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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_0Distribution 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 L 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.^{T} | ||

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.
| ||

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, ρ
_{max}]; 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. | ||

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. |

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