Essentials of Computational Chemistry

394 11 IMPLICIT MODELS FOR CONDENSED PHASES
explicitly, we replace it by a continuous electric field that represents a statistical average
over all solvent degrees of freedom at thermal equilibrium. This field is usually called the
‘reaction field’ in the regions of space occupied by the solute, since it derives from reaction
of the solvent to the presence of the solute. The electric field at a given point in space is
the gradient of the electrostatic potential φ at that point, and the work required to create the
charge distribution may be determined from the interaction of the solute charge density ρ
with the electrostatic potential according to
G =− 1
ρ (r)φ (r) dr (11.3)
2
The charge density ρ of the solute may be expressed either as some continuous function
of r or as discrete point charges, depending on the theoretical model used to represent the
solute. The polarization energy, GP, discussed above, is simply the difference in the work
of charging the system in the gas phase and in solution. Thus, in order to compute the
polarization free energy, all that is needed is the electrostatic potential in solution and in the
gas phase (the latter may be regarded as a dielectric medium characterized by a dielectric
constant of 1).
11.2.1 The Poisson Equation
At the heart of all continuum solvent models is a reliance on the Poisson equation of classical
electrostatics to express the electrostatic potential as a function of the charge density and the
dielectric constant. The Poisson equation, valid for situations where a surrounding dielectric
medium responds in a linear fashion to the embedding of charge, is written
∇ 2 4πρ (r)
φ (r) =−
ε
(11.4)
where ε is the dielectric constant of the medium. Insofar as continuum solvation involves
representing the solute explicitly and the solvent implicitly, the charge distribution of the
solute is thought of as being inside a cavity that displaces an otherwise homogeneous dielectric
medium. Thus, there are really two regions, one inside and one outside the cavity, in
which case the Poisson equation is properly written as
∇ε (r) ·∇φ (r) =−4πρ (r) (11.5)
The Poisson equation is valid under conditions of zero ionic strength. If dissolved, mobile
electrolytes are present in the solvent, the Poisson–Boltzmann (PB) equation applies instead
2 kBT
∇ε (r) ·∇φ (r) − ε (r) λ (r) κ
q sinh
qφ (r)
=−4πρ (r) (11.6)
kBT
where q is the magnitude of the charge of the electrolyte ions, λ is a simple switching
function which is zero in regions inaccessible to the electrolyte and one otherwise, and κ 2