Handbook of Solvents - George Wypych - ChemTech - Ventech!

662 Mati Karelson
where ri stands for the i-th electron position vector operator, Ra is the position vector of the
a-th nucleus with the charge Za in the solute and δ(r) is the Dirac’s delta function. The first
0
term in square brackets in Eq. [11.1.71], Vm ( r ) , represents the electrostatic potential created
by the solvent in the absence of the solute and the second, integral term corresponds to the
reaction potential response function of the polarizable solvent. Together these terms produce
the reaction field potential applying to the solvent molecule in the polarizable dielectric
medium. Notably, a principal part of the Hamiltonian [11.1.71] is the solute charge
density that can be represented using different approximations of which the multipolar expansion
has been mostly applied. By using the distributed multipole model, it is possible to
obtain the GSCRF equations for the molecules of a complex shape. However, it has been
mentioned that the use of multipole expansions of the solvent electrostatic and reaction potentials
in Eq. [11.1.71] may cause this Hamiltonian to become unbound and special damping
procedures have been invented to overcome this difficulty.
The GSCRF total energy of the solute is given by the following equation
EGSCRF = ψH� ψ + ψ∫drΩs() rVm() r ψ + ψ∫drΩs() r ψ ∫dr′
G(
r, r′ ) Ω s(
r′
) [11.1.73]
2
0 0 1
This energy expression can be used to build up the respective variational functional to get
the molecular orbitals [above]. A crucial step in the general self-consistent reaction field
procedure is the estimation of the solvent charge density needed to obtain the response function
G(r,r�) and the reaction potential. The use of Monte Carlo or molecular dynamics simulations
of the system consisting the solute and surrounding solvent molecules has been
proposed to find the respective solvent static and polarization densities.
Several methods have been developed to account for the solute cavities of arbitrary
shape in the solution. The polarizable continuum model (PCM) is based on the numerical
integration of the relevant electrostatic equations describing the electrostatic interaction between
the molecular charge distribution and the charge created on the boundary surface between
the solute molecule and surrounding dielectric continuum. 40-45 Within this method,
the solute cavity is usually constructed from the overlapping van der Waals spheres of constituent
atoms in the solute molecule and the solvent reaction field arising from the solute
charge distribution is calculated numerically. Alternatively, the cavity can be defined as
constructed from the electron isodensity surface around the solute molecule (IPCM). 46 According
to the classical electrostatics, the electrostatic potential at any point in the space can
be described in terms of the apparent charge distribution, σ, on the cavity surface. It consists
of two terms
Φs = ΦM + Φσ
[11.1.74]
the first of which (Φ M ) corresponds to the electrostatic potential created by the charge distribution
of the solute and the second (Φ σ ) is due to the reaction potential by the solvent. The
latter is directly connected with the apparent charge distribution on the surface of the cavity
as follows:
Φ
σ
() r
=
∫
Σ
() s
σ
r −s
d 2
s
[11.1.75]