Handbook of Solvents - George Wypych - ChemTech - Ventech!

486 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
also effective way of seeing this problem is to partition the polarization vector in terms accounting
for the main degrees of freedom of the solvent molecules, each one determined by
a proper relaxation time (or by the related frequency).
In most applications, this spectral decomposition is limited to two terms representing
“fast” and “slow” phenomena, respectively 106,107
()
Pt = P + P
[8.125]
fast slow
In particular, the fast term is easily associated with the polarization due to the bound
electrons of the solvent molecules which can instantaneously adjust themselves to any
change of the inducing field. The slow term, even if it is often referred to as a general
orientational polarization, has a less definite nature. Roughly speaking, it collects many different
nuclear and molecular motions (vibrational relaxations, rotational and translational
diffusion, etc.) related to generally much longer times. Only when the fast and the slow
terms are adjusted to the actual description of the solute and/or the external field, on one
hand, and to each other on the other hand, do we have solvation systems corresponding to
full equilibrium. Nonequilibrium solvation systems, on the contrary, are characterized by
only partial response of the solvent. A very explicative example of this condition, but not
the only one possible, is represented by a vertical electronic excitation in the solute molecule.
In this case, in fact, the immediate solvent response will be limited to its electronic polarization
only, as the slower terms will not able to follow such fast change but will remain
frozen in the equilibrium status existing immediately before the transition.
This kind of analysis is easily shifted to BE-ASC models in which the polarization
vector is substituted by the apparent surface charge in the representation of the solvent reaction
field. 107 In this framework, the previous partition of the polarization vector into fast and
slow components leads to two corresponding surface apparent charges, σ f and σ s, the sum of
which gives the total apparent charge σ. The definition of these charges is in turn related to
the static dielectric constant ε(0) (for the full equilibrium total charge σ), to the frequency
dependent dielectric constant ε(ω) (for the fast component σ f), and to a combination of them
(for the remaining slow component σ s).
The analytical form of the frequency dependence of ε(ω) in general is not known, but
different reliable approximations can be exploited. For example, if we assume that the solvent
is polar and follows a Debye-like behavior, we have the general relation:
() 0 − ( ∞)
ε ε
εω () = ε()
∞ +
1−iωτ D
[8.126]
with ε(0)=78.5, ε(∞)=1,7756, and τ D=0.85×10 -11 s in the specific case of water.
It is clear that at the optical frequencies usually involved in the main physical processes,
the value of ε(ω) is practically equal to ε(∞); variations with respect to this, in fact,
become important only at ωτ D