Elsevier Science Home
Computer Physics Communications Program Library
Full text online from Science Direct
Programs in Physics & Physical Chemistry
CPC Home

[Licence| Download | New Version Template] adzh_v2_0.tar.gz(144 Kbytes)
Manuscript Title: KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach.
Authors: O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich
Program title: KANTBP
Catalogue identifier: ADZH_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 179(2008)685
Programming language: FORTRAN 77.
Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV.
Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP.
RAM: This depends on
  1. the number of differential equations;
  2. the number and order of finite elements;
  3. the number of hyperradial points; and
  4. the number of eigensolutions required
The test run requires 2 MB
Keywords: eigenvalue and multi-channel scattering problems, Kantorovich method, finite element method, R-matrix calculations, hyperspherical coordinates, multi-channel adiabatic approximation, ordinary differential equations, high-order accuracy approximations.
PACS: 02.30.Hq, 02.60.Jh, 02.60.Lj, 03.65.Nk, 31.15.Ja, 31.15.Pf, 34.50.-s, 34.80.Bm.
Classification: 2.1, 2.4.

External routines: GAULEG and GAUSSJ [2]

Nature of problem:
In the hyperspherical adiabatic approach [3 5], a multidimensional Schrödinger equation for a two-electron system [6] or a hydrogen atom in magnetic field [7 9] is reduced by separating radial coordinate ρ from the angular variables to a system of the second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions of the continuum spectrum for such systems of coupled differential equations on finite intervals of the radial variable ρ ∈ [ρmin, ρmax]. This approach can be used in the calculations of effects of electron screening on low-energy fusion cross sections [10 12].

Solution method:
The boundary problems for the coupled second-order differential equations are solved by the finite element method using high-order accuracy approximations [13]. The generalized algebraic eigenvalue problem AF = EBF with respect to pair unknowns (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [14]. The generalized algebraic eigenvalue problem (A -EB)F = λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDLT factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [14]. As a test desk, the program is applied to the calculation of the reaction matrix and corresponding radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field described in [9] on finite intervals of the radial variable ρ ∈ [ρmin, ρmax]. For this benchmark model the required analytical expressions for asymptotics of the potential matrix elements and first-derivative coupling terms, and also asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system.

Restrictions:
The computer memory requirements depend on:
  1. the number of differential equations;
  2. the number and order of finite elements;
  3. the total number of hyperradial points; and
  4. the number of eigensolutions required.
Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write Up and listing of [1] for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMS0 and ASYMSC (when solving the scattering problem) which evaluate asymptotics of the radial wave functions at left and right boundary points in case of a boundary condition of the third type for the above problems.

Running time:
The running time depends critically upon:
  1. the number of differential equations;
  2. the number and order of finite elements;
  3. the total number of hyperradial points on interval [ρmin, ρmax]; and
  4. the number of eigensolutions required.
The test run which accompanies this paper took 2s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.

References:
[1] O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen and S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649 675; http://cpc.cs.qub.ac.uk/summaries/ADZH v1 0.html.
[2] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
[3] J. Macek, J. Phys. B 1 (1968) 831 843.
[4] U. Fano, Rep. Progr. Phys. 46 (1983) 97 165.
[5] C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77 142. 3
[6] A.G. Abrashkevich, D.G. Abrashkevich and M. Shapiro, Comput. Phys. Commun. 90 (1995) 311 339.
[7] M.G. Dimova, M.S. Kaschiev and S.I. Vinitsky, J. Phys. B 38 (2005) 2337 2352.
[8] O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov and S.I. Vinitsky, J. Phys. A 40 (2007) 11485 11524.
[9] O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev and V.V. Serov, Comput. Phys. Commun. 178 (2007) 301 330; http://cpc.cs.qub.ac.uk/summaries/AEAA v1 0.html.
[10] H.J. Assenbaum, K. Langanke and C. Rolfs, Z. Phys. A 327 (1987) 461 468.
[11] V. Melezhik, Nucl. Phys. A 550 (1992) 223 234.
[12] L. Bracci, G. Fiorentini, V.S. Melezhik, G. Mezzorani and P. Pasini, Phys. Lett. A 153 (1991) 456 460.
[13] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40 64.
[14] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cli s, Prentice Hall, New York, 1982.