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 (small changes

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

Spatial patterns of large African cats : a large-scale study on density, home range size, and home range overlap of lions Panthera leo and leopards Panthera pardus

SUPPORTING INFORMATION : APPENDIX S1. Site information. APPENDIX S2. Intuitive explanation of the autocorrelated kernel density estimator. APPENDIX S3. Sources of density data. APPENDIX S4. Mathematical modifications of Jetz et al.’s (2014) overlap equation. APPENDIX S5. Lion pride size data.1. Spatial patterns of and competition for resources by territorial carnivores are typically explained by two hypotheses: 1) the territorial defence hypothesis and 2) the searching efficiency hypothesis. 2. According to the territorial defence hypothesis, when food resources are abundant, carnivore densities will be high and home ranges small. In addition, carnivores can maximise their necessary energy intake with minimal territorial defence. At medium resource levels, larger ranges will be needed, and it will become more economically beneficial to defend resources against a lower density of competitors. At low resource levels, carnivore densities will be low and home ranges large, but resources will be too scarce to make it beneficial to defend such large territories. Thus, home range overlap will be minimal at intermediate carnivore densities. 3. According to the searching efficiency hypothesis, there is a cost to knowing a home range. Larger areas are harder to learn and easier to forget, so carnivores constantly need to keep their cognitive map updated by regularly revisiting parts of their home ranges. Consequently, when resources are scarce, carnivores require larger home ranges to acquire sufficient food. These larger home ranges lead to more overlap among individuals’ ranges, so that overlap in home ranges is largest when food availability is the lowest. Since conspecific density is low when food availability is low, this hypothesis predicts that overlap is largest when densities are the lowest. 4. We measured home range overlap and used a novel method to compare intraspecific home range overlaps for lions Panthera leo (n = 149) and leopards Panthera pardus (n = 111) in Africa. We estimated home range sizes from telemetry location data and gathered carnivore density data from the literature. 5. Our results did not support the territorial defence hypothesis for either species. Lion prides increased their home range overlap at conspecific lower densities whereas leopards did not. Lion pride changes in overlap were primarily due to increases in group size at lower densities. By contrast, the unique dispersal strategies of leopards led to reduced overlap at lower densities. However, when human-caused mortality was higher, leopards increased their home range overlap. Although lions and leopards are territorial, their territorial behaviour was less important than the acquisition of food in determining their space use. Such information is crucial for the future conservation of these two iconic African carnivores.The Natural Sciences and Engineering Research Council of Canada and a Hugh Kelly Fellowship from Rhodes University, Grahamstown, SA.https://onlinelibrary.wiley.com/journal/13652907am2024Centre for Wildlife ManagementMammal Research InstituteZoology and EntomologySDG-15:Life on lan

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