Chemistry Reference
In-Depth Information
set convergence issues; a similar strategy has been employed in the SIESTA
code (below).
191-193
When comparing the various codes and sizes of applica-
tions, it is important to remember that different systems examined can lead to
very different grid size requirements; for example, O atoms may require a grid
spacing three times smaller than Si atoms due to the harsher pseudopotential
near the O nucleus, and this, in turn, implies 27 times more grid points.
Tsuchida and Tsukada have developed an adaptive FE algorithm for
large-scale electronic structure calculations.
194-196
They utilize a three-dimen-
sional cubic Hermite basis to represent the electron orbitals and have extended
their method to linear-scaling complexity.
197
MG techniques were employed
in the solution of the Poisson problem, and accurate forces suitable for mole-
cular dynamics simulations were computed. The method was tested on dia-
mond and cubic BN lattices and the C
60
molecule. The FE code has also
been applied to simulate liquid formamide at the ab initio level.
196
Those
simulations were used to assess the accuracy of previous simulations using
empirical force fields. The current version of the Tsuchida-Tsukada code is
called FEMTECK;
197
this code incorporates their recently developed linear-
scaling technology.
The MIKA project has developed FE solvers in addition to FD methods.
73
The FE portion of their code is based on the Elmer package. The
p
-element basis
employed is based on underlying Legendre polynomials. It is pointed out in Ref.
73 that high-order polynomials are better suited for smooth functions. Near a
nucleus, for example, where the wave function varies rapidly, it is better to use
lower order elements. The MIKA project FE code has been tested successfully on
small-to-medium-sized molecular cases. More recently it has been applied to
examine electron currents through nanostructures.
198
In other FE work, the Kaxiras
199
and Vashishta
200,201
groups have devel-
oped multiscale methods that seamlessly link a central ab initio DFT region with
coarser classical molecular dynamics levels and terminate with a continuum FE
domain. These new algorithms allow for the accurate modeling of a central
atomic resolution domain while still incorporating forces from more distant
regions treated at a continuum mechanics level; crucial applications of this
kind of work are stress distributions and the propagation of cracks through
solids. Finally, Yamakawa and Hyodo
202
have developed a hybrid Gaussian/
FE mixed basis for all-electron DFT calculations, and the new method was suc-
cessfully tested on several small-molecule cases. The use of Gaussians for the
core electrons allows for a coarser FE grid for the valence electrons.
We complete this section with a listing of other algorithmic develop-
ments in real-space electronic structure. As mentioned above, the PARSEC
code has incorporated alternative techniques for accelerating the solution of
the eigenvalue problem based on Chebyshev-filtered subspace methods,
165,166
thus circumventing the need for multiscale methods. Jordan and Mazziotti
167
have developed new spectral difference methods for real-space electronic
structure that can yield the same accuracies as the FD representation with
