Real-Space and Multigrid Methods in Computational Chemistry - Reviews in Computational Chemistry - page 248

over the domain (when discretized on a lattice). A lattice field theory 136-138
includes a normalization step that does lead to exact charge balance in the
examined domain, by adjusting the bulk densities self-consistently. This
is an important point because the lack of conservation may become more
pronounced on coarser levels without the normalization. Third, the defect
correction must contain terms related to the nonlinearity. Fourth, the driving
term f h (see Eq. [40]) includes the nonlinear terms during the relaxation
steps. Finally, discontinuities in the dielectric constant can lead to numerical
difficulties. We have found, however, that for typical dielectric constants in
biological problems (80 for water, 2 for the internal parts of membranes,
and 4-20 for protein interiors), these issues can be handled directly in FAS
solvers 58,142,148 without the special techniques developed in Ref. 149.

Poisson-Nernst-Planck (PNP) Equations for Ion Transport
All of the problems discussed so far pertain to static or equilibrium solu-
tions of partial differential equations. Many problems in chemistry are none-
quilibrium in nature, however, and here we discuss briefly one approximate
means of modeling transport (the PNP theory), 4,19,150 and techniques used
for solving the relevant equations with MG methods. The underlying equa-
tions first involve solution of the diffusion equation in the presence of an exter-
nal potential, and then solution of the Poisson equation to generate a new
potential. The two equations must be solved repeatedly—one equation affects
the other. If the transport process is steady state (i. e., the particle densities do
not change with time), the problem can then be recast as two Poisson-type
equations that must be solved self-consistently. The nonlinearity arises from
the coupling of the two equations. Typical applications would include the
study of ion transport through a semiconductor device or through an ion chan-
nel in biology.
The conservation law for the ion transport is

qr ð r ; t Þ

¼r j ð r ; t Þ ½ 51

q t

where is the current. There is a

conservation equation for each of the chemical species, but we omit those
labels here. The diffusion (or Smoluchowski) equation results from the
assumption that the particles move in a medium where their motion is rapidly
damped and thermalized; 151 this is the Brownian motion assumption. Then

r ð r ; t Þ is the particle number density, and j ð r ; t Þ

j ð r ;

Þ¼ ð r Þ½r r ð r ; Þþ b r f ð r Þ r ð r ; Þ ½ 52a

t D t t

¼ D ð r Þ exp ½ bf ð r Þr exp ½ bf ð r Þ r ð r ; t Þ ½ 52b