Error estimation of recursive orthogonal polynomial expansion method for large Hamiltonian system

Recursive polynomial expansion method is an efficient scheme to evaluate Green functions for large systems without direct diagonalization of the Hamiltonian. It is based on a polynomial expansion of the Green function, and has many advantages compared with other methods. However, there are little reports on its error estimations. In this paper, the cut-off error of the method is estimated analytically, which results from the truncation of expansion at finite orders. It is found that the error is inversely proportional to the number of expansion order N except for the singular points for the system with point spectrum. For the system with continuous spectrum, the error is inversely proportional to N^3/2 and decreases much faster in terms of the expansion order.     

Efficient first-principles calculations of the electronic structure of periodic systems

We have recently presented a real-space method for electronic-structure calculations of periodic systems that is based on the Hohenberg-Kohn-Sham density-functional theory. The method allows the computation of electronic properties of periodic systems in the spirit of traditional plane-wave approaches. In addition, it can be implemented efficiently on parallel computers. Here we will show that the method's inherent parallelism, in conjunction with a newly designed approach for solving the Kohn-Sham equations, enables the accurate study of the ionic and electronic properties of periodic systems containing thousands of atoms from first principles.     

Computation of Maximally Localized Wannier Functions using a simultaneous diagonalization algorithm

We show that a simultaneous diagonalization algorithm used in signal processing applications can be used in the context of electronic structure calculations to efficiently compute Maximally Localized Wannier Functions (MLWFs). Applications to calculations of MLWFs in molecular and solid systems demonstrate the efficiency of the approach. We also present and discuss a parallel version of the algorithm. An extension of the concept of MLWF to generalized minimum spread wavefunctions is proposed.     

Radial numerical integrations based on the sinc function

An algorithm for the generation of adaptive radial grids used in density functional theory or quantum chemical calculations is described. Our approach is general and can be applied for the integration over Slater or Gaussian type functions with only minor modifications. The relative error of the integration is fully controlled by the algorithm within a specified range of exponential parameters and for a given principal quantum number.     

Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach

We present the Gaussian and plane waves (GPW) method and its implementation in Quickstep which is part of the freely available program package CP2K. The GPW method allows for accurate density functional calculations in gas and condensed phases and can be effectively used for molecular dynamics simulations. We show how derivatives of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calculations for single molecules and plane wave based calculations in the condensed phase is excellent. Wave function optimisation with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the density matrix. We illustrate these findings with calculations using commodity PCs as well as supercomputers.     

Linear-scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP

ONETEP is an ab initio electronic structure package for total energy calculations within density-functional theory. It combines ‘linear scaling’, in that the total computational effort scales only linearly with system size, with ‘plane-wave’ accuracy, in that the convergence of the total energy is systematically improvable in the manner typical of conventional plane-wave pseudopotential methods. We present recent progress on improving the performance, and thus in effect the feasible scope and scale, of calculations with ONETEP on parallel computers comprising large clusters of commodity servers. Our recent improvements make calculations of tens of thousands of atoms feasible, even on fewer than 100 cores. Efficient scaling with number of atoms and number of cores is demonstrated up to 32,768 atoms on 64 cores.     

JuNoLo - Jülich nonlocal code for parallel post-processing evaluation of vdW-DF correlation energy

Nowadays the state of the art Density Functional Theory (DFT) codes are based on local (LDA) or semilocal (GGA) energy functionals. Recently the theory of a truly nonlocal energy functional has been developed. It has been used mostly as a post-DFT calculation approach, i.e. by applying the functional to the charge density calculated using any standard DFT code, thus obtaining a new improved value for the total energy of the system. Nonlocal calculation is computationally quite expensive and scales as N² where N is the number of points in which the density is defined, and a massively parallel calculation is welcome for a wider applicability of the new approach. In this article we present a code which accomplishes this goal.

Program summary

Program title: JuNoLoCatalogue identifier: AEFM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 176 980No. of bytes in distributed program, including test data, etc.: 2 126 072Distribution format: tar.gzProgramming language: Fortran 90Computer: any architecture with a Fortran 90 compilerOperating system: Linux, AIXHas the code been vectorised or parallelized?: Yes, from 1 to 65536 processors may be used.RAM: depends strongly on the problem's size.Classification: 7.3External routines:• FFTW (http://www.tw.org/)• MPI (http://www.mcs.anl.gov/research/projects/mpich2/ or http://www.lam-mpi.org/)Nature of problem: Obtaining the value of the nonlocal vdW-DF energy based on the charge density distribution obtained from some Density Functional Theory code.Solution method: Numerical calculation of the double sum is implemented in a parallel F90 code. Calculation of this sum yields the required nonlocal vdW-DF energy.Unusual features: Binds to virtually any DFT program.Additional comments: Excellent parallelization features.Running time: Depends strongly on the size of the problem and the number of CPUs used.     

The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge-Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed.     

Generating relativistic pseudo-potentials with explicit incorporation of semi-core states using APE, the Atomic Pseudo-potentials Engine

We present a computer package designed to generate and test norm-conserving pseudo-potentials within Density Functional Theory. The generated pseudo-potentials can be either non-relativistic, scalar relativistic or fully relativistic and can explicitly include semi-core states. A wide range of exchange-correlation functionals is included.

Program summary

Program title: Atomic Pseudo-potentials Engine (APE)Catalogue identifier: AEAC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 88 287No. of bytes in distributed program, including test data, etc.: 649 959Distribution format: tar.gzProgramming language: Fortran 90, CComputer: any computer architecture, running any flavor of UNIXOperating system: GNU/LinuxRAM: <5 MbClassification: 7.3External routines: GSL (http://www.gnu.org/software/gsl/)Nature of problem: Determination of atomic eigenvalues and wave-functions using relativistic and nonrelativistic Density-Functional Theory. Construction of pseudo-potentials for use in ab-initio simulations.Solution method: Grid-based integration of the Kohn-Sham equations.Restrictions: Relativistic spin-polarized calculations are not possible. The set of exchange-correlation functionals implemented in the code does not include orbital-dependent functionals.Unusual features: The program creates pseudo-potential files suitable for the most widely used ab-initio packages and, besides the standard non-relativistic Hamann and Troullier-Martins potentials, it can generate pseudo-potentials using the relativistic and semi-core extensions to the Troullier-Martins scheme. APE also has a very sophisticated and user-friendly input system.Running time: The example given in this paper (Si) takes 10 s to run on a Pentium IV machine clocked at 2 GHz.     

Linear optical properties of solids within the full-potential linearized augmented planewave method

We present a scheme for the calculation of linear optical properties by the all-electron full-potential linearized augmented planewave (LAPW) method. A summary of the theoretical background for the derivation of the dielectric tensor within the random-phase approximation is provided. The momentum matrix elements are evaluated in detail for the LAPW basis, and the interband as well as the intra-band contributions to the dielectric tensor are given. As an example the formalism is applied to Aluminum. The program is available as a module within the WIEN2k code.     

Optimal linear filtering and extrapolation of realizations of a vector random function observed with errors

A solution to an optimal linear filtering and extrapolation problem is proposed. The investigation object is a realization of multidimensional random function, which is measured with random errors. The solution method does not put any essential constraints on the investigated random fimction, measurment error characteristics, and on the character of relations between them. The solution is based on Pugachov’s canonical decomposition of the investigated random function, and its form is convenient for use with a computer. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 125–132, July–August, 2000.     

Localization of current dipole within a sphere by magnetic measurements

The principal objective of this study is the development of computer programs to determine the location and strength of neural electric activity within the brain from noninvasive magnetic field measurements at the surface of the head. This report presents theoretical calculations and computer programs derived from the method described by Williamson and Kaufman to determine the depth and strength of a current dipole in a sphere. From the location of the magnetic field radial component extremes, Br maximum and Br minimum, the orientation and location of the current dipole can be determined. The accuracy of the solution is dependent on precise location of of the magnetic field extremes as measured from the surface of a sphere, e.g. the head. To validate the program for locating the dipole, theoretical calculations and computer programs related to the total magnetic field vector resulting from a hypothetical current source within a homogeneous sphere were generated. The errors in calculations of the current dipole depth and strength are presented.     

Optimal digital filtering of continuous signals in time lag systems

The problem of the optimal estimation of continuous processes by discrete measurements in the presence of time lag (delay) is considered. On the basis of the theory of parametric transfer functions, an optimal, periodically nonstationary filter is developed, which affords a minimum of the estimation error variance at any instant of time. The comparison is performed of the obtained solution with the optimal stationary filter, which ensures a minimum of the mean (by continuous time) error variance. It is shown that in the problem of estimation of the Markov process of the first order, a simpler stationary filter with the fixer of order zero is insignificantly inferior to the optimal filter.     

Scattering of electromagnetic radiation by a multilayered sphere

The computer implementation of the algorithm for the calculation of electromagnetic radiation scattering by a multilayered sphere developed by Yang, is presented. It has been shown that the program is effective, resulting in very accurate values of scattering efficiencies for a wide range of size parameters, which is a considerable improvement over previous implementations of similar algorithms. The program, named scattnlay, would be the first of its kind to be publicly available.

Program summary

Program title: scattnlayCatalogue identifier: AEEY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEY_1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Gnu General Public License (GPL)No. of lines in distributed program, including test data, etc.: 8932No. of bytes in distributed program, including test data, etc.: 175 276Distribution format: tar.gzProgramming language: ANSI CComputer: Any with a C compilerOperating system: Linux (any), Windows, SolarisRAM: ∼1-100 MBClassification: 1.3Nature of problem: The scattering of electromagnetic (EM) radiation by a multilayered sphere is an interesting phenomenon to study for the application of such materials in several fields. Just to mention two examples, metal nanoshells (a dielectric core surrounded by a metallic shell) are a class of nanoparticles with tunable optical resonances that can be used, among others, in medicine for optical imaging and photothermal cancer therapy; while in the field of atmospheric sciences, light absorption by aerosols has a heating effect in the atmosphere that is of great interest to study several climatic effects. Although at first glance the expressions of the scattering coefficients seem simple and straightforward to implement, they involve several numerical difficulties which make most of the existent algorithms inapplicable to several extreme cases. More recently, Yang [1] has developed an improved recursive algorithm that circumvents most of the numerical problems present in previous algorithms, which is implemented in the current program.Solution method: Calculations of Mie scattering coefficients and efficiency factors for a multilayered sphere as described by Yang [1], combined with standard solutions of the scattering amplitude functions.Restrictions: Single scattering, permeability of the layers is always unity.Running time: Seconds to minutesReferences:

[1]

W. Yang, Appl. Opt. 42 (2003) 1710-1720.

     

Optimal filtering of a random background in image processing problems

We describe a recurrent construction procedure for mean-square optimal linear spatio-temporal filtering of a random background, which makes it possible to construct filtered frames using explicitly written compact analytical expressions.     

The optimal filtering of a class of dynamic multiscale systems

Inthelasttwodecades,themultiscaleautoregressive(MAR)framework[1—6]wasdevelopedtomodelavarietyofrandomprocessescompactlyandestimatethemeffi-ciently.TheMARwasfirstmotivatedbyBassevilleetal.[1],andbasedontheirworkChouetal.proposedthemultiscalestochasticmodelsandoptimalestimationalgorithmsforarichclassofprocesseswhitenedbywavelettransforms(WT)[7—11].Luettgen[5]andFrakt[6]contributedalottothestochasticrealizationoftheMAR.MARestimatesthestationaryprocessesfromnumerousmeasurementsbymultiscale…