Chemistry Reference
In-Depth Information
chemistry. Because the applications have covered a wide range of topics over the
last decade, this section cannot review the whole field. Rather, some representa-
tive examples will be considered from the electronic structure, electrostatics, and
transport areas. The focus will be on methods that utilize MG techniques in con-
junction with real-space representations, but several of the publications dis-
cussed have employed alternative real-space techniques for solving the
eigenvalue (and/or total energy minimization), electrostatic, and transport pro-
blems.
Electronic Structure
In June of 2005 a CECAM workshop was held in Lyon, France, on the
topics of real-space, embedding, and linear-scaling techniques for electronic
structure. A special issue of
Physica Status Solidi B
157
is devoted to the proceed-
ings of that meeting, and several review articles appear there that thoroughly
cover current methods development activities and applications. Accordingly,
that is a good place to start when entering the literature of real-space methods
for electronic structure. Another review covers earlier developments in the
field,
58
and a recent topic provides an excellent introduction to real-space meth-
ods applied to nanostructure modeling.
158
The initial stages of real-space methods development focused on the
underlying representations (FD and FE) and test studies on small sys-
tems
59,63,159-163
(e.g., diatomic molecules) to determine the accuracy of
the methods, convergence with approximation order, and the rate of con-
vergence to the exact numerical result as a function of the number of self-
consistency iterations. Here we discuss the substantial progress over the last
several years, which has centered on linear-scaling algorithms, alternative
representations, and applications to large systems at the nanoscale. The
review will be organized based on the methods of representation: finite dif-
ferences, finite elements, and other approaches (such as wavelets, atomic
orbitals, etc.).
In the discussion to follow, we note the difference between solving the
eigenvalue problem and minimizing the total electronic energy in some orbital
or density matrix representation. This distinction is important for labeling the
various algorithms. If we solve the Kohn-Sham equations of DFT as an
eigen-
value
problem in real space, the wave functions must satisfy restrictions (as
discussed above), namely they cover the whole physical domain and they
must be normalized and orthogonal since they are the eigenfunctions of a Her-
mitian operator. Given those restrictions, the scaling of any real-space numer-
ical method must go formally as
cq
2
N
g
, where
c
is a constant prefactor,
q
is
the number of eigenfunctions (typically equal to
N
e
=
2), and
N
g
is the number
of grid points in the domain. With specialized techniques (below), this scaling
may be effectively reduced to
cqN
g
, which corresponds to the cost of updating
all the wave functions on the grid.
