Essentials of Computational Chemistry

11.2 ELECTROSTATIC INTERACTIONS WITH A CONTINUUM 395
is the Debye–Hückel parameter given by
κ 2 = 8πq2 I
εkBT
(11.7)
where I is the ionic strength of the electrolyte solution. The inverse of κ is also called the
Debye length.
Thus, in order to determine the electrostatic potential in solvents containing either nonelectrolytes
or electrolytes, we need only solve Eqs. (11.5) or (11.6), respectively, using the
known charge density of the solute and some cavity defining how the dielectric constant
varies about the solute. As differential equations go, Eq. (11.5) is straightforward, but
Eq. (11.6) is fairly unpleasant. As a result, it is often simplified at low ionic strength by
using a truncated power expansion for the hyperbolic sine, giving the so-called linearized
PB equation
∇ε (r) ·∇φ (r) − ε (r) λ (r) κ 2 φ (r) =−4πρ (r) (11.8)
Note that it is fairly common in the literature for continuum solvation calculations to be
reported as having been carried out using Poisson–Boltzmann electrostatics even when no
electrolyte concentration is being considered, i.e., the Poisson equation is considered a special
case of the PB equation and not named separately.
For certain ideal cavity shapes, the relevant PB equations have particularly simple analytic
solutions. While such ideal cavities are not typically to be expected for arbitrary solute
molecules, consideration of some examples is instructive in illustrating how more sophisticated
modeling may be undertaken by generalization therefrom.
11.2.1.1 Ideal cavities
Consider a conducting sphere bearing charge q, which may be taken as an approximation to
a monatomic ion. The charge on such an object spreads out uniformly on the surface, and
the charge density at any point on the surface may thus be expressed as
ρ (s) = q
4πa 2
(11.9)
where s is a surface point and a is the radius of the sphere. So, in order to evaluate Eq. (11.3),
we will need to integrate only on the surface of the sphere (since the charge density is zero
everywhere else). To determine the electrostatic potential at the surface we must approach
from the outside (the dielectric constant of a conductor is infinite and the electrostatic potential
everywhere inside is zero, so there is a formal discontinuity in the potential at the
surface). From the outside, the electrostatic potential is well known to be equivalent to that
for a point charge q at the origin, giving the central field result
φ (r) =− q
ε |r|
(11.10)