Maple procedures for the coupling of angular momenta. VII. Extended and accelerated computations

During recent years, much attention in developing general-purpose, computer-algebra systems was focused not only on better symbolic algorithms but, to a very similar extent, also on fast numerical computations and improved tools for visualization. Behind this development, of course, the main idea is to provide the users with a single environment for the solution of their scientific or engineering tasks. In a revised version of the Racah program, we follow this idea and provide a fast and much extended access to the standard quantities from the theory of angular momentum within the framework of Maple. In this revision, emphasis is paid to the efficient computation of the standard quantities by supporting both, the default software model as well as fast (hardware) floating-point computations. Moreover, Racah is now organized and distributed as a Maplemodule which can be installed and utilized like any other module, including help pages and the use of internally recognized data structures. The present extension of the Racah program may therefore enlarge the range of applications considerably towards problems from quantum optics, collision theory or even solid-state physics.     

Maple procedures for the coupling of angular momenta. VIII. Spin-angular coefficients for single-shell configurations

Matrix elements of physical operators are required when the accurate theoretical determination of atomic energy levels, orbitals and radiative transition data need to be obtained for open-shell atoms and ions. The spin-angular part for these matrix elements is typically based on standard quantities such as matrix elements of the unit tensor, the (reduced) coefficients of fractional parentage as well as a number of other reduced matrix elements concerning various products of electron creation and annihilation operators. Therefore, in order to facilitate the access to the matrix elements of one- and two-particle scalar operators, we present here an extension to the Racah program for the full set of standard quantities and the pure spin-angular coefficients in LS- and jj-couplings. A flexible notation is introduced for defining and manipulating the electron creation and the electron annihilation operators. This will allow us to solve successfully various angular momentum problems in atomic physics.

Program summary

Title of program:RacahCatalogue number: ADURProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADURProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: NoneComputers for which the program is designed: All computers with a valid license of the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.]Installations: University of Kassel (Germany)Operating systems under which the program has been tested: Linux 8.1+Program language used:Maple, Release 8 and 9Memory required to execute with typical data: 30 MBNumber of lines in distributed program, including test data, etc.:36 875Number of bytes in distributed program, including test data, etc.: 1 104 604Distribution format: tar.gzNature of the physical problem: The accurate computation of atomic properties and level structures requires a good understanding and implementation of the atomic shell model and, hence, a fast and reliable access to its standard quantities. Apart from various coefficients of fractional parentage and the reduced matrix elements of the unit tensors, these quantities include the so-called spin-angular coefficients, i.e. the spin-angular parts of the many-electron matrix elements of physical operators, taken in respect of a basis of symmetry-adapted subshell and configuration state functions.Method of solution: The concepts of quasispin and second quantized (creation and annihilation) operators in a spherical tensorial form are used to evaluate and calculate the spin-angular coefficients of one- and two-particle physical operators [G. Gaigalas, Lithuanian J. Phys. 39 (1999) 79, http://arXiv.org/physics/0405078; G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B: At. Mol. Phys. 30 (1997) 3747]. Moreover, the same concepts are applied to support the computation of the coefficients of fractional grandparentage, i.e. the simultaneous de-coupling of two electrons from a single-shell configuration. All these coefficients are now implemented consistently within the framework of the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219].Restrictions on the complexity of the problem: In the present version of the Racah program, all spin-angular coefficients are restricted to the case of a single open shell. For the symmetry-adapted subshell states of such single-shell configurations, the spin-angular coefficients can be calculated for (tensorial coupled) one-particle operators of arbitrary rank as well as for scalar two-particle operators. As previously [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219], the Racah program supports all atomic shells with l?3 in LS-coupling (i.e. s-, p-, d- and f-shells) and all subshells with j?9/2 in jj-coupling, respectively.Unusual features of the program: From the very beginning, the Racah program has been designed as an interactive environment for the (symbolic) manipulation and computation of expressions from the theories of angular momentum and the atomic shell model. With the present extension of the program, we provide the user with a simple access to the coefficients of fractional grandparentage (CFGP) as well as to the spin-angular coefficients of one- and two-particle physical operators. To facilitate the specification of the tensorial form of the operators, a short but powerful notation has been introduced for the creation and annihilation operators as well as for the products of such operators as required for the development of many-body perturbation theory in a symmetry-adapted basis. All the coefficients and the matrix elements from above are equally supported for both LS- and jj-coupled operators and functions. The main procedures of the present extension are described below in Appendix B. In addition, a list of all available commands of the Racah program can be found in the file Racah-commands.ps which is distributed together with the code.Typical running time: The program replies promptly on most requests. Even large tabulations of standard quantities and pure spin-angular coefficients for one- and two-particle scalar operators in LS- and jj-coupling can be carried out in a few (tens of) seconds.     

Maple procedures for the coupling of angular momenta. VI. LS-jj transformations

Transformation matrices between different coupling schemes are required, if a reliable classification of the level structure is to be obtained for open-shell atoms and ions. While, for instance, relativistic computations are traditionally carried out in jj-coupling, a LSJ coupling notation often occurs much more appropriate for classifying the valence-shell structure of atoms. Apart from the (known) transformation of single open shells, however, further demand on proper transformation coefficients has recently arose from the study of open d- and f-shell elements, the analysis of multiple-excited levels, or the investigation on inner-shell phenomena. Therefore, in order to facilitate a simple access to LS↔jj transformation matrices, here we present an extension to the Racah program for the set-up and the transformation of symmetry-adapted functions. A flexible notation is introduced for defining and for manipulating open-shell configurations at different level of complexity which can be extended also to other coupling schemes and, hence, may help determine an optimum classification of atomic levels and processes in the future.     

Jahn—A program for representing atomic and nuclear states within an isospin basis

A computer program is presented to deal with atomic and nuclear state functions within an isospin-coupled basis. Apart from the classification of the isospin bases states, the program Jahn supports the computation of the corresponding coefficients of fractional parentage as well as of the transformation matrices going from a LS-coupled to an isospin-coupled basis. In the future, these features may facilitate the treatment of atomic systems in order to obtain a deeper insight into the coupling of open-shell atoms and ions. The Jahn program has been designed for interactive work and is distributed as a Maple module.

Program summary

Title of program:JahnCatalogue identifier:ADXA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXA_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:NoneComputers for which the program is designed: All computers with a valid license of the computer algebra package Maple which is a registered trademark of Waterloo Maple Inc.Installations: University of Kassel (Germany)Operating systems under which the program has been tested: Linux 8.1+Program language used:Maple, Release 8 and 9Memory required to execute with typical data: 30 MBNumber of lines in distributed program, including test data, etc.: 38 158Number of bytes in distributed program, including test data, etc.: 743 689Distribution format: tar.gzNature of the physical problem: The accurate computation of atomic (nuclear) properties and level structures requires a good understanding and implementation of the atomic (nuclear) shell model and, hence, a fast and reliable access to its classification, the coefficients of fractional parentage and the coefficients of fractional grandparentage. For open-shell atoms and ions, moreover, a reliable classification of the level structure often requires the knowledge of some transformation matrices in order to find the main components of the wave functions as well as their proper spectroscopic notation. In particular, the transformation from a LS-coupled to an isospin-coupled basis is important for atoms and ions with the two open shells n₁lN₁n₂lN₂.Method of solution: The concept of the isospin formalism is used and explained in [V. Šimonis, PhD Thesis, Institute of Physics, Vilnius, 1982 (in Russian); Z. Rudzikas, J. Kaniauskas, Quasispin and Isospin in the Theory of Atom, Mokslas, Vilnius, 1984 (in Russian); J.M. Kaniauskas, V.?. Šimonis, Z.B. Rudzikas, J. Phys. B: At. Mol. Phys. 20 (1987) 3267]. The coefficients of fractional parentage (CFP) in the isospin basis, the coefficients of fractional grandparentage (CFGP) in the isospin basis and the transformation matrices from a LS-coupled to an isospin-coupled basis are provided for s-, p-, d-shells. These matrices are utilized to transform symmetry-adapted configuration state functions (CSF) as obtained from the coupling of two open shells n₁lN₁n₂lN₂. Moreover, a simple notation is introduced to handle such symmetry functions interactively.Restrictions onto the complexity of the problem: The classification of the n₁lN₁n₂lN₂ electron configurations provides support for the subshell angular momentum l=0,…,2 and for the occupation numbers N₁ and N₂, where N₁ and N₂ must be in the range N₁=0,…,(2l+1) and N₂=0,…,(2l+1), respectively. The program provides the CFP and CFGP for isospin-coupled subshell states for the orbital angular momenta l=0,1 and occupation numbers N?2(2l+1) and for l = 2 with N?4, respectively. It also evaluates the transformation matrices for l=0,1 and occupation numbers N₁, N₂ and N in the range N₁=0,…,2l; N₂=1,2; N=N₁+N₂?2(2l+1) and for l=2 and occupation numbers N₁, N₂ and N in the range N₁=0,…,3; N₂=1,2; N=N₁+N₂?4, respectively. The transformation of an atomic state function (ASF) or configuration state function (CSF) from an LS-coupled to an isospin-coupled basis can be obtained for these orbital momenta and occupation numbers.Unusual features of the program: The program is designed as an interactive environment for the (symbolic) manipulation and computation of expressions from theory of atomic and nuclear shell model. Here we provide the user with a simple access to the coefficients of fractional parentage as well as to the transformation matrices . A complete transformation of LS-coupled CSF or ASF into an isospin-coupled basis can be carried out just by typing a few lines at Maple's prompt. These coefficients and transformation matrices enable the user to make a more detailed analysis of matrix elements of the operators of physical quantities within the isospin basis. The (main) commands of the Jahn program are described in detail in Appendices A and B.Typical running time: The program replies promptly on most requests. Even large tabulations of CFP or transformation matrices can be obtained within a few seconds.     

Maple procedures for the coupling of angular momenta. An up-date of the Racah module

An up-date of the Racah module is presented, adopted to Maple 11 and 12, which supports both, algebraic manipulations of expressions from Racah's algebra as well as numerical computations of many functions and symbols from the theory of angular momentum. The functions that are known to the program include the Wigner rotation matrices and n-j symbols, Clebsch-Gordan and Gaunt coefficients, spherical harmonics of various kinds as well as several others.

Program summary

Program title:RacahCatalogue identifier: ADFV_v10_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADFV_v10_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 436No. of bytes in distributed program, including test data, etc.: 544 866Distribution format: tar.gzProgramming language: Maple 11 and 12Computer: All computers with a license for the computer algebra package Maple [1]Operating system: Suse Linux 10.2+ and Ubuntu 8.10Classification: 4.1, 5Catalogue identifier of previous version: ADFV_v9_0Journal reference of previous version: Comput. Phys. Comm. 174 (2006) 616Does the new version supersede the previous version?: YesNature of problem: The theories of angular momentum and spherical tensor operators, sometimes known also as Racah's algebra, provide a powerful calculus for studying spin networks and (quantum) many-particle systems. For an efficient use of these theories, however, one requires not only a reliable handling of a large number of algebraic transformations and rules but, more often than not, also a fast access to their standard quantities, such as the Wigner n-j symbols, Clebsch-Gordan coefficients, spherical harmonics of various kinds, the rotation matrices, and many others.Solution method: A set of Maple procedures has been developed and maintained during the last decade which supports both, algebraic manipulations as well as fast computations of the standard expressions and symbols from the theory of angular momentum [2,3]. These procedures are based on a sizeable set of group-theoretical (and often rather sophisticated) relations which has been discussed and proven in the literature; see the monograph by Varshalovich et al. [4] for a comprehensive compilation. In particular the algebraic manipulation of complex (Racah) expressions may result in considerable simplifications, thus reducing the ‘numerical costs’, and often help obtain further insight into the behaviour of physical systems.Reasons for new version: A revision of the Racah module became necessary for mainly three reasons: (i) Since the last extension of the Racah procedures [5], which was developed within the framework of Maple 8, several updates of Maple were distributed by the vendors (currently Maple 13) and required a number of adaptations to the source code; (ii) the increasing size and program structure of the Racah module made it advisible to separate the (procedures for the treatment of the) atomic shell model from the manipulation and computation of Racah expressions. Therefore, the computation of angular coefficients for different coupling schemes, (grand) coefficients of fractional parentage as well as the matrix elements (of various irreducible tensors from the shell model) is to be maintained from now on independently within the Jucys module; (iii) a number of bugs and inconsistencies have been reported to us and corrected in the present version.Summary of revisions: In more detail, the following changes have been made:

1.

Since recent versions of Maple now support the automatic type checking of all incoming arguments and the definition of user-defined types; we have adapted most of the code to take advantage of these features, and especially those commands that are accessible by the user.

2.

In the computation of the Wigner n-j symbols and Clebsch-Gordan coefficients, we now return a ‘0’ in all cases in which the triangular rules are not fulfilled, for example, if δ(a,b,c)=0 for or . This change in the program saves the user making these tests on the quantum numbers explicitly everytime (in the summation over more complex expressions) that such a symbol or coefficient is invoked. The program still terminates with an error message if the (half-integer and integer) angular momentum quantum numbers appear in an inproper combination.

3.

While a recursive generation of the Wigner 3-j and 6-j symbols [6] may reduce the costs of some computations (and has thus been utilized in the past), it also makes the program rather sophisticated, especially if an algebraic evaluation or computations with a high number of Digits need to be supported by the same generic commands. The following procedures are therefore no longer supported by the Racah module:Racah_compute_w3j_jrange(), Racah_compute_w3j_mrange(),Racah_compute_w3j_recursive(), Racah_compute_w6j_range(), andRacah_compute_w6j_recursive().On most PCs, a sequential computation of all requested symbols is carried out within the same time basically.

4.

Because the module Jucys has grown to a size of about 35,000 lines of code and data, it appears helpful and necessary to maintain it independently. The procedures from the Jucys modules were designed to facilitate the computation of matrix elements of the unit tensors, the coefficients of fractional parentage (of various types) as well as transformation matrices between different coupling schemes [7] and are, thus, independent of the Racah module (although they typically require that the Racah code is available). The Jucys module is no longer distributed together with the present code.

5.

Apart from the Wigner n-j symbols (see above), some minor bugs have been reported and corrected in Racah_expand() and Racah_set().

6.

To facilitate the test of the installation and as a first tutorial on the module, we now provide the Maple worksheet Racah-tests-2009-maple12.mw in the Racah2009 root directory. This worksheet contains the examples and test cases from the previous versions. For the test of the installation, it is recommended that a ‘copy’ of this worksheet is saved and compared to the results from the re-run. It can be used also as a helpful source to define new examples in interactive work with the Racah module.

The Racah module is distributed in a tar file ADFV_v10_0.tar.gz from which the RACAH2009 root directory is (re-)generated by the command tar -zxvf ADFV_v10_0.tar.gz. This directory contains the source code libraries (tested for Maple 11 and 12), a Read.me for the installation of the program, the worksheet Racah-tests-2009-maple12.mw as well as the document Racah-commands-2009.pdf. This .pdf document serves as a Short Reference Manual and provides the definition of all the data structures of the Racah program together with an alphabetic list of all user relevant (and exported) commands. Although emphasis was placed on preserving the compatibility of the program with earlier releases of Maple, this cannot always be guaranteed due to changes in the Maple syntax. The Racah2009 root also contains an example of a .mapleinit file that can be modified and incorporated into the user's home directory to make the Racah module accessible like any other module of Maple. As mentioned above, the worksheet Racah-tests-2009-maple12.mw, help test the installation and may serve as a first tutorial.Restrictions: The (Racah) program is based on the concept of Racah expressions [cf. Fig. 1 in Ref. [4]] which, in principle, may contain any number of Wigner n-j symbols (n?9), Clebsch-Gordan coefficients, spherical harmonics and/or rotation matrices. In practise, of course, the required time and the success of an evaluation procedure depends on the complexity of the expressions and on the storage available, sometimes also on Maple's internal garbage treatment. In some cases, it is advisable to attempt first a simplification of the magnetic quantum numbers for a given expression before the summation over further 6-j and 9-j symbols should be taken into account. For all other quantities (that are compiled in Ref. [8], Tables 1 and 2, and explained in more detail in the Short Reference Manual, Racah-commands-2009.pdf), we currently just facilitate fast numerical computations by exploiting, as far as possible, Maple's hardware floating-point model. The program also supports simplifications on the Wigner rotation matrices. In integrals over the rotation matrices, products of up to three Wigner D-functions or reduced matrices (with the same angular arguments) are recognized; for the integration over a solid angle, however, the domain of integration must be specified explicitly for the Euler angles α and γ in order to force Maple to generate a constant of integration. In the course of the evaluation of Racah expressions, it is, in practice, often difficult to check internally whether all substructures of an expression are defined properly. Therefore, the user must ensure that all angular momenta (if given explicitly) must finally evaluate to integer and half-integer values and that they satisfy proper coupling conditions.Unusual features: The Racah program is designed for interactive use and for providing a quick and algebraic evaluation of (complex) expressions from Racah's algebra. In the evaluation, it exploits a large set of sum rules which are known from Racah's algebra and which may include (multiple) summations over dummy indices; see Varshalovich et al. [5] for a more detailed account of the theory. One strength of the program is that it recognizes automatically the symmetries of the symbols and functions, and that it applies also (some of) the graphical rules due to Yutsis and coworkers [9]. As before, the result of the evaluation process will be provided as Racah expressions, if a further simplification could be achieved, and may hence be used for further derivations and calculations within the given framework. In dealing with recoupling coefficients, these coefficients can be entered simply as a string of angular momenta (variables), separated by commas, and very similar to how they appear in mathematical texts. This is a crucial advantage of the program, compared with previous developments, for which the angular momenta and coupling coefficients had often to be given in a very detailed format. A Short Reference Manual to all procedures of the Racah program is provided by this distribution; it also contains the worksheet Racah-tests-2009-maple12.mw that contains the examples from all previous versions and may help test the installation. This worksheet can serve as a first tutorial to the Racah procedures. In the past, the Racah program has been utilized extensively in a number of applications including angular and polarization studies of heavy ions [10], angular distributions and correlation functions following photon-induced excitation processes [11], entanglement studies [12], in application of point-group symmetries and several others.Running time: The worksheet supplied with the distribution takes about 1 minute to run.References:

[1] Maple is a registered trademark of Waterloo Maple Inc.

[2] S. Fritzsche, Comp. Phys. Commun. 103 (1997) 51.

[3] S. Fritzsche, S. Varga, D. Geschke, B. Fricke, Comp. Phys. Commun. 111 (1998) 167;

T. Ingho, S. Fritzsche, B. Fricke, Comp. Phys. Commun. 139 (2001) 297;

S. Fritzsche, T. Ingho, T. Bastug, M. Tomaselli, Comp. Phys. Commun. 139 (2001) 314.

[4] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore a.o., 1988.

[5] J. Pagaran, S. Fritzsche, G. Gaigalas, Comp. Phys. Commun. 174 (2006) 616.

[6] K. Schulten, R.G. Gordon, Comp. Phys. Commun. 11 (1976) 269.

[7] G. Gaigalas, S. Fritzsche, B. Fricke, Comp. Phys. Commun. 135 (2001) 219;

G. Gaigalas, S. Fritzsche, Comp. Phys. Commun. 149 (2002) 39;

G. Gaigalas, O. Scharf, S. Fritzsche, Comp. Phys. Commun. 166 (2005) 141.

[8] S. Fritzsche, T. Ingho, M. Tomaselli, Comp. Phys. Commun. 153 (2003) 424.

[9] A.P. Yutsis, I.B. Levinson, V.V. Vanagas, The Theory of Angular Momentum, Israel Program for Scientific Translation, Jerusalem, 1962.

[10] S. Fritzsche, P. Indelicato, T. Stöhlker, J. Phys. B 38 (2005) S707.

[11] M. Kitajima, M. Okamoto, M. Hoshino, et al., J. Phys. B 35 (2002) 3327;

N.M. Kabachnik, S. Fritzsche, A.N. Grum-Grzhimailo, et al., Phys. Reports 451 (2007) 155;

S. Fritzsche, A.N. Grum-Grzhimailo, E.V. Gryzlova, N.M. Kabachnik, J. Phys. B 41 (2008) 165601;

T. Radtke, et al., Phys. Rev. A 77 (2008) 022507.

[12] T. Radtke, S. Fritzsche, Comp. Phys. Commun. 175 (2006) 145.

     

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).     

RAEEM: A Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations

In Maple 8, by taking advantage of the package RIF contained in DEtools, we developed a package RAEEM which is a comprehensive and complete implementation of such methods as the tanh-method, the extended tanh-method, the Jacobi elliptic function method and the elliptic equation method. RAEEM can entirely automatically output a series of exact traveling wave solutions, including those of polynomial, exponential, triangular, hyperbolic, rational, Jacobi elliptic, Weierstrass elliptic type. The effectiveness of the package is illustrated by applying it to a large variety of equations. In addition to recovering previously known solutions, we also obtain more general forms of some solutions and new solutions.

Program summary

Title of program: RAEEMCatalogue identifier: ADUPProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUPProgram obtained from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: PC Pentium IVInstallations: CopyOperating systems: Windows 98/2000/XPProgram language used: Maple 8Memory required to execute with typical data: depends on the problem, minimum about 8M wordsNo. of bits in a word: 8No. of lines in distributed program, including test data, etc.: 3163No. of bytes in distributed program, including the test data, etc.: 26 720Distribution format: tar.gzNature of physical problem: Our program provides exact traveling wave solutions, which describe various phenomena in nature, and thus can give more insight into the physical aspects of problems. These solutions may be easily used in further applications.Restriction on the complexity of the problem: The program can handle system of nonlinear evolution equations with any number of independent and dependent variables, in which each equation is a polynomial (or can be converted to a polynomial) in the dependent variables and their derivatives.Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the equations we have computed, the running time is no more than 100 s.     

A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations

The tanh function expansion method for finding traveling solitary wave solutions to coupled nonlinear evolution equations is described. A complete implementation RATHS written in Maple is presented, in which the operator mains can output exact solitary wave solutions entirely automatically. Furthermore, RATHS can handle any number of dependent variables u_i as well as any number of independent variables x_j contained in the input system. This package can also be applied to ODEs. The effectiveness of RATHS is illustrated by applying it to a variety of equations.

Program summary

Title of program: RATHSCatalogue identifier: ADSD (also ADQK)Program Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADRY (also ADQR)Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputers: PC Pentium IVInstallations: CopyOperating systems: Windows 98/2000/XPProgram language used: Maple V R6Memory required to execute with typical data: depends on the problem, minimum about 8M wordsNo. of bits in a word: 8No. of bytes in distributed program, including the test data, etc.: 16 608Distribution format:tar gzip fileKeywords: Coupled nonlinear evolution equations, traveling solitary wave solutions, dependent variable, independent variableNature of physical problem: Our program give out exact solitary wave solutions, which can describe various phenomena in nature, and thus can give more insight into the physical aspects of problems and may be easily used in further applications.Restriction on the complexity of the problem: The program can handle coupled nonlinear evolution equations, in which every equation is a polynomial (or can be converted to a polynomial) in the unknown functions and their derivatives.Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the coupled equations which we have computed, the running time is no more than 20 seconds.     

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)     

Package for calculations and simplifications of expressions with Dirac matrices (MatrixExp)

This paper describes a package for calculations of expressions with Dirac matrices. Advantages over existing similar packages are described. MatrixExp package is intended for simplification of complex expressions involving γ-matrices, providing such tools as automatic Feynman parameterization, integration in d-dimensional space, sorting and grouping of results in a given order. Also, in comparison with the existing similar package Tracer, the presented package MatrixExp has more enhanced input possibility. User-available functions of MatrixExp package are described in detail. Also an example of calculation of Feynman diagram for process b→sγg with application of functions of MatrixExp package is presented.

Program summary

Title of program:MatrixExpCatalogue identifier:ADWBProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWBProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneProgramming language:MATHEMATICAComputer:PC PentiumOperating system:WindowsNo. of lines in distributed program, including test data, etc.: 1551No. of bytes in distributed program, including test data, etc.: 16 040Distribution format:tar.gzRAM:loading the package uses approx. 3 500 000 bytes of RAM. However memory required for calculations depends heavily on the expressions in the view, as the package uses recursive functions, and MATHEMATICA dynamically allocates memory. Package has been tested to work on PC Pentium II 233 MHz with 128 Mb of memory calculating typical diagrams of contemporary calculations.Nature of problem:Feynman diagram calculation, simplification of expressions with γ-matricesSolution method:Analytic transformations, dimensional regularization, Feynman parameterizationRestrictions:MatrixExp package works only with single line of expressions (G[l1,]), in contrast to the Tracer package that works with multiple lines, i.e., the following is possible in Tracer, but not in MatrixExp: G[l1,]**G[l2,]**G[l3,], which will return the result of G[l1,]**G[l1,]**G[l1,]….Unusual features:noneRunning time:Seconds for expressions with several different γ-matrices on Pentium IV 1.8 GHz and of the order of a minute on Pentium II 233 MHz. Calculation times rise with the number of matrices.     

Comment on “A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations” by Engui Fan and H.H. Dai: [Comput. Phys. Commun. 153 (2003) 17]

Fan and Dai [Comput. Phys. Commun. 153 (2003) 17] have found a series of traveling wave solutions for nonlinear equations by applying a direct approach with computerized symbolic computations. They have claimed that the proposed method, in comparison with most existing symbolic computation methods such as a tanh method and Jacobi function method, not only give new and more general solutions, but also provides a guideline to classify various types of the solution according to some parameters. We show that the claims by Fan and Dai are wrong since some of the solutions do not satisfy the differential equation that they have adopted for the algebraic method.     

A REDUCE package for finding conserved densities of systems of nonlinear difference-difference equations

A computer algebra program for finding polynomial conserved densities of nonlinear difference-difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference-difference equations.     

A REDUCE package for finding conserved densities of systems of implicit difference-difference equations

A computer algebra program for finding polynomial conserved densities of implicit difference-difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference-difference equations of polynomial type.

Program summary

Title of program: TXCDCatalogue identifier: ADTSProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTSProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: PC/AT compatible machineOperating systems: Windows 2000Programming language used: REDUCE 3.6, RLISPMemory required to execute with typical data: Depends on the problem, minimum about 2 M bytes.No. of bits in a word: 32No. of bytes in distributed program, including test data, etc.: 10 005No. of lines in distributed program, including test data, etc.: 1739Distribution format: tar gzip fileNature of physical problem: The existence of conserved densities for difference-difference equations is of interest for their classification and for understanding the stability of their solutions.Restriction on the complexity of the problem: The program can handle difference-difference equations which can be transformed to polynomial ones, and determine the homogeneous conservation laws.Typical running time: It depends on the equation and the rank of the conserved density. It increases exponentially with the rank of the conserved density. The running times on the PC Pentium with operating systems Windows 2000 (Xeon, 1.7 GHz) are shown in the table below. Timings are given in milliseconds.

Performance on Windows
Example	Rank
	0	1	2	3	4	5	6	7	8	9
1(i)	15	15	15	31	150	718	5483	28176	127914	493092
1(ii)	15	15	16	63	170	2264	11171	103938	299001	1386468
1(iii)	15	15	15	46	250	5686	29210	203190	924372
1(iv)	15	15	15	31	47	156	1031	8905	40485	194595
2	15	15	45	187	2358	36673	433794
3(i)	15	63	1780	66518	1390030	∗∗
3(ii)	15	47	829	37640	786594	∗∗
The cases ∗∗ were rejected by memory error.

Full-size table

     

Programs for the approximation of real and imaginary single- and multi-valued functions by means of Hermite-Padé-approximants

We present programs for the calculation and evaluation of special type Hermite-Padé-approximations. They allow the user to either numerically approximate multi-valued functions represented by a formal series expansion or to compute explicit approximants for them. The approximation scheme is based on Hermite-Padé polynomials and includes both Padé and algebraic approximants as limiting cases. The algorithm for the computation of the Hermite-Padé polynomials is based on a set of recursive equations which were derived from a generalization of continued fractions. The approximations retain their validity even on the cuts of the complex Riemann surface which allows for example the calculation of resonances in quantum mechanical problems. The programs also allow for the construction of multi-series approximations which can be more powerful than most summation methods.

Program summary

Title of program: hp.srCatalogue identifier: ADSOProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSOProgram obtainable from: CPC Program Library, Queen's University Belfast, Northern IrelandLicensing provisions: Persons requesting the program must sign the standard CPC non-profit use licenseComputer: Sun Ultra 10Installation: Computing Center, University of Regensburg, GermanyOperating System: Sun Solaris 7.0Program language used: MapleV.5Distribution format: tar gzip fileMemory required to execute with typical data: 32 MB; the program itself needs only about 20 kBNumber of bits in a word: 32No. of processors used: 1Has the code been vectorized?: noNo. of bytes in distributed program, including test data etc.: 38194No. of lines in distributed program, including test data, etc.: 4258Nature of physical problem: Many physical and chemical quantum systems lead to the problem of evaluating a function for which only a limited series expansion is known. These functions can be numerically approximated by summation methods even if the corresponding series is only asymptotic. With the help of Hermite-Padé-approximants many different approximation schemes can be realized. Padé and algebraic approximants are just well-known examples. Hermite-Padé-approximants combine the advantages of highly accurate numerical results with the additional advantage of being able to sum complex multi-valued functions.Method of solution: Special type Hermite-Padé polynomials are calculated for a set of divergent series. These polynomials are then used to implicitly define approximants for one of the functions of this set. This approximant can be numerically evaluated at any point of the Riemann surface of this function. For an approximation order not greater than 3 the approximants can alternatively be expressed in closed form and then be used to approximate the desired function on its complete Riemann surface.Restriction on the complexity of the problem: In principle, the algorithm is only limited by the available memory and speed of the underlying computer system. Furthermore the achievable accuracy of the approximation only depends on the number of known series coefficients of the function to be approximated assuming of course that these coefficients are known with enough accuracy.Typical running time: 10 minutes with parameters comparable to the testrunsUnusual features of the program: none     

Sixth-order symmetric and symplectic exponentially fitted modified Runge-Kutta methods of Gauss type

The construction of symmetric and symplectic exponentially fitted modified Runge-Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. In a previous paper [H. Van de Vyver, A fourth order symplectic exponentially fitted integrator, Comput. Phys. Comm. 176 (2006) 255-262] a two-stage fourth-order symplectic exponentially fitted modified RK method has been proposed. Here, two three-stage symmetric and symplectic exponentially fitted integrators of Gauss type, either with fixed nodes or variable nodes, are derived. The algebraic order of the new integrators is also analyzed, obtaining that they possess sixth-order as the classical three-stage RK Gauss method. Numerical experiments with some oscillatory problems are presented to show that the new methods are more efficient than other symplectic RK Gauss codes proposed in the scientific literature.     

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Through discussion and example, parallels are drawn to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.

Program summary

Title of program: DDESpecialSolutions.mCatalogue identifier:ADUJProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUJProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: Created using a PC, but can be run on UNIX and Apple machinesOperating systems under which the program has been tested: Windows 2000 and Windows XPProgramming language used: Mathematica, version 3.0 or higherMemory required to execute with typical data: 9 MBNumber of processors used: 1Has the code been vectorised or parallelized?: NoNumber of lines in distributed program, including test data, etc.: 3221Number of bytes in distributed program, including test data, etc.: 23 745Nature of physical problem: The program computes exact solutions to differential-difference equations in terms of the tanh function. Such solutions describe particle vibrations in lattices, currents in electrical networks, pulses in biological chains, etc.Method of solution: After the differential-difference equation is put in a traveling frame of reference, the coefficients of a candidate polynomial solution in tanh are solved for. The resulting traveling wave solutions are tested by substitution into the original differential-difference equation.Restrictions on the complexity of the program: The system of differential-difference equations must be polynomial. Solutions are polynomial in tanh.Typical running time: The average run time of 16 cases (including the Toda, Volterra, and Ablowitz-Ladik lattices) is 0.228 seconds with a standard deviation of 0.165 seconds on a 2.4 GHz Pentium 4 with 512 MB RAM running Mathematica 4.1. The running time may vary considerably, depending on the complexity of the problem.     

Global optimization and finite temperature simulations of atomic clusters: Use of XenArm clusters as test systems

The large potential energy barriers separating local minima on the potential energy surface of cluster systems pose serious problems for optimization and simulation methods. This article discusses algorithms for dealing with these problems. Lennard-Jones clusters are used to illustrate the important issues. In addition, the complexities in going from one-component to binary Lennard-Jones clusters are explored.     

?-SHAKE: An extension to SHAKE for the explicit treatment of angular constraints

This paper presents ?-SHAKE, an extension to SHAKE, an algorithm for the resolution of holonomic constraints in molecular dynamics simulations, which allows for the explicit treatment of angular constraints. We show that this treatment is more efficient than the use of fictitious bonds, significantly reducing the overlap between the individual constraints and thus accelerating convergence. The new algorithm is compared with SHAKE, M-SHAKE, the matrix-based approach described by Ciccotti and Ryckaert and P-SHAKE for rigid water and octane.