Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 673
A scheme has been developed that reduces the spatial representation of the dispersion
interaction into a surface representation of this interaction. 4 According to this approach, the
average dispersion-repulsion energy of a solute-solvent system has been written as follows:
disp− rep = ∫ ∫
( ) ( )
E � U Ω g Ω dΩ
[11.1.126]
where Ω stands for the set of all coordinates of the molecules involved, g(Ω) is the solute-solvent
pair distribution function and U(Ω) is expressed as a sum of two-body dispersion-repulsion
potentials. In the case of the fixed geometry of the solute molecule
∑
( r )
( k ) ( k )
E disp−rep = nSNSdms ∫rms
g dr
s∈S ∑∑
m∈M k
ms ms
3
ms
[11.1.127]
The integrals in the last formula can be limited only to a certain minimum distance defined,
for instance, by the van-der-Waals envelopes of interacting molecules. By introduc-
(k)
ing the auxiliary vector functions A ms ( rms ) such that
� ( k ) ( k ) −k
∇ A r = d r g r [11.1.128]
ms
( ) ( )
ms ms
ms
ms ms
the average dispersion-repulsion energy between the solute and solvent molecules in solution
may be written as follows
( k )
E disp−rep = nS∑NS∑∑∫ Amsn d
s∈S m∈M k
Σs
σ σ
[11.1.129]
where n σ is the outer normal to the surface Σ s at the position σ. The integral in the last equation
may be calculated numerically using an appropriate partitioning (tessellation) of the
surface.
A quantum-mechanical method of calculation of the dispersion energy has been developed
on the basis of the above-cited semiclassical Abe’s theory. 81 According to this
method, the dispersion energy, E disp , for a solute molecule in a spherical cavity is given as
follows
E
disp
M S ( μ IJ ) ( μ KO )
2 1
=−
3 3∑∑
S S M
3a
a E − E + E −E
S M
J≠I K≠O K
O
2 2
J
M
I
[11.1.130]
where the superscript S refers to the solvent molecule and the superscript M to the solute
M S
molecule. Thus, μ IJ and μ KO are the transition dipoles between the respective states of the
S S
solute (I and J) and the solvent (K and O) molecules. In equation [11.1.130], E K,
EOand
M M
E J,
EIdenote
the energies of the K-th and O-th state of the solvent and of the J-th and I-th
state of the solute molecule, respectively. The cavity radii for the solvent and solute molecules
are denoted as aS and aM, respectively.