Ground state of the time-independent Gross-Pitaevskii equation

We present a suite of programs to determine the ground state of the time-independent Gross-Pitaevskii equation, used in the simulation of Bose-Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast convergence to the true ground state. Versions are given for the one-, two-, and three-dimensional equation, using either a spectral method, well suited for harmonic trapping potentials, or a spatial grid.

Program summary

Program title: GPODACatalogue identifier: ADZN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZN_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.: 5339No. of bytes in distributed program, including test data, etc.: 19 426Distribution format: tar.gzProgramming language: Fortran 90Computer: ANY (Compilers under which the program has been tested: Absoft Pro Fortran, The Portland Group Fortran 90/95 compiler, Intel Fortran Compiler)RAM: From <1 MB in 1D to ∼¹⁰² MB for a large 3D gridClassification: 2.7, 4.9External routines: LAPACK, BLAS, DFFTPACKNature of problem: The order parameter (or wave function) of a Bose-Einstein condensate (BEC) is obtained, in a mean field approximation, by the Gross-Pitaevskii equation (GPE) [F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71 (1999) 463]. The GPE is a nonlinear Schrödinger-like equation, including here a confining potential. The stationary state of a BEC is obtained by finding the ground state of the time-independent GPE, i.e., the order parameter that minimizes the energy. In addition to the standard three-dimensional GPE, tight traps can lead to effective two- or even one-dimensional BECs, so the 2D and 1D GPEs are also considered.Solution method: The ground state of the time-independent of the GPE is calculated using the Optimal Damping Algorithm [E. Cancès, C. Le Bris, Int. J. Quantum Chem. 79 (2000) 82]. Two sets of programs are given, using either a spectral representation of the order parameter [C.M. Dion, E. Cancès, Phys. Rev. E 67 (2003) 046706], suitable for a (quasi) harmonic trapping potential, or by discretizing the order parameter on a spatial grid.Running time: From seconds in 1D to a few hours for large 3D grids     

Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap

Here we develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional, two-dimensional circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, Fortran 90/95 versions provide some simplification over the Fortran 77 programs, and these programs are also included (six programs in all).

Program summary

Program title: (i) imagetime1d, (ii) imagetime2d, (iii) imagetime3d, (iv) imagetimecir, (v) imagetimesph, (vi) imagetimeaxial, (vii) realtime1d, (viii) realtime2d, (ix) realtime3d, (x) realtimecir, (xi) realtimesph, (xii) realtimeaxialCatalogue identifier: AEDU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDU_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.: 122 907No. of bytes in distributed program, including test data, etc.: 609 662Distribution format: tar.gzProgramming language: FORTRAN 77 and Fortran 90/95Computer: PCOperating system: Linux, UnixRAM: 1 GByte (i, iv, v), 2 GByte (ii, vi, vii, x, xi), 4 GByte (iii, viii, xii), 8 GByte (ix)Classification: 2.9, 4.3, 4.12Nature of problem: These programs are designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-, two- or three-space dimensions with a harmonic, circularly-symmetric, spherically-symmetric, axially-symmetric or anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Solution method: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation, in either imaginary or real time, over small time steps. The method yields the solution of stationary and/or non-stationary problems.Additional comments: This package consists of 12 programs, see “Program title”, above. FORTRAN77 versions are provided for each of the 12 and, in addition, Fortran 90/95 versions are included for ii, iii, vi, viii, ix, xii. For the particular purpose of each program please see the below.Running time: Minutes on a medium PC (i, iv, v, vii, x, xi), a few hours on a medium PC (ii, vi, viii, xii), days on a medium PC (iii, ix).

Program summary (1)

Title of program: imagtime1d.FTitle of electronic file: imagtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (2)

Title of program: imagtimecir.FTitle of electronic file: imagtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (3)

Title of program: imagtimesph.FTitle of electronic file: imagtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (4)

Title of program: realtime1d.FTitle of electronic file: realtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (5)

Title of program: realtimecir.FTitle of electronic file: realtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (6)

Title of program: realtimesph.FTitle of electronic file: realtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (7)

Title of programs: imagtimeaxial.F and imagtimeaxial.f90Title of electronic file: imagtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (8)

Title of program: imagtime2d.F and imagtime2d.f90Title of electronic file: imagtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (9)

Title of program: realtimeaxial.F and realtimeaxial.f90Title of electronic file: realtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (10)

Title of program: realtime2d.F and realtime2d.f90Title of electronic file: realtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (11)

Title of program: imagtime3d.F and imagtime3d.f90Title of electronic file: imagtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (12)

Title of program: realtime3d.F and realtime3d.f90Title of electronic file: realtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum Ram Memory: 8 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.     

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as

     

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.

     

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.References: [1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.     

A program for the simulation and analysis of open atmospheric balloon soundings

The equation of motion for a balloon in an atmosphere is generalized but placed in proper context by taking into account some fluid theory results and a few factors not considered in previous works. The design of a computer program becomes necessary to find solutions. A code that allows to perform 2D simulations of open balloons flights is developed. The coupled integrodifferential nature of the problem represented a significant challenge for a satisfactory implementation.     

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.     

A trigonometrically-fitted one-step method with multi-derivative for the numerical solution to the one-dimensional Schrödinger equation

In this paper we present a new multi-derivative or Obrechkoff one-step method for the numerical solution to an one-dimensional Schrödinger equation. By using trigonometrically-fitting method (TFM), we overcome the traditional Obrechkoff one-step method (or called as the non-TFM) for its poor-accuracy in the resonant state. In order to demonstrate the excellent performance for the resonant state, we consider only the simplest TFM, of which the local truncation error (LTE) is of O(h⁷), a little higher than the one of the traditional Numerov method of O(h⁶), and only the first- and second-order derivatives of the potential function are needed. In the new method, in order to solve two unknowns, wave function and its first-order derivative, we use a pair of two symmetrically linear-independent one-step difference equations. By applying it to the well-known Woods-Saxon's potential problem, we find that the TFM can surpass the non-TFM by five orders for the highest resonant state, and surpass Numerov method by eight orders. On the other hand, because of the small error constant, the accuracy improvement to the ground state is also remarkable, and the numerical result obtained by TFM can be four to five orders higher than the one by Numerov method.     

A domain decomposition molecular dynamics program for the simulation of flexible molecules of spherically-symmetrical and nonspherical sites. II. Extension to NVT and NPT ensembles

We describe the algorithms for NVT and NPT-ensemble simulations developed within the parallel molecular dynamics program GBMOLDD. This program uses the domain decomposition algorithm and is targeted at large-scale simulations of molecular systems (particularly polymers and liquid crystals) composed of both spherically-symmetric and nonspherical sites. The nonspherical sites can be described either by a Gay-Berne potential or by soft repulsive spherocylinders. The molecules can be of arbitrary topology and the intramolecular forces are described via standard force fields. We tested the stability of both leap-frog and velocity-Verlet integrators on two “real-life” systems—a nematic liquid crystal phase of 1944 one-site Gay-Berne molecules and on 512 flexible liquid-crystalline dimers. In both cases the algorithm demonstrates good stability over the typical simulation times required for new phase formation and/or molecular relaxation processes.     

A variational approach to multi-phase motion of gas, liquid and solid based on the level set method

We propose a simple and robust numerical algorithm to deal with multi-phase motion of gas, liquid and solid based on the level set method [S. Osher, J.A. Sethian, Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulation, J. Comput. Phys. 79 (1988) 12; M. Sussman, P. Smereka, S. Osher, A level set approach for capturing solution to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146; J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999; S. Osher, R. Fedkiw, Level Set Methods and Dynamics Implicit Surface, Applied Mathematical Sciences, vol. 153, Springer, 2003]. In Eulerian framework, to simulate interaction between a moving solid object and an interfacial flow, we need to define at least two functions (level set functions) to distinguish three materials. In such simulations, in general two functions overlap and/or disagree due to numerical errors such as numerical diffusion. In this paper, we resolved the problem using the idea of the active contour model [M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision 1 (1988) 321; V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, International Journal of Computer Vision 22 (1997) 61; G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001; R. Kimmel, Numerical Geometry of Images: Theory, Algorithms, and Applications, Springer-Verlag, 2003] introduced in the field of image processing.     

A GPGPU based program to solve the TDSE in intense laser fields through the finite difference approach

We present a General-purpose computing on graphics processing units (GPGPU) based computational program and framework for the electronic dynamics of atomic systems under intense laser fields. We present our results using the case of hydrogen, however the code is trivially extensible to tackle problems within the single-active electron (SAE) approximation. Building on our previous work, we introduce the first available GPGPU based implementation of the Taylor, Runge–Kutta and Lanczos based methods created with strong field ab-initio simulations specifically in mind; CLTDSE. The code makes use of finite difference methods and the OpenCL framework for GPU acceleration. The specific example system used is the classic test system; Hydrogen. After introducing the standard theory, and specific quantities which are calculated, the code, including installation and usage, is discussed in-depth. This is followed by some examples and a short benchmark between an 8 hardware thread (i.e. logical core) Intel Xeon CPU and an AMD 6970 GPU, where the parallel algorithm runs 10 times faster on the GPU than the CPU.     

SuSpect: A Fortran code for the Supersymmetric and Higgs particle spectrum in the MSSM

We present the Fortran code SuSpect version 2.3, which calculates the Supersymmetric and Higgs particle spectrum in the Minimal Supersymmetric Standard Model (MSSM). The calculation can be performed in constrained models with universal boundary conditions at high scales such as the gravity (mSUGRA), anomaly (AMSB) or gauge (GMSB) mediated supersymmetry breaking models, but also in the non-universal MSSM case with R-parity and CP conservation. Care has been taken to treat important features such as the renormalization group evolution of parameters between low and high energy scales, the consistent implementation of radiative electroweak symmetry breaking and the calculation of the physical masses of the Higgs bosons and supersymmetric particles taking into account the dominant radiative corrections. Some checks of important theoretical and experimental features, such as the absence of non-desired minima, large fine-tuning in the electroweak symmetry breaking condition, as well as agreement with precision measurements can be performed. The program is simple to use, self-contained and can easily be linked to other codes; it is rather fast and flexible, thus allowing scans of the parameter space with several possible options and choices for model assumptions and approximations.

Program summary

Title of program:SuSpectCatalogue identifier:ADYR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYR_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneProgramming language used:FORTRAN 77Computer:Unix machines, PCNo. of lines in distributed program, including test data, etc.:21 821No. of bytes in distributed program, including test data, etc.:249 657Distribution format:tar.gzOperating system:Unix (or Linux)RAM:approximately 2500 KbytesNumber of processors used:1 processorNature of problem:SuSpect calculates the supersymmetric and Higgs particle spectrum (masses and some other relevant parameters) in the unconstrained Minimal Supersymmetric Standard Model (MSSM), as well as in constrained models (cMSSMs) such as the minimal Supergravity (mSUGRA), the gauge mediated (GMSB) and anomaly mediated (AMSB) Supersymmetry breaking scenarii. The following features and ingredients are included: renormalization group evolution between low and high energy scales, consistent implementation of radiative electroweak symmetry breaking, calculation of the physical particle masses with radiative corrections at the one- and two-loop level.Solution method:The main methods used in the code are: (1) an (adaptative fourth-order) Runge-Kutta type algorithm (following a standard algorithm described in “Numerical Recipes”), used to solve numerically a set of coupled differential equations resulting from the renormalization group equations at the two-loop level of the perturbative expansions; (2) diagonalizations of mass matrices; (3) some mathematical (Spence, etc) functions resulting from the evaluation of one and two-loop integrals using the Feynman graphs techniques for radiative corrections to the particle masses; (4) finally, some fixed-point iterative algorithms to solve non-linear equations for some of the relevant output parameters.Restrictions:(1) The code is limited at the moment to real input parameters. (2) It also does not include flavor non-diagonal terms which are possible in the most general soft supersymmetry breaking Lagrangian. (3) There are some (mild) limitations on the possible range of values of input parameter, i.e. not any arbitrary values of some input parameters are allowed: these limitations are essentially based on physical rather than algorithmic issues, and warning flags and other protections are installed to avoid as much as possible execution failure if unappropriate input values are used.Running time:between 1 and 3 seconds depending on options, with a 1 GHz processor.     

A program of generation and selection of configurations for the configuration interaction method in atomic calculations SELECTCONF

This program written in FORTRAN is aimed at generation and selection of the admixed configurations which are used in the theoretical calculations of atomic states by the configuration interaction (CI) method. The admixed configurations are generated and selected using the file of radial orbitals written down in the form adopted in the code [C. Froese Fischer, Comput. Phys. Comm. 43 (1987) 355] and other analogous codes. Selection of configurations is performed on the ground of evaluations in the second order of the perturbation theory [P. Bogdanovich, R. Karpuškien?, Comput. Phys. Comm. 134 (2001) 321; R. Karpuškien?, R. Karazija, P. Bogdanovich, Phys. Scripta 64 (2001) 333]. Output of selected configurations is arranged in a format suitable for the codes generating the configuration states [C. Froese Fischer, B. Liu, Comput. Phys. Comm. 64 (1991) 406; P. Bogdanovich, A. Momkauskait?, Comput. Phys. Comm. 157 (2004) 217].

Program summary

Title of program:SELECTCONFCatalogue identifier:ADWDProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWDProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:NoneComputers:Any computer with a FORTRAN 77 compilerOperating systems under which the program has been tested:LinuxProgramming language used:FORTRAN 77Memory required to execute with typical data:4 MBNo. of lines in distributed program, including test data, etc.:7459No. of bytes in distributed program, including test data, etc.:108 420Distribution format:gzip fileNature of the physical problem:Due to the restricted possibilities of the computers and codes, which are employed, the practice of CI requires one to select and superpose those configurations the usage of which happens to be the most effective. This program is designed for the selection of such admixed configurations.Method of solution:All admixed configurations possible in the specified basis set of radial orbitals (RO) are constructed using the one-electron and two-electron virtual excitations. Then the averaged evaluation of their influence on the energy or wave function of the adjusted configuration is performed in the second order of perturbation theory. The results of this evaluation are used for the selection of admixed configurations.Restrictions onto the complexity of the problem:In the present version of the program the number of passive shells is restricted by MIUZ=20; the number of active shells by MIAT=10; the number of generated admixed configurations, by MECO=10000; the number of RO used, by MOR=MRO=99. All these limitations are not hard-coded and can be changed by substituting the values of the corresponding parameters.Unusual features of the program:The possibility of carrying out the averaged evaluation of the influence of admixed configurations in the second order of perturbation theory and to perform their selection on this ground.Typical running time:Several seconds. This time depends on the size of the problem: the computation time depends approximately linearly on the number of possible admixed configurations, which increases rapidly with a growing number of active shells and an extending RO basis set.     

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem

A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [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; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere.

Program summary

Program title: ODPEVPCatalogue identifier: AEDV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3001No. of bytes in distributed program, including test data, etc.: 24 195Distribution 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: depends on

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Test run requires 4 MBClassification: 2.1, 2.4External routines: GAULEG [3]Nature of problem: The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate ) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2].Solution method: The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF=EBF with respect to a pair of unknown (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 [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a three-dimensional sphere [20].Restrictions: The computer memory requirements depend on:

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see sections below and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z) of Eq. (1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f₁(z) and f₂(z) of Eq. (1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions.Running time: The running time depends critically upon:

1.

the number and order of finite elements;

2.

the number of points on interval [zmin,zmax]; and

3.

the number of eigenfunctions required.

The test run which accompanies this paper took 2 s with 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. Comm. 177 (2007) 649-675

[2]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Comm. 179 (2008) 685-693.

[3]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[4]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov, V.V. Serov, Phys. Rev. A 77 (2008) 034702-1-4.

[5]

E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430.

[6]

Yu.N. Demkov, J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365.

[7]

P.M. Krassovitskiy, N.Zh. Takibaev, Bull. Russian Acad. Sci. Phys. 70 (2006) 815-818.

[8]

V.S. Melezhik, J.I. Kim, P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15.

[9]

F.M. Pen'kov, Phys. Rev. A 62 (2000) 044701-1-4.

[10]

M. Born, X. Huang, Dynamical Theory of Crystal Lattices, The Clarendon Press, Oxford, England, 1954.

[11]

L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964.

[12]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127;

A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640.

[13]

C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320.

[14]

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.

[15]

A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64.

[16]

K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982.

[17]

O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269.

[18]

Yu.A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361.

[19]

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. Comm. 178 (2008) 301-330.

[20]

A.G. Abrashkevich, M.S. Kaschiev, S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.

     

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains 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 energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials.

Program summary

Program title: KANTBPCatalogue identifier: ADZH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_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.: 4224No. of bytes in distributed program, including test data, etc.: 31 232Distribution 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: depends on (a) the number of differential equations; (b) the number and order of finite-elements; (c) the number of hyperradial points; and (d) the number of eigensolutions required. Test run requires 30 MBClassification: 2.1, 2.4External routines: GAULEG and GAUSSJ [W.H. Press, B.F. Flanery, S.A. Teukolsky, W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986]Nature of problem: In the hyperspherical adiabatic approach [J. Macek, J. Phys. B 1 (1968) 831-843; U. Fano, Rep. Progr. Phys. 46 (1983) 97-165; C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142], a multi-dimensional Schrödinger equation for a two-electron system [A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Comm. 90 (1995) 311-339] or a hydrogen atom in magnetic field [M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352] is reduced by separating the radial coordinate ρ from the angular variables to a system of second-order ordinary differential equations which contain 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 for such systems of coupled differential equations.Solution method: The boundary problems for coupled differential equations are solved by the finite-element method using high-order accuracy approximations [A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64]. 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 [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. 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 [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials described in [Yu. A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361; O. Chuluunbaatar, A.A. Gusev, S.Y. Larsen, S.I. Vinitsky, J. Phys. A 35 (2002) L513-L525; N.P. Mehta, J.R. Shepard, Phys. Rev. A 72 (2005) 032728-1-11; O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269]. For this benchmark model the needed analytical expressions for the potential matrix elements and first-derivative coupling terms, their asymptotics and 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:

•

(a) the number of differential equations;

•

(b) the number and order of finite-elements;

•

(c) the total number of hyperradial points; and

•

(d) the number of eigensolutions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSC (when solving the scattering problem) that evaluate the asymptotics of the radial wave functions at the right boundary point in case of a boundary condition of the third type, respectively.Running time: The running time depends critically upon:

•

(a) the number of differential equations;

•

(b) the number and order of finite-elements;

•

(c) the total number of hyperradial points on interval [0,ρmax]; and

•

(d) the number of eigensolutions required.

The test run which accompanies this paper took 28.48 s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.     

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.

Program summary

Program title:POTHMFCatalogue identifier:AEAA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAA_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.:8123No. of bytes in distributed program, including test data, etc.:131 396Distribution 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:Depends on

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the number of radial points.

Test run requires 4 MBClassification:2.5External routines:POTHMF uses some Lapack routines, copies of which are included in the distribution (see README file for details).Nature of problem:In the multi-channel adiabatic approach the Schrödinger equation for a hydrogen-like atom in a homogeneous magnetic field of strength γ (γ=B/B₀, B₀≅2.35×¹⁰⁵ T is a dimensionless parameter which determines the field strength B) is reduced by separating the radial coordinate, r, from the angular variables, (θ,φ), and using a basis of the angular oblate spheroidal functions [3] to a system of second-order ordinary differential equations which contain first-derivative coupling terms [4]. The purpose of this program is to calculate potential curves and matrix elements of radial coupling needed for calculating the low-lying bound and scattering states of hydrogen-like atoms in a homogeneous magnetic field of strength 0<γ?1000 within the adiabatic approach [5]. The program evaluates also asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem needed to extract from the R-matrix a required symmetric shortrange open-channel reaction matrix K [6] independent from matching point [7]. In addition, the program computes the dipole transition matrix elements in the length form between the basis functions that are needed for calculating the dipole transitions between the low-lying bound and scattering states and photoionization cross sections [8].Solution method:The angular oblate spheroidal eigenvalue problem depending on the radial variable is solved using a series expansion in the Legendre polynomials [3]. The resulting tridiagonal symmetric algebraic eigenvalue problem for the evaluation of selected eigenvalues, i.e. the potential curves, is solved by the LDL^T factorization using the DSTEVR program [2]. Derivatives of the eigenfunctions with respect to the radial variable which are contained in matrix elements of the coupled radial equations are obtained by solving the inhomogeneous algebraic equations. The corresponding algebraic problem is solved by using the LDL^T factorization with the help of the DPTTRS program [2]. Asymptotics of the matrix elements at large values of radial variable are computed using a series expansion in the associated Laguerre polynomials [9]. The corresponding matching points between the numeric and asymptotic solutions are found automatically. These asymptotics are used for the evaluation of the asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem [7]. As a test desk, the program is applied to the calculation of the energy values of the ground and excited bound states and reaction matrix of multi-channel scattering problem for a hydrogen atom in a homogeneous magnetic field using the KANTBP program [10].Restrictions:The computer memory requirements depend on:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points.

Restrictions due to dimension sizes can be changed by resetting a small number of PARAMETER statements before recompiling (see Introduction and listing for details).Running time:The running time depends critically upon:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points on interval [rmin,rmax].

The test run which accompanies this paper took 7 s required for calculating of potential curves, radial matrix elements, and dipole transition matrix elements on a finite-element grid on interval [rmin=0, rmax=100] used for solving discrete and continuous spectrum problems and obtaining asymptotic regular and irregular matrix radial solutions at rmax=100 for continuous spectrum problem on the Intel Pentium IV 2.4 GHz. The number of radial differential equations was equal to 6. The accompanying test run using the KANTBP program took 2 s for solving discrete and continuous spectrum problems using the above calculated potential curves, matrix elements and asymptotic regular and irregular matrix radial solutions. Note, that in the accompanied benchmark calculations of the photoionization cross-sections from the bound states of a hydrogen atom in a homogeneous magnetic field to continuum we have used interval [rmin=0, rmax=1000] for continuous spectrum problem. The total number of radial differential equations was varied from 10 to 18.References:

[1]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[2]

http://www.netlib.org/lapack/.

[3]

M. Abramovits, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

[4]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640; C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (Eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320; U. Fano, A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986.

[5]

M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352; O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, V.V. Serov, T.V. Tupikova, S.I. Vinitsky, Proc. SPIE 6537 (2007) 653706-1-18.

[6]

M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167-257.

[7]

M. Gailitis, J. Phys. B 9 (1976) 843-854; J. Macek, Phys. Rev. A 30 (1984) 1277-1278; S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova, O. Chuluunbaatar, Programming and Computer Software 33 (2007) 105-116.

[8]

H. Friedrich, Theoretical Atomic Physics, Springer, New York, 1991.

[9]

R.J. Damburg, R.Kh. Propin, J. Phys. B 1 (1968) 681-691; J.D. Power, Phil. Trans. Roy. Soc. London A 274 (1973) 663-702.

[10]

O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675.