GYutsis: heuristic based calculation of general recoupling coefficients

General angular momentum recoupling coefficients can be expressed as a summation formula over products of 6-j coefficients. Yutsis, Levinson and Vanagas developed graphical techniques for representing the general recoupling coefficient as a cubic graph and they describe a set of reduction rules allowing a stepwise generation of the corresponding summation formula. This paper is a follow up to [Van Dyck and Fack, Comput. Phys. Comm. 151 (2003) 353-368] where we described a heuristic algorithm based on these techniques. In this article we separate the heuristic from the algorithm and describe some new heuristic approaches which can be plugged into the generic algorithm. We show that these new heuristics lead to good results: in many cases we get a more efficient summation formula than our previous approach, in particular for problems of higher order. In addition the new features and the use of our program GYutsis, which implements these techniques, is described both for end users and application programmers.

Program summary

Title of program: CycleCostAlgorithm, GYutsisCatalogue number: ADSAProgram Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSAProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Users may obtain the program also by downloading either the compressed tar file gyutsis.tgz (for Unix and Linux) or the zip file gyutsis.zip (for Windows) from our website (http://caagt.rug.ac.be/yutsis/). An applet version of the program is also available on our website and can be run in a web browser from the URL http://caagt.rug.ac.be/yutsis/GYutsisApplet.html.Licensing provisions: noneComputers for which the program is designed: any computer with Sun's Java Runtime Environment 1.4 or higher installed.Programming language used: Java 1.2 (Compiler: Sun's SDK 1.4.0)No. of lines in program: approximately 9400No. of bytes in distributed program, including test data, etc.: 544 117Distribution format: tar gzip fileNature of physical problem: A general recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta can be expressed as a formula consisting of products of 6-j coefficients summed over a certain number of variables. Such a formula can be generated using the program GYutsis (with a graphical user front end) or CycleCostAlgorithm (with a text-mode user front end).Method of solution: Using the graphical techniques of Yutsis, Levinson and Vanagas (1962) a summation formula for a general recoupling coefficient is obtained by representing the coefficient as a Yutsis graph and by performing a selection of reduction rules valid for such graphs. Each reduction rule contributes to the final summation formula by a numerical factor or by an additional summation variable. Whereas an optimal summation formula (i.e. with a minimum number of summation variables) is hard to obtain, we present here some new heuristic approaches for selecting an edge from a k-cycle in order to transform it into an (k−1)-cycle (k>3) in such a way that a ‘good’ summation formula is obtained.Typical running time: From instantaneously for the typical problems to 30 s for the heaviest problems on a Pentium II-350 Linux-system with 256 MB RAM.     

The analytical Scheme calculator for angular momentum coupling and recoupling coefficients

We describe a Scheme implementation of the interactive environment to calculate analytically the Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients that are used in the quantum theory of angular momentum. The orthogonality conditions for considered coefficients are implemented. The program provides a fast and exact calculation of the coefficients for large values of quantum angular momenta.

Program summary

Title of program:Scheme2ClebschCatalogue number:ADWCProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWCProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneComputer for which the program is designed:Any Scheme-capable platformOperating systems under which the program has been tested: Windows 2000Programming language used:SchemeMemory required to execute with typical data:50 MB (≈ size of DrScheme, version 204)No. of lines in distributed program, including test data, etc.: 2872No. of bytes in distributed program, including test data, etc.: 109 396Distribution format:tar.gzNature of physical problem:The accurate and fast calculation of the angular momentum coupling and recoupling coefficients is required in various branches of quantum many-particle physics. The presented code provides a fast and exact calculation of the angular momentum coupling and recoupling coefficients for large values of quantum angular momenta and is based on the GNU Library General Public License PLT software http://www.plt-scheme.org/.Method of solution:A direct evaluation of sum formulas. A general angular momentum recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta is expressed as a sum over products of the Clebsch-Gordan coefficients.Restrictions on the complexity of the problem:Limited only by the DrScheme implementation used to run the program. No limitation inherent in the code.Typical running time:The Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients with small angular momenta are computed almost instantaneously. The running time for large-scale calculations depends strongly on the number and magnitude of arguments' values (i.e., of the angular momenta).     

A computer program for two-particle generalized coefficients of fractional parentage

We present a FORTRAN90 program GCFP for the calculation of the generalized coefficients of fractional parentage (generalized CFPs or GCFP). The approach is based on the observation that the multi-shell CFPs can be expressed in terms of single-shell CFPs, while the latter can be readily calculated employing a simple enumeration scheme of antisymmetric A-particle states and an efficient method of construction of the idempotent matrix eigenvectors. The program provides fast calculation of GCFPs for a given particle number and produces results possessing numerical uncertainties below the desired tolerance. A single j-shell is defined by four quantum numbers, (e,l,j,t).A supplemental C++ program parGCFP allows calculation to be done in batches and/or in parallel.

Program summary

Program title:GCFP, parGCFPCatalogue identifier: AEBI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBI_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.: 17 199No. of bytes in distributed program, including test data, etc.: 88 658Distribution format: tar.gzProgramming language: FORTRAN 77/90 (GCFP), C++ (parGCFP)Computer: Any computer with suitable compilers. The program GCFP requires a FORTRAN 77/90 compiler. The auxiliary program parGCFP requires GNU-C++ compatible compiler, while its parallel version additionally requires MPI-1 standard librariesOperating system: Linux (Ubuntu, Scientific) (all programs), also checked on Windows XP (GCFP, serial version of parGCFP)RAM: The memory demand depends on the computation and output mode. If this mode is not 4, the program GCFP demands the following amounts of memory on a computer with Linux operating system. It requires around 2 MB of RAM for the A=12 system at Ex?2. Computation of the A=50 particle system requires around 60 MB of RAM at Ex=0 and ∼70 MB at Ex=2 (note, however, that the calculation of this system will take a very long time). If the computation and output mode is set to 4, the memory demands by GCFP are significantly larger. Calculation of GCFPs of A=12 system at Ex=1 requires 145 MB. The program parGCFP requires additional 2.5 and 4.5 MB of memory for the serial and parallel version, respectively.Classification: 17.18Nature of problem: The program GCFP generates a list of two-particle coefficients of fractional parentage for several j-shells with isospin.Solution method: The method is based on the observation that multishell coefficients of fractional parentage can be expressed in terms of single-shell CFPs [1]. The latter are calculated using the algorithm [2,3] for a spectral decomposition of an antisymmetrization operator matrix Y. The coefficients of fractional parentage are those eigenvectors of the antisymmetrization operator matrix Y that correspond to unit eigenvalues. A computer code for these coefficients is available [4]. The program GCFP offers computation of two-particle multishell coefficients of fractional parentage. The program parGCFP allows a batch calculation using one input file. Sets of GCFPs are independent and can be calculated in parallel.Restrictions:A<86 when Ex=0 (due to the memory constraints); small numbers of particles allow significantly higher excitations, though the shell with j?11/2 cannot get full (it is the implementation constraint).Unusual features: Using the program GCFP it is possible to determine allowed particle configurations without the GCFP computation. The GCFPs can be calculated either for all particle configurations at once or for a specified particle configuration. The values of GCFPs can be printed out with a complete specification in either one file or with the parent and daughter configurations printed in separate files. The latter output mode requires additional time and RAM memory. It is possible to restrict the (J,T) values of the considered particle configurations. (Here J is the total angular momentum and T is the total isospin of the system.) The program parGCFP produces several result files the number of which equals to the number of particle configurations. To work correctly, the program GCFP needs to be compiled to read parameters from the standard input (the default setting).Running time: It depends on the size of the problem. The minimum time is required, if the computation and output mode (CompMode) is not 4, but the resulting file is larger. A system with A=12 particles at Ex=0 (all 9411 GCFPs) took around 1 sec on a Pentium4 2.8 GHz processor with 1 MB L2 cache. The program required about 14 min to calculate all 1.3×¹⁰⁶ GCFPs of Ex=1. The time for all 5.5×¹⁰⁷ GCFPs of Ex=2 was about 53 hours. For this number of particles, the calculation time of both Ex=0 and Ex=1 with CompMode = 1 and 4 is nearly the same, when no other processes are running. The case of Ex=2 could not be calculated with CompMode = 4, because the RAM memory was insufficient. In general, the latter CompMode requires a longer computation time, although the resulting files are smaller in size. The program parGCFP puts virtually no time overhead. Its parallel version speeds-up the calculation. However, the results need to be collected from several files created for each configuration.References:[1] J. Levinsonas, Works of Lithuanian SSR Academy of Sciences 4 (1957) 17.[2] A. Deveikis, A. Bon?kus, R. Kalinauskas, Lithuanian Phys. J. 41 (2001) 3.[3] A. Deveikis, R.K. Kalinauskas, B.R. Barrett, Ann. Phys. 296 (2002) 287.[4] A. Deveikis, Comput. Phys. Comm. 173 (2005) 186. (CPC Catalogue ID. ADWI_v1_0)     

An object-oriented C++ implementation of Davidson method for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix

A C++ class named Davidson is presented for determining a few eigenpairs with lowest or alternatively highest values of a large, real, symmetric matrix. The algorithm described by Stathopoulos and Fischer is used. The exception mechanism is involved to report the errors. The class is written in ANSI C++, so it is fully portable. In addition a console program as well as a program with graphical user interface for Microsoft Windows is attached, which allow one to calculate the lowest eigenstates of time-independent Schrödinger equation for a given binding potential in one, two or three spatial dimensions. The package contains the classes providing often used potential functions (model atom potential, Coulomb potential, square well potential and Kramers-Henneberger well potential) as well as a possibility to use any potential stored in a file (then any dimensionality of the problem is allowed).The described code is the subject of M.Sc. thesis of T.D. prepared under the supervision of J.M.

Program summary

Program title: DavidsonCatalogue identifier: ADZM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZM_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.: 3 037 055No. of bytes in distributed program, including test data, etc.: 20 002 609Distribution format: tar.gzProgramming language: C++Computer: AllOperating system: AnyRAM: User's parameters dependentWord size: 32 and 64 bitsSupplementary material: Test results for the 2D and 3D cases is availableClassification: 4, 4.8Nature of problem: Finding a few extreme eigenpairs of a real, symmetric, sparse matrix. Examples in quantum optics (interaction of matter with a laser field).Solution method: Davidson algorithmRunning time: The test example included in the distribution package (1D matrix) takes approximately 30 minutes to run. 2D matrix calculations can take hours and 3D, days, to run.     

New efficient programs to calculate general recoupling coefficients: Part I: Generation of a summation formula

A program to generate a summation formula in terms of 6-j coefficients for a general angular momentum recoupling coefficient is described. This algorithms makes use of binary tree transformations as introduced by Burke [Comput. Phys. Commun. 1 (1970) 241] in the program NJSYM. Due attention is paid to finding an optimal summation formula with a minimal number of summation variables, thereby improving the results of NJSYM. The results obtained here are at least as good as (and often better than) the results of the alternative approach NJGRAF, introduced by Bar-Shalom and Klapisch [Comput. Phys. Commun. 50 (1998) 375], using more advanced graphical methods.     

The MATHSCOUT Mathematica package to postprocess the output of other scientific programs

mathscout is a mathematica¹ package to postprocess the output of other programs for scientific calculations. We wrote mathscout to import data from a major program for ab initio computational chemistry into mathematica, so that we could postprocess the chemical results. It can be used to import the output of many other packages that are used, e.g. in molecular dynamics, crystallography, spectroscopic analysis, metabolic and physiological modeling, meteorology and other areas of environmental science, cosmology and particle physics. mathscout assigns a name to each table and non-tabular datum that it extracts. This name is constructed mechanically from the identifier or phrase that precedes or follows or embeds the item in the output that mathscout processes. A selection of non-contiguous items, or all the items in a section of the file, or in the entire file are extracted using simple commands. So far, we have focused on our immediate needs to postprocess the output of the Gaussian² program. Calculations on several molecules that illustrate the usage of the package are presented here and in the Supplementary Information. mathscout is shortened to msct in the software.

Program summary

Program title: msct.mCatalogue identifier: ADZQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZQ_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.: 30 396No. of bytes in distributed program, including test data, etc.: 1 799 469Distribution format: tar.gzProgramming language: MathematicaComputer: Any computer running unix and MathematicaOperating system: UnixSupplementary material: The Development guideClassification: 4.14, 5, 16.1, 20Nature of problem: Import data from output files of scientific computing packages, such as Gaussian, into Mathematica for symbolic calculation and production of publication quality tables and plots.Solution method: Provision of mnemonic top-down parsing procedures, functional programming.Running time: The complete extraction of data from a small basis density functional calculation on the water molecule, and from a larger basis density functional calculation on the zinc hydrate ion, that ran to 33 iterations, took 1 second and 23 seconds, respectively, on a Dell Poweredge 1750.     

A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of Δx2Nq+2 when a (2Nq+1)-point formula is used for any positive integer Nq with Δx=Δy, while Nq=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of Nq, which is more efficient than the traditional methods through the decrease of the step size.     

An application of the interpolating scaling functions to wave packet propagation

The wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the order of the underlying interpolating polynomial. In this basis the potential energy matrix is diagonal. The kinetic energy matrix is sparse, and in the 1D case, has a band-diagonal structure. An important future of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore, the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to the high order finite difference methods. The results for the calculation of the H+H₂ collinear collision shows that the ISF provide one with an accurate and efficient representation for use in wave packet propagation method.     

Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission

We present a grid-based procedure to solve the eigenvalue problem for the two-dimensional Schrödinger equation in cylindrical coordinates. The Hamiltonian is discretized by using adapted finite difference approximations of the derivatives and this leads to an algebraic eigenvalue problem with a large (sparse) matrix, which is solved by the method of Arnoldi. By this procedure the single particle eigenstates of nuclear systems with arbitrary deformations can be obtained. As an application we have considered the emission of scission neutrons from fissioning nuclei.     

Optimized evaluation of a large sum of functions using a three-grid approach

The efficient evaluation of numerous values of functions that vary rapidly only in a small part of the region of interest is presented. An optimized algorithm using a sequence of grids with increasing resolution is developed. The algorithm does not make any assumptions about special properties of the function to be evaluated, e.g., symmetry. An additional speed-up is obtained by exploiting the asymptotic behaviour of the functions to be summed. Two applications from high resolution atmospheric radiative transfer modelling in the infrared and microwave are presented. In a third example asymmetric Rautian line shapes important for high resolution molecular spectroscopy are considered. Computational gains by more than two orders of magnitude with relative errors less than 10⁻³ have been achieved.     

Deterministic event-based simulation of quantum phenomena

We propose and analyse simple deterministic algorithms that can be used to construct machines that have primitive learning capabilities. We demonstrate that locally connected networks of these machines can be used to perform blind classification on an event-by-event basis, without storing the information of the individual events. We also demonstrate that properly designed networks of these machines exhibit behavior that is usually only attributed to quantum systems. We present networks that simulate quantum interference on an event-by-event basis. In particular we show that by using simple geometry and the learning capabilities of the machines it is possible to simulate single-photon interference in a Mach-Zehnder interferometer. The interference pattern generated by the network of deterministic learning machines is in perfect agreement with the quantum theoretical result for the single-photon Mach-Zehnder interferometer. To illustrate that networks of these machines are indeed capable of simulating quantum interference we simulate, event-by-event, a setup involving two chained Mach-Zehnder interferometers, and demonstrate that also in this case the simulation results agree with quantum theory.     

Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r²+λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.     

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

A FORTRAN 77 program for calculating energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach is presented. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with homogeneous boundary conditions: (i) the Dirichlet, Neumann and third type at the left and right boundary points for continuous spectrum problem, (ii) the Dirichlet and Neumann type conditions at left boundary point and Dirichlet, Neumann and third type at the right boundary point for the discrete spectrum problem. The resulting system of radial equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. As a test desk, the program is applied to the calculation of the reaction matrix and radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field. This version extends the previous version 1.0 of the KANTBP program [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675].

Program summary

Program title: KANTBPCatalogue identifier: ADZH_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_v2_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.: 20 403No. of bytes in distributed program, including test data, etc.: 147 563Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM: This depends on

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the number of hyperradial points; and

4.

the number of eigensolutions required.

The test run requires 2 MBClassification: 2.1, 2.4External routines: GAULEG and GAUSSJ [2]Nature of problem: In the hyperspherical adiabatic approach [3-5], a multidimensional Schrödinger equation for a two-electron system [6] or a hydrogen atom in magnetic field [7-9] is reduced by separating radial coordinate ρ from the angular variables to a system of the second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions of the continuum spectrum for such systems of coupled differential equations on finite intervals of the radial variable ρ∈[ρmin,ρmax]. This approach can be used in the calculations of effects of electron screening on low-energy fusion cross sections [10-12].Solution method: The boundary problems for the coupled second-order differential equations are solved by the finite element method using high-order accuracy approximations [13]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [14]. The generalized algebraic eigenvalue problem (A−EB)F=λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDLT factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [14]. As a test desk, the program is applied to the calculation of the reaction matrix and corresponding radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field described in [9] on finite intervals of the radial variable ρ∈[ρmin,ρmax]. For this benchmark model the required analytical expressions for asymptotics of the potential matrix elements and first-derivative coupling terms, and also asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system.Restrictions: The computer memory requirements depend on:

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the total number of hyperradial points; and

4.

the number of eigensolutions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Section 3 and [1] for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMS0 and ASYMSC (when solving the scattering problem) which evaluate asymptotics of the radial wave functions at left and right boundary points in case of a boundary condition of the third type for the above problems.Running time: The running time depends critically upon:

1.

the number of differential equations;

2.

the number and order of finite elements;

3.

the total number of hyperradial points on interval [ρmin,ρmax]; and

4.

the number of eigensolutions required.

The test run which accompanies this paper took 2 s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.References:[1] O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; http://cpc.cs.qub.ac.uk/summaries/ADZHv10.html.[2] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.[3] J. Macek, J. Phys. B 1 (1968) 831-843.[4] U. Fano, Rep. Progr. Phys. 46 (1983) 97-165.[5] C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142.[6] A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Commun. 90 (1995) 311-339.[7] M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352.[8] O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524.[9] O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev, V.V. Serov, Comput. Phys. Commun. 178 (2007) 301 330; http://cpc.cs.qub.ac.uk/summaries/AEAAv10.html.[10] H.J. Assenbaum, K. Langanke, C. Rolfs, Z. Phys. A 327 (1987) 461-468.[11] V. Melezhik, Nucl. Phys. A 550 (1992) 223-234.[12] L. Bracci, G. Fiorentini, V.S. Melezhik, G. Mezzorani, P. Pasini, Phys. Lett. A 153 (1991) 456-460.[13] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40-64.[14] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982.     

A B-spline Galerkin method for the Dirac equation

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y^″=−λ²y, that can also be written as a pair of first-order equations y^′=λz, z^′=−λy. Expanding both y(r) and z(r) in the Bk basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the Bk basis and z(r) in the dBk/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method (Bk,Bk±1) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.     

A new simulation approach to the freezing transition

The task of accurately locating fluid-crystal phase boundaries by computer simulation is hampered by problems associated with traversing mixed-phase states. We describe a recently introduced Monte Carlo (MC) method that circumvents this problem by implementing a global coordinate transformation (“phase switch”) which takes the system from one pure phase to the other in a single MC step. The method is potentially quite general. We illustrate it by application to the freezing of hard spheres.     

FFT-split-operator code for solving the Dirac equation in 2+1 dimensions

The main part of the code presented in this work represents an implementation of the split-operator method [J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129-160; R. Heather, Comput. Phys. Comm. 63 (1991) 446] for calculating the time-evolution of Dirac wave functions. It allows to study the dynamics of electronic Dirac wave packets under the influence of any number of laser pulses and its interaction with any number of charged ion potentials. The initial wave function can be either a free Gaussian wave packet or an arbitrary discretized spinor function that is loaded from a file provided by the user. The latter option includes Dirac bound state wave functions. The code itself contains the necessary tools for constructing such wave functions for a single-electron ion. With the help of self-adaptive numerical grids, we are able to study the electron dynamics for various problems in 2+1 dimensions at high spatial and temporal resolutions that are otherwise unachievable.Along with the position and momentum space probability density distributions, various physical observables, such as the expectation values of position and momentum, can be recorded in a time-dependent way. The electromagnetic spectrum that is emitted by the evolving particle can also be calculated with this code. Finally, for planning and comparison purposes, both the time-evolution and the emission spectrum can also be treated in an entirely classical relativistic way.Besides the implementation of the above-mentioned algorithms, the program also contains a large C++ class library to model the geometric algebra representation of spinors that we use for representing the Dirac wave function. This is why the code is called “Dirac++”.

Program summary

Program title: Dirac++ or (abbreviated) d++Catalogue identifier: AEAS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAS_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.: 474 937No. of bytes in distributed program, including test data, etc.: 4 128 347Distribution format: tar.gzProgramming language: C++Computer: Any, but SMP systems are preferredOperating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIXNumber of processors used: Generally unlimited, but best scaling with 2-4 processors for typical problemsRAM: 160 Megabytes minimum for the examples given hereClassification: 2.7External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists.Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies.Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem.Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called “checkpointing”) which allows the resumption of an interrupted calculation.Additional comments: Along with the program's source code, we provide several sample configuration files, a pre-calculated bound state wave function, and template files for the analysis of the results with both MatLab and Igor Pro.Running time: Running time ranges from a few minutes for simple tests up to several days, even weeks for real-world physical problems that require very large grids or very small time steps.References:

[1]

J.A. Fleck, J.R. Morris, M.D. Feit, Time-dependent propagation of high energy laser beams through the atmosphere, Appl. Phys. 10 (1976) 129-160.

[2]

R. Heather, An asymptotic wavefunction splitting procedure for propagating spatially extended wavefunctions: Application to intense field photodissociation of H⁺₂, Comput. Phys. Comm. 63 (1991) 446.

[3]

M. Frigo, S.G. Johnson, FFTW: An adaptive software architecture for the FFT, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 3, IEEE, 1998, pp. 1381-1384.

[4]

M. Frigo, S.G. Johnson, The design and implementation of FFTW3, in: Proceedings of the IEEE, vol. 93, IEEE, 2005, pp. 216-231. URL: http://www.fftw.org/.

[5]

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, F. Rossi, GNU Scientific Library Reference Manual, second ed., Network Theory Limited, 2006. URL: http://www.gnu.org/software/gsl/.

[6]

M.D. Feit, J.A. Fleck, A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47 (1982) 412-433.

     

Highly precise solutions of the one-dimensional Schrödinger equation with an arbitrary potential

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