Chemistry Reference
In-Depth Information
convergence rates should not depend on size for a properly functioning multi-
scale solver. An excellent discussion of the various linear-scaling approaches
to electronic structure is presented in the topic by Martin.
55
OTHER NONLINEAR PROBLEMS: THE POISSON—
BOLTZMANN AND POISSON—NERNST—PLANCK
EQUATIONS
Poisson-Boltzmann Equation
A second nonlinear problem to consider is the Poisson-Boltzmann (PB)
mean-field theory of ionic solution electrostatics.
129-134
This approximate
theory has found wide application in biophysics involving problems such as
protein-protein and protein-membrane interactions. At a qualitative level,
the PB equation arises from replacement of the exact mobile-ion charge distri-
bution by its average, assuming that the average is given by the Boltzmann
factor for the ions interacting with the mean electrostatic potential:
r½
e
ð
r
Þr
f
ð
r
Þ¼
4
p
½
r
f
ð
r
Þþ
zn
þ
exp
ð
b
z
f
ð
r
Þ
v
ð
r
ÞÞ
zn
exp
ð
b
z
f
ð
r
Þ
v
ð
r
ÞÞ
½
47
where a three-dimensional representation has been used, and
kT
.
Besides assuming an average distribution of mobile ions determined by the
Boltzmann factor, this equation assumes that the underlying medium (water,
membrane, protein) is a continuum dielectric with spatially varying dielectric
constant
b
¼
1
=
. For example, in a membrane protein calculation, it is common
to use values of 80 for water, 4-20 for the protein interior, and 2 for the
hydrophobic part of the membrane. On the right-hand side (rhs) of Eq. [47]
there are three contributions to the charge density: one for fixed, discrete
charges (the partial or full charges on the protein atoms), and one each
for the distributed positive and negative mobile ions. The charge
z
is the mag-
nitude of the charge on the mobile ions. (The physical assumptions underlying
the PB equation generally imply accurate results only for
z
e
ð
r
Þ
¼
1.) The added
potential
v
excludes ions from parts of space such as the membrane or pro-
tein interior domains. The variables
n
þ
and
n
are the bulk concentrations of
the positive and negative ions in the aqueous solution far from the region of
interest (where the potentials have decayed to zero due to screening).
If the electric potential happens to be small, the exponential terms in
Eq. [47] can be linearized to produce
ð
r
Þ
z
2
r½
e
ð
r
Þr
f
ð
r
Þ¼
4
p
½
r
f
ð
r
Þþ
b
ð
n
þ
þ
n
Þ
f
ð
r
Þ
½
48
