Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 651
α
s
( p)
( p)
ψ μ� ψ ψ μ� ψ
= 2∑
p≠0 E −E
0 0
s s s s s s
0s
ps
[11.1.22]
is the polarizability tensor of the solvent molecule in the ground state, respectively, and
and
( 0p) 0
( p)
μ = ψ μ� ψ
[11.1.23]
s() i
s() i a s() i
( λν ) ( λ) ( ν )
μ = ψ μ� ψ
[11.1.24]
a a a a
are the transition dipoles between the two states (0 → p) in the solvent and in the solute
(λ→ν) molecules, respectively. In the last formulae, Eν and Eλ denote the energy of the solute
molecule in the respective (ν-th and λ-th) states, and Eps and E0s - the energy of a solvent
molecule in the p-th and in the ground state, respectively. The first term in equation
[11.1.18] represents therefore the electrostatic interaction of the unperturbed charge distribution
of the two molecules, given as the interaction between the respective permanent
point dipoles. The second term in this equation corresponds to the interaction of the permanent
dipole of the solute with the dipole induced in the solvent whereas the third term reflects
the interaction of the permanent dipole of the solvent with the induced dipole of the
solute. The last term represents the second-order interaction of both molecules in excited
states and quantifies thus effectively the dispersion interaction in the solute-solvent system.
The equation [11.1.18] refers, of course, to a single fixed configuration of the solute
and the solvent molecules. In order to find the effective interaction energy in the liquid medium,
an appropriate statistical averaging over all configurations has to be carried out. In
most practical applications, this procedure is very complicated and thus the semiclassical
continuum approaches are employed to describe the solvent. The description of the electrostatic
interactions between the solute and the solvent has been based on the Onsager’s reaction
field model. According to this model, the energy of electrostatic interaction between an
arbitrary charge distribution inside the solute molecule and the surrounding polarizable dielectric
medium is given by the following equation 16
1
E el = ∫ EE(
ε−1)
dV
8π
V
s o [11.1.25]
where E o is the electrostatic field of the charges in the molecule in vacuo and E s is the modified
field in the presence of dielectric medium. Notably, within the formalism of the last
[11.1.25], the dielectric constant ε of the medium is still a function of the space coordinates,
i.e., both the interior of the molecule and the surrounding medium are treated by the same
equation. However, the integral in the last equation cannot be found analytically and even
the numerical integration over the space presents a difficult mathematical task. Therefore,
the electrostatic equation is usually simplified by the application of the Gauss divergence
theorem. According to this theorem, the volume integral in [11.1.25] is transformed into a
surface integral over some boundary