Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.3 Polar solvation dynamics 137
ations: The first fluctuation dissipation theorem states that following a step function change
in F:
() ()
F t = 0, t < 0; F t = F , t ≥0
[4.3.21]
j j j
the corresponding averaged system’s observable relaxes to its final equilibrium value as
t →∞according to
( )
1
X j() t − X j(
∞ ) = ∑F
X X () t − X X
kT
B
l
l j l j l
[4.3.22]
where all averages are calculated with the equilibrium ensemble of H 0. Applying Eq.
[4.3.22] to the case where a sudden switch of a point charge q → q+Δq takes place, we have
2 Δq
( ΦΦ Φ ) ΦΦ()
t
Δq
Φ() t − Φ(
∞ ) = () t − =
kT
kT
B B
δ δ [4.3.23]
The left hand side of [4.3.23], normalized to 1 att=0,isalinear approximation to the solvation
function
St ()
E t E
t
solv () − solv ( ∞)
Φ() − Φ(
∞)
=
LR
E ( 0)
−E ( ∞)
Φ( 0) − Φ(
∞)
solv solv
[4.3.24]
and Eq. [4.3.23] shows that in linear response theory this non equilibrium relaxation function
is identical to the equilibrium correlation function
St () LRCt ()
δΦ( 0)
δΦ(
t)
≡
2
δΦ
[4.3.25]
C(t) is the time correlation function of equilibrium fluctuations of the solvent response potential
at the position of the solute ion. The electrostatic potential in C(t) will be replaced by
the electric field or by higher gradients of the electrostatic potential when solvation of
higher moments of the charge distribution is considered.
The time dependent solvation function S(t) is a directly observed quantity as well as a
convenient tool for numerical simulation studies. The corresponding linear response approximation
C(t) is also easily computed from numerical simulations, and can also be studied
using suitable theoretical models. Computer simulations are very valuable both in
exploring the validity of such theoretical calculations, as well as the validity of linear response
theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization
of the solute and solvent motions that dominate the solvation process. Many
such simulations were published in the past decade, using different models for solvents such
as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the
close qualitative similarity between the time evolution of solvation in different simple solvents,
and second, the marked deviation from the simple exponential relaxation predicted
by the Debye relaxation model (cf. Eq. [4.3.18]). At least two distinct relaxation modes are