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

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• 作者：O. Chuluunbaatar ; A.A. Gusev ; S.I. Vinitsky ; A.G. Abrashkevich
• 关键词：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
• 刊名：Computer Physics Communications
• 出版年：2008
• 出版时间：1 November 2008
• 年：2008
• 卷：179
• 期：9
• 页码：685-693
• 全文大小：189 K

文摘

A FORTRAN 77 program for calculating energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach is presented. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with homogeneous boundary conditions: (i) the Dirichlet, Neumann and third type at the left and right boundary points for continuous spectrum problem, (ii) the Dirichlet and Neumann type conditions at left boundary point and Dirichlet, Neumann and third type at the right boundary point for the discrete spectrum problem. The resulting system of radial equations containing 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 reaction matrix and radial wave functions for 3D-model of a hydrogen-like atom in a homogeneous magnetic field. This version extends the previous version 1.0 of the KANTBP program [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].

Program summary

Program title: KANTBP

Catalogue identifier: ADZH_v2_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZH_v2_0.html

Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland

Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html

No. of lines in distributed program, including test data, etc.: 20 403

No. of bytes in distributed program, including test data, etc.: 147 563

Distribution format: tar.gz

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

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[ρ_min,ρ_max]. 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 LDL^T 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[ρ_min,ρ_max]. 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 Section 3 and [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 [ρ_min,ρ_max]; and

4. the number of eigensolutions required. The test run which accompanies this paper took 2 s 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, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649–675; http://cpc.cs.qub.ac.uk/summaries/ADZHv10.html.

[2] W.H. Press, S.A. Teukolsky, W.T. Vetterling, 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.

[6] A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Commun. 90 (1995) 311–339.

[7] M.G. Dimova, M.S. Kaschiev, 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, 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, V.V. Serov, Comput. Phys. Commun. 178 (2007) 301 330; http://cpc.cs.qub.ac.uk/summaries/AEAAv10.html.

[10] H.J. Assenbaum, K. Langanke, 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, P. Pasini, Phys. Lett. A 153 (1991) 456–460.

[13] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40–64.

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