8,435 research outputs found
Analysis of the divide-and-conquer method for electronic structure calculations
We study the accuracy of the divide-and-conquer method for electronic
structure calculations. The analysis is conducted for a prototypical subdomain
problem in the method. We prove that the pointwise difference between electron
densities of the global system and the subsystem decays exponentially as a
function of the distance away from the boundary of the subsystem, under the gap
assumption of both the global system and the subsystem. We show that gap
assumption is crucial for the accuracy of the divide-and-conquer method by
numerical examples. In particular, we show examples with the loss of accuracy
when the gap assumption of the subsystem is invalid
SCDM-k: Localized orbitals for solids via selected columns of the density matrix
The recently developed selected columns of the density matrix (SCDM) method
[J. Chem. Theory Comput. 11, 1463, 2015] is a simple, robust, efficient and
highly parallelizable method for constructing localized orbitals from a set of
delocalized Kohn-Sham orbitals for insulators and semiconductors with
point sampling of the Brillouin zone. In this work we generalize the SCDM
method to Kohn-Sham density functional theory calculations with k-point
sampling of the Brillouin zone, which is needed for more general electronic
structure calculations for solids. We demonstrate that our new method, called
SCDM-k, is by construction gauge independent and is a natural way to describe
localized orbitals. SCDM-k computes localized orbitals without the use of an
optimization procedure, and thus does not suffer from the possibility of being
trapped in a local minimum. Furthermore, the computational complexity of using
SCDM-k to construct orthogonal and localized orbitals scales as O(N log N )
where N is the total number of k-points in the Brillouin zone. SCDM-k is
therefore efficient even when a large number of k-points are used for Brillouin
zone sampling. We demonstrate the numerical performance of SCDM-k using systems
with model potentials in two and three dimensions.Comment: 25 pages, 7 figures; added more background sections, clarified
presentation of the algorithm, revised the presentation of previous work,
added a more high level overview of the new algorithm, and mildly clarified
the presentation of the results (there were no changes to the numerical
results themselves
O(N) methods in electronic structure calculations
Linear scaling methods, or O(N) methods, have computational and memory
requirements which scale linearly with the number of atoms in the system, N, in
contrast to standard approaches which scale with the cube of the number of
atoms. These methods, which rely on the short-ranged nature of electronic
structure, will allow accurate, ab initio simulations of systems of
unprecedented size. The theory behind the locality of electronic structure is
described and related to physical properties of systems to be modelled, along
with a survey of recent developments in real-space methods which are important
for efficient use of high performance computers. The linear scaling methods
proposed to date can be divided into seven different areas, and the
applicability, efficiency and advantages of the methods proposed in these areas
is then discussed. The applications of linear scaling methods, as well as the
implementations available as computer programs, are considered. Finally, the
prospects for and the challenges facing linear scaling methods are discussed.Comment: 85 pages, 15 figures, 488 references. Resubmitted to Rep. Prog. Phys
(small changes
Cyclic Density Functional Theory : A route to the first principles simulation of bending in nanostructures
We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) -- a
self-consistent first principles simulation method for nanostructures with
cyclic symmetries. Using arguments based on Group Representation Theory, we
rigorously demonstrate that the Kohn-Sham eigenvalue problem for such systems
can be reduced to a fundamental domain (or cyclic unit cell) augmented with
cyclic-Bloch boundary conditions. Analogously, the equations of electrostatics
appearing in Kohn-Sham theory can be reduced to the fundamental domain
augmented with cyclic boundary conditions. By making use of this symmetry cell
reduction, we show that the electronic ground-state energy and the
Hellmann-Feynman forces on the atoms can be calculated using quantities defined
over the fundamental domain. We develop a symmetry-adapted finite-difference
discretization scheme to obtain a fully functional numerical realization of the
proposed approach. We verify that our formulation and implementation of Cyclic
DFT is both accurate and efficient through selected examples.
The connection of cyclic symmetries with uniform bending deformations
provides an elegant route to the ab-initio study of bending in nanostructures
using Cyclic DFT. As a demonstration of this capability, we simulate the
uniform bending of a silicene nanoribbon and obtain its energy-curvature
relationship from first principles. A self-consistent ab-initio simulation of
this nature is unprecedented and well outside the scope of any other systematic
first principles method in existence. Our simulations reveal that the bending
stiffness of the silicene nanoribbon is intermediate between that of graphene
and molybdenum disulphide. We describe several future avenues and applications
of Cyclic DFT, including its extension to the study of non-uniform bending
deformations and its possible use in the study of the nanoscale flexoelectric
effect.Comment: Version 3 of the manuscript, Accepted for publication in Journal of
the Mechanics and Physics of Solids,
http://www.sciencedirect.com/science/article/pii/S002250961630368
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
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