Chemistry Reference
In-Depth Information
CHAPTER 5
Real-Space and Multigrid Methods
in Computational Chemistry
Thomas L. Beck
Departments of Chemistry and Physics, University of Cincinnati,
Cincinnati, Ohio
INTRODUCTION
Real-space methods are iterative numerical techniques for solving partial
differential equations on grids in coordinate space. The physical responses due to
many chemical phenomena are restricted to domains that are relatively local in
space, and real-space methods are well suited to exploit that physical locality.
In real-space methods, the iterative updates of the desired functions require
information only in a small neighborhood near the updated point. The draw-
back with this approach is that, if the iterations are performed only on a single
(finest) grid, the solver tends to stall due to the long-wavelength components of
the errors. Multigrid methods overcome this stalling by utilizing information
from a wide range of length scales. With the incorporation of multiscale ideas,
solvers can often be designed for which the cost scales linearly with system size.
In the last 15 years, real-space and multigrid methods have been developed for a
wide range of problems in computational chemistry. Those problems include
electronic structure, Poisson and Poisson-Boltzmann equations, and transport
models. This reviewwill first give a tutorial introduction to real-space andmulti-
grid methods and then present some case studies illustrating how these techni-
ques have been applied in chemistry, nanomaterials, and biology.
