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Manuscript Title: KANTBP: 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, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky
Program title: KANTBP
Catalogue identifier: ADZH_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 177(2007)649
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: 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 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 [1]

Nature of problem:
In the hyperspherical adiabatic approach [2-4], a multi-dimensional Schrödinger equation for a two-electron system [5] or a hydrogen atom in magnetic field [6] 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 [7]. The generalized algebraic eigenvalue problem A F = E B F 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 [8]. The generalized algebraic eigenvalue problem (A - EB)F = λD F 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 L D LT factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [8]. 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 [9-12]. 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 28.48s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.

References:
[1] W.H. Press, B.F. Flanery, S.A. Teukolsky and W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
[2] J. Macek, J. Phys. B 1 (1968) 831-843.
[3] U. Fano, Rep. Progr. Phys. 46 (1983) 97-165.
[4] C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142.
[5] A.G. Abrashkevich, D.G. Abrashkevich and M. Shapiro, Comput. Phys. Commun. 90 (1995) 311-339.
[6] M.G. Dimova, M.S. Kaschiev and S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352.
[7] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40-64.
[8] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982.
[9] Yu. A. Kuperin, P. B. Kurasov, Yu. B. Melnikov and S. P. Merkuriev, Annals of Physics 205 (1991) 330-361.
[10] O. Chuluunbaatar, A.A. Gusev, S.Y. Larsen and S.I. Vinitsky, J. Phys. A 35 (2002) L513-L525.
[11] N.P. Mehta and J.R. Shepard, Phys. Rev. A 72 (2005) 032728-1-11.
[12] O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen and S.I. Vinitsky, J. Phys. B 39 (2006) 243-269.