2ε+ 1)
�
[11.1.32]
0
where � μ is the dipole moment of the solute. In most applications, only these two terms that
are the largest by size are considered in the calculation of the electrostatic interaction energy.
However, depending on system studied, the interaction of higher electrical moments
with the corresponding reaction field may become also significant and the terms corresponding
to higher moments of order 2 l (quadruple, octuple, hexadecuple, etc.) should be
taken into account.
In many cases, the shape of the solute molecule may be very different from the sphere
and therefore, it is necessary to develop the methods of calculation of the electrostatic solvation
energy for more complex cavities. In the case of the ellipsoidal cavity with main
semiaxes a, b, and c, the analytical formulas are still available for the calculation of the
charge and dipolar terms of the electrostatic interaction with the reaction field. The charge
term is simply
2
ell ( 1−ε)
Q
E Born =
2ε
abc
whereas in the respective dipolar term 19
ell
E Onsager
[11.1.33]
= R � μ [11.1.34]
the reaction field R is presented using a special tensor as follows
⎛3Aa(
1− Aa)(
1−
εμ )
⎞
a
⎜
0 0 ⎟
⎜ abc[
ε+ ( 1−
ε)
Aa]
⎟
⎜
3Ab( 1− Ab)(
1−
εμ )
⎟
b
�
R = ⎜ 0
0 ⎟μ
⎜
abc[
ε+ ( 1−
ε)
Ab]
⎟
⎜
3Ac( 1− Ac)(
1−
εμ ) ⎟
c
⎜
0 0
⎟
⎝
abc[
ε+ ( 1−
ε)
Ac]
⎟
⎠
[11.1.35]
where A a,A band A c are the standard ellipsoidal shape factor integrals, and μ a, μ b and μ c are
the dipole moment components along the main semiaxes of the ellipsoid. Several methods
have been developed to define the semiaxes of the ellipsoidal cavity. For instance, these
have been taken collinear with the axes of the solute dipole polarizability tensor, and their
lengths proportional to the respective eigenvalues. 20 Another definition proceeds from the
inertia tensor of the van-der-Waals solid, i.e., a solid or uniform density composed of interlocking
van-der-Waals spheres. 21 Also, the ellipsoidal surface has been defined in terms of
the best fitting of a given molecular electrostatic isopotential surface. 22
The above-discussed theoretical formulation of the electrostatic solute-solvent interaction
is applicable for the fixed charge distribution inside the solute molecule. However,
the solvent reaction field may cause a redistribution of the charge inside the solute. The