Handbook of Solvents - George Wypych - ChemTech - Ventech!

652 Mati Karelson
E el = SE o dS
− ε 1 �
8π
∫ Φ n [11.1.26]
S
where S is the boundary surface, � n the outward normal unit vector on S and the reaction potential
Φ s is defined as follows: E s = -grad Φ S. Depending on the shape of solute molecular
cavity, different approaches have been applied for the calculation of the electrostatic solvation
energies of compounds in liquids. Within the classical reaction field theory of
Kirkwood and Onsager, 17,18 the solute molecule is represented by a set of point charges fixed
inside of sphere of a radius a 0 and the electrostatic equation [11.1.26] is solved by applying
the appropriate boundary conditions inside and outside the sphere. It is also assumed that
the dielectric constant inside the cavity (sphere) is equal to unity (vacuum) and outside the
cavity has a constant value, corresponding to the macroscopic dielectric constant of the medium
studied. In that case, the energy of the electrostatic interaction between the solute
charge distribution and the surrounding dielectric medium is given by the following infinite
expansion
E
1
∞
el = ∑ ee i j ∑ 2 i, j l=
0
( l + 1)( 1−ε)
ε(
+ ) +
⎡
⎢
⎣
l 1
l
⎤ rr i
1⎦a0 ⎥
l
j
P
2l+ 1 l ij
( cosθ
)
[11.1.27]
where e i and e j are the charges inside the sphere at positions r i and r j, respectively, and θ ij is
the angle at the center of the sphere between the vectors r i and r j. In the last equation, the
summation proceeds over all charged particles (nuclei and electrons) of the solute and
P 1(cosθ ij) are the Legendre polynomials of l-th order. By expressing the Legendre polynomials
as the products of the respective spherical harmonics of order m (-l ≤m ≤l), equation
[11.1.27] can be rewritten as
where
and
1
∞ l
m
E el =− ∑ ∑ l
2 l = 0 m=−1
R M [11.1.28]
m
l
m m
R = f M
[11.1.29]
f
l
=
l l
( l + 1)( ε−1)
( + 1) ε+
1
1
l a
l 2l+ 1
0
[11.1.30]
m m
In these equations, Ml and R l represent the electrical momentum and the respective
reaction field component. The first term (l = 0) in the expansion [11.1.27] gives the interaction
of the excess (ionic) charge of the solute with the respective reaction field created in the
dielectric medium (Born term)
E
Born
− Q
=
a
1 ε
2ε
2
0
[11.1.31]