65 research outputs found
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O(N) complexity algorithms for First-Principles Electronic Structure Calculations
The fundamental equation governing a non-relativistic quantum system of N particles is the time-dependant Schroedinger Equation [Schroedinger, 1926]. In 1965, Kohn and Sham proposed to replace this original many-body problem by an auxiliary independent-particles problem that can be solved more easily (Density Functional Theory). Solving this simplified problem requires to find the subspace of dimension N spanned by the N eigenfunctions {Psi}{sub i} corresponding to the N lowest eigenvalues {var_epsilon}{sub i} of a non-linear Hamiltonian operator {cflx H} determined from first-principles. From the solution of the Kohn-Sham equations, forces acting on atoms can be derived to optimize geometries and simulate finite temperature phenomenon by molecular dynamics. This technique is used at LLNL to determine the Equation of State of various materials, and to study biomolecules and nanomaterials
A novel multigrid method for electronic structure calculations
A general real-space multigrid algorithm for the self-consistent solution of
the Kohn-Sham equations appearing in the state-of-the-art electronic-structure
calculations is described. The most important part of the method is the
multigrid solver for the Schroedinger equation. Our choice is the Rayleigh
quotient multigrid method (RQMG), which applies directly to the minimization of
the Rayleigh quotient on the finest level. Very coarse correction grids can be
used, because there is no need to be able to represent the states on the coarse
levels. The RQMG method is generalized for the simultaneous solution of all the
states of the system using a penalty functional to keep the states orthogonal.
The performance of the scheme is demonstrated by applying it in a few molecular
and solid-state systems described by non-local norm-conserving
pseudopotentials.Comment: 9 pages, 3 figure
O(N) algorithms in tight-binding molecular-dynamics simulations of the electronic structure of carbon nanotubes
The O(N) and parallelization techniques have been successfully applied in
tight-binding molecular-dynamics simulations of single-walled carbon nanotubes
(SWNTs) of various chiralities. The accuracy of the O(N) description is found
to be enhanced by the use of basis functions of neighboring atoms (buffer). The
importance of buffer size in evaluating the simulation time, total energy, and
force values together with electronic temperature has been shown. Finally,
through the local density of state results, the metallic and semiconducting
behavior of (10x10) armchair and (17x0) zigzag SWNT s, respectively, has been
demonstrated.Comment: 15 pages, 10 figure
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
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Maximally localized Wannier functions: Theory and applications
The electronic ground state of a periodic system is usually described in
terms of extended Bloch orbitals, but an alternative representation in terms of
localized "Wannier functions" was introduced by Gregory Wannier in 1937. The
connection between the Bloch and Wannier representations is realized by
families of transformations in a continuous space of unitary matrices, carrying
a large degree of arbitrariness. Since 1997, methods have been developed that
allow one to iteratively transform the extended Bloch orbitals of a
first-principles calculation into a unique set of maximally localized Wannier
functions, accomplishing the solid-state equivalent of constructing localized
molecular orbitals, or "Boys orbitals" as previously known from the chemistry
literature. These developments are reviewed here, and a survey of the
applications of these methods is presented. This latter includes a description
of their use in analyzing the nature of chemical bonding, or as a local probe
of phenomena related to electric polarization and orbital magnetization.
Wannier interpolation schemes are also reviewed, by which quantities computed
on a coarse reciprocal-space mesh can be used to interpolate onto much finer
meshes at low cost, and applications in which Wannier functions are used as
efficient basis functions are discussed. Finally the construction and use of
Wannier functions outside the context of electronic-structure theory is
presented, for cases that include phonon excitations, photonic crystals, and
cold-atom optical lattices.Comment: 62 pages. Accepted for publication in Reviews of Modern Physic
Real-Space Mesh Techniques in Density Functional Theory
This review discusses progress in efficient solvers which have as their
foundation a representation in real space, either through finite-difference or
finite-element formulations. The relationship of real-space approaches to
linear-scaling electrostatics and electronic structure methods is first
discussed. Then the basic aspects of real-space representations are presented.
Multigrid techniques for solving the discretized problems are covered; these
numerical schemes allow for highly efficient solution of the grid-based
equations. Applications to problems in electrostatics are discussed, in
particular numerical solutions of Poisson and Poisson-Boltzmann equations.
Next, methods for solving self-consistent eigenvalue problems in real space are
presented; these techniques have been extensively applied to solutions of the
Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue
problems arising in semiconductor and polymer physics. Finally, real-space
methods have found recent application in computations of optical response and
excited states in time-dependent density functional theory, and these
computational developments are summarized. Multiscale solvers are competitive
with the most efficient available plane-wave techniques in terms of the number
of self-consistency steps required to reach the ground state, and they require
less work in each self-consistency update on a uniform grid. Besides excellent
efficiencies, the decided advantages of the real-space multiscale approach are
1) the near-locality of each function update, 2) the ability to handle global
eigenfunction constraints and potential updates on coarse levels, and 3) the
ability to incorporate adaptive local mesh refinements without loss of optimal
multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic
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Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations
Ab initio DFT electronic structure calculations involve an iterative process to solve the Kohn-Sham equations for an Hamiltonian depending on the electronic density. We discretize these equations on a grid by finite differences. Trial eigenfunctions are improved at each step of the algorithm using multigrid techniques to efficiently reduce the error at all length scale, until self-consistency is achieved. In this paper we focus on an iterative eigensolver based on the idea of inexact inverse iteration, using multigrid as a preconditioner. We also discuss how this technique can be used for electrons described by general non-orthogonal wave functions, and how that leads to a linear scaling with the system size for the computational cost of the most expensive parts of the algorithm
Linear scaling first-principles molecular dynamics with controlled accuracy
We propose a real-space finite differences approach for accurate and unbiased O(N) Density Functional Theory molecular dynamics simulations based on a localized orbitals representation of the electronic structure. The discretization error can be reduced systematically by adapting the mesh spacing, while the orbitals truncation error decreases exponentially with the radius of the localization regions. For regions large enough, energy conservation in microcanonical simulations is demonstrated for liquid water. We propose an explanation for the energy drift observed for smaller regions
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