Handbook of Solvents - George Wypych - ChemTech - Ventech!

138 Abraham Nitzan
observed, a fast Gaussian-like component and a slower relaxation mode of an exponential
character which may correspond to the expected Debye relaxation. In what follows we describe
these and other features observed in computer simulations of solvation dynamics using
simple generic model dielectric solvents.
4.3.4 NUMERICAL SIMULATIONS OF SOLVATION IN SIMPLE POLAR
SOLVENTS: THE SIMULATION MODEL 11a
The simplest simulated system is a Stockmayer fluid: structureless particles characterized
by dipole-dipole and Lennard-Jones interactions, moving in a box (size L) with periodic
boundary conditions. The results described below were obtained using 400 such particles
and in addition a solute atom A which can become an ion of charge q embedded in this solvent.
The long range nature of the electrostatic interactions is handled within the effective
dielectric environment scheme. 15 In this approach the simulated system is taken to be surrounded
by a continuum dielectric environment whose dielectric constant ε′ is to be chosen
self consistently with that computed from the simulation. Accordingly, the electrostatic potential
between any two particles is supplemented by the image interaction associated with a
spherical dielectric boundary of radius Rc (taken equal to L/2) placed so that one of these
particles is at its center. The Lagrangian of the system is given by
( μμ)
N
I
I
L R, R MARA M Ri M
i Vij
, ,
μ μ
N
N
N
1 2 1
2 1
2 1 J
IJ
= + ∑ + ∑ − 2 ∑ ( Rij ) −∑ViA ( RiA)
−
2 2 2
2
i=
1
i=
1
N 1 DD
− ∑ V R R − V R R − −
2
i≠j
i≠j N
N
AD
2 2
( i, j, μ i, μ j)
∑ ( A, i, μ i) ∑λi(
μ i μ)
i=
1
i=
1
i = 1
[4.3.26]
where N is the number of solvent molecules of mass M, μ dipole moment, and I moment of
inertia. R A and R i are positions of the impurity atom (that becomes an ion with charge q)
and a solvent molecule, respectively, and R ij is |R i -R j|. V LJ ,V DD and V AD are, respectively,
Lennard-Jones, dipole-dipole, and charge-dipole potentials, given by
12 6
[ ]
( ) = ε ( σ / ) −(
σ / )
LJ
V R e R R
ij
D D D
LJ
(Vij is of the same form with σA and εA replacing σD and εD) and
( i, j, μ i, μ j)
DD
V R R
where n=(R i-R j)/R ij,
( n )( n ) ( ′−)
μμ i j −3
μ i μ j 2 ε 1
=
−
R R
( ε′
− )
( 2ε′ + 1)
⎛
AD
V ( Ri, μ i, RA) = q⎜ 1 2 1
−
⎜
⎝R
R
3 3
ij C
( 2ε′+ 1)
3 3
ij C
⎞
⎟μ
⎟
⎠
( R −R
)
i i A
μμ
i j
[4.3.27]
[4.3.28]
[4.3.29]
The terms containing ε′ in the electrostatic potentials V DD and V AD are the reaction
field image terms. 15,16 The last term in Eq. [4.3.26] is included in the Lagrangian as a constraint,
in order to preserve the magnitude of the dipole moments( μ i = μ)
with a SHAKE