Handbook of Solvents - George Wypych - ChemTech - Ventech!

136 Abraham Nitzan
where A is an integration constant and τ L is the longitudinal Debye relaxation time
τ
L
ε
ε τ
e
= D
[4.3.17]
s
The integration constant A is determined from the initial conditions: Immediately following
the switch-on of the charge distribution, i.e. of D, E is given by E(t=0) = D/εe,so -1
A= ε −ε D
−1
. Thus, finally,
( e s )
1 ⎛ 1 1 ⎞
Et () = D+ ⎜ − ⎟De
ε
⎟
s ⎝εe
εs
⎠
−t/
τL
[4.3.18]
We see that in this case the relaxation is characterized by the time τL which can be very
different from τD. For example, in water εe / εs
≅1/ 40, and while τD ≅ 10 ps,
τL is of the order
of 0.25ps.
4.3.3 LINEAR RESPONSE THEORY OF SOLVATION DYNAMICS
The continuum dielectric theory of solvation dynamics is a linear response theory, as expressed
by the linear relation between the perturbation D and the response of E, Eq. [4.3.2].
Linear response theory of solvation dynamics may be cast in a general form that does not
depend on the model used for the dielectric environment and can therefore be applied also in
molecular theories. 13,14 Let
H = H + H′
0
[4.3.19]
where H0 describes the unperturbed system that is characterized by a given potential surface
on which the nuclei move, and where
H′=∑X jFj() t
[4.3.20]
j
is some perturbation written as a sum of products of system variables Xj and external time
dependent perturbations Fj(t). The nature of X and F depend on the particular experiment: If
for example the perturbation is caused by a point charge q(t) at position rj, q(t)δ(r-rj), we may
identify F(t) with this charge and the corresponding Xj is the electrostatic potential operator
at the charge position. For a continuous distribution ρ(r,t) of such charge we may write
3
H′= ∫d
rΦ() r ρ ( r,t)
, and for ρ( r,t ) = ∑ qj() t δ(
r−r j
j ) this becomes ∑ Φ ( rj ) qj( t)
.
j
Alternatively we may find it convenient to express the charge distribution in terms of point
moments (dipoles, quadrupoles, etc.) coupled to the corresponding local potential gradient
tensors, e.g. H′ will contain terms of the form μ∇Φ and Q:∇∇Φ where μ and Q are point dipoles
and quadrupoles respectively.
In linear response theory the corresponding solvation energies are proportional to the
corresponding products q, μ and Q: where denotes the usual observable
average. For example, the average potential is proportional in linear response to
q
the perturbation source q. The energy needed to create the charge q is therefore∫ dq′
2
0
≈( 1/ 2) q ≈(
1/ 2)
q.
Going back to the general expressions [4.3.19] and [4.3.20], linear response theory relates
non-equilibrium relaxation close to equilibrium to the dynamics of equilibrium fluctu-