Chemistry Reference
In-Depth Information
Feng, and Beck
119
have developed MG methods that allow for the simulta-
neous update of the potential on coarse levels. Wang et al.
120
found rapid con-
vergence for a method in which the charge density and resulting potential were
updated on coarse levels. Wijesekera, Feng, and Beck employed the simulta-
neous algorithm proposed by Costiner and Ta'asan but did not observe a sig-
nificant efficiency gain in relation to the sequential approach. This topic
deserves further exploration for large-scale calculations.
There have been several other advances in real-space FD methodology.
The Multigrid Instead of k-Space (MIKA) project has developed several var-
iants of real-space solvers, including FD representations and MG methods.
The review article by Torsti et al.
73
discusses recent algorithm development
work in that group thorougly; here we just mention their large-scale applica-
tions concerning quantum dots, quantum Hall effects, surface nanostructures,
positron interactions with matter, all-electron finite elements, electron trans-
port through nanodevices, and new wavelet algorithms. Hirose, Ono, and
co-workers have developed alternative large-scale (order-
N
e
) FD methods
for modeling nanostructures
158,175,176
and have applied these methods to
study electron transport through sodium bulk/wire contacts and C
60
bridges.
They have also proposed a ''double-grid'' technique for efficient pseudopoten-
tial calculations. Hoshi and Fujiwara
177-179
have developed linear-scaling FD
methods and, more recently, highly efficient numerical schemes for simulating
nanostructure processes.
One advantage of real-space calculations is the ability to place domains of
higher resolution in localized regions of space
84,85
without sacrificing algorithm
efficiency. This is particularly difficult to accomplish in the context of plane-wave
calculations due to their nonlocality. As an alternative to placing a
localized
mesh
refinement at a specific locationon a coarsermesh,
84,85
several groups have devel-
oped methods for curving the grids to generate higher resolution near nuclei.
87,88
It is sensible to have higher resolution near the nuclear locations because, even
with pseudopotentials to remove the core electrons, the potential varies most
rapidly there. The grid-curving methods have been found to enhance accuracy
without a large gain in computational cost. Two potential disadvantages of these
methods are that the Laplacian operator can become less banded relative to FD
discretization on a Cartesian grid, and the grid curving transformation can be
rather nonlocal (see the grid pictures in Ref. 88). These drawbacks appear to
have been enough to limit the further development of the grid curving
approaches. Local refinements embedded in coarser domains, which maintain
the same Cartesian structure, are likely a better alternative.
An ambitious goal is to model materials at the all-electron level. Typi-
cally, for large-scale materials calculations in DFT, the core electrons are
removed and replaced by a nonlocal pseuodopotential. Mortensen, Hansen,
and Jacobsen
180
have taken the first steps to perform all-electron DFT calcula-
tions with FD difference representations and MG acceleration. They employ
the projection augmented wave method in the frozen-core approximation.
