KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach

doi:10.1016/j.cpc.2014.08.002

Computer Physics Communications

Volume 185, Issue 12, December 2014, Pages 3341-3343

https://doi.org/10.1016/j.cpc.2014.08.002 Get rights and content

Abstract

A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at the left- and right-boundary points for continuous spectrum problem. The resulting system of these 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 reflection and transmission matrices and corresponding wave functions for the two-dimensional problem with different barrier potentials.

Program summary

Program title: KANTBP

Catalogue identifier: ADZH_v3_0

Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_v3_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.: 81813

No. of bytes in distributed program, including test data, etc.: 276779

Distribution format: tar.gz

Programming language: FORTRAN 90/95. Compilers: Intel Fortran 8.0+, GNU Fortran 95 4.4.5+.

Computer: Personal computer.

Operating system: Unix/Linux, Window.

RAM: Depends on

(a)
the number of differential equations
(b)
the number and order of finite elements
(c)
the number of longitudinal points
(d)
the number of eigensolutions required.

Classification: 2.7.

Does the new version supersede the previous version?: No

Catalogue identifier of previous version: ADZH_v2_0

Journal reference of previous version: Comput. Phys. Comm. 179 (2008) 685

Nature of problem:

In the adiabatic approach [1], a multidimensional Schrödinger equation for quantum reflection [2], three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [3], fission model of collision of heavy ions [4] or the photoionization of a hydrogen atom in magnetic field [5] is reduced by separating the longitudinal coordinate, labeled as $z$ , from the transversal 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 new version of the program based on the use of the finite element method of high-order accuracy approximations for calculating reflection and transmission matrices and wave functions for such systems of coupled differential equations on finite intervals of the variable $z \in [z_{m i n}, z_{m a x}]$ with homogeneous boundary conditions of the third-type at the left- and right-boundary points following from the above scattering problems.

Solution method:

The boundary-value problems for the coupled second-order differential equations are solved by the finite element method using high-order accuracy approximations [6–8]. The generalized algebraic eigenvalue problem $A F = E$ B F with respect to pair unknowns ( $E$ , F) arising after the replacement of the differential eigenvalue problem by the finite-element approximation is solved by the subspace iteration method [6]. The generalized algebraic eigenvalue problem ( $A - E$ B) F = D F with respect to pair unknowns (D, F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, $E$ , is solved by the ${L D L}^{T}$ factorization of symmetric matrix and back-substitution methods [6].

Reasons for new version:

The previous versions of KANTBP were intended only to calculate the energy levels, reaction matrix and radial wave functions of the bound state problem and scattering problem in the coupled-channel hyperspherical adiabatic approach, in which original problems were reduced to a set of coupled-channel second order differential equations with respect to radial variable in a semi-axis. However a wider range of physical scattering problems are reduced to a set of coupled-channel second order differential equations with respect to the longitudinal variable on the whole axis. In this case one needs to formulate the third-type boundary conditions for systems of coupled differential equations on a finite interval and calculate a desirable scattering matrix which is expressed via unknown reflection and transmission amplitude matrices of asymptotes of solutions in the open channels. The purpose of this new version is to provide a program for calculating the reflection and transmission amplitude matrices and corresponding wave functions of the continuous spectrum problem thus covering a wider range of physical scattering problems.

Summary of revisions:

The KANTBP 3.0 extends the framework of the previous versions, KANTBP 1.0 and KANTBP 2.0. It calculates the reflection and transmission amplitude matrices and corresponding wave functions of the continuous spectrum for systems of coupled differential equations on finite intervals of the variable $z \in$ [ $z_{m i n}, z_{m a x}$ ] using a general homogeneous boundary condition of the third-type at $z = z_{m i n} < 0$ and $z = z_{m a x} > 0$ . The third-type boundary conditions are formulated for the continuous problems under consideration by using known asymptotes for a set of linear independent asymptotic regular and irregular solutions in the open channels and a set of linear independent regular asymptotic solutions in the closed channels, respectively. The program is applied to the computation of the penetration coefficient for 2D-model of pair particles connected by the oscillator interaction potential (throughout symmetric or nonsymmetric) as well as the Coulomb-like barriers.

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 longitudinal points
(d)
the number of eigensolutions required.

The user must supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutine ASYMEV (when solving the eigenvalue problem) or ASYMSL and ASYMSR (when solving the scattering problem) which evaluate asymptotics of the wave functions at boundary points in case of a boundary condition of the third-type for the above problems.

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 longitudinal points on interval [ $z_{m i n}, z_{m a x}$ ]
(d)
the number of eigensolutions required.

As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions of the boundary-value problem for a set of

N

coupled-channel ordinary second order differential equations which follows from the two-dimensional problem describing a quantum tunneling of two particles

i = 1, 2

with masses

m_{i}

and effective charges

Z_{i}

, interacted by a harmonic oscillator potential through the repulsive Coulomb-like barrier potential

U_{i} (x_{i}) = Z_{i} {(x_{i}^{s} + {\bar{x}}_{m i n}^{s})}^{- 1 / s}

[8]. The following values of parameters were used:

m_{1} = 1, m_{2} = 3, {\bar{x}}_{m i n} = 0.1

Z_{1} = Z_{2} = 0.1, s = 8, N = 4

. The test run took 25 s with calculation of matrix potentials on the Intel Core i5 CPU 3.33 GHz, 4 GB RAM, Windows 7. This test run requires 5 MB of disk storage. The program KANTBP was tested on the JINR Central Information and Computer Complex.

The work was supported partially by RFBR Grants Nos. 14-01-00420 and 13-01-00668 and the JINR theme 05-6-1119-2014/2016 “Methods, Algorithms and Software for Modeling Physical Systems, Mathematical Processing and Analysis of Experimental Data”.

References:

[1]
M. Born, Festschrift Goett. Nach. Math. Phys. K1 (1951) 1–6.
[2]
H. Friedrich, Theoretical Atomic Physics, third ed., Springer, Berlin, 2006, p. 416.
[3]
G.L. Goodvin, M.R.A. Shegelski, Phys. Rev. A 72 (2005) 042713-1-7.
[4]
P. Ring, H. Massmann, J.O. Rasmussen, Nuclear Phys. A 296 (1978) 50–76.
[5]
A. Alijah, J. Hinze, J.T. Broad, J. Phys. B 23 (1990) 45–60.
[6]
K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982.
[7]
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.
[8]
A.A. Gusev, S.I. Vinitsky, O. Chuluunbaatar, V.P. Gerdt, V.A. Rostovtsev, Lect. Notes Comput. Sci. 6885 (2011) 175–191.

References (0)

Cited by (22)

KANTBP 3.1: A program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel and adiabatic approaches
2022, Computer Physics Communications
Citation Excerpt :
The KANTBP (KANTorovich Boundary Problem) versions 1.0 [1] and 2.0 [2] were intended only to calculate the energy levels, reaction matrix and radial wave functions of the bound state problem and the elastic scattering problem in the coupled-channel hyperspherical adiabatic approach, in which the original problems were reduced to a system of coupled-channel second order differential equations (CCSODEs) with respect to a radial variable on a semi-axis. The KANTBP version 3.0 [3] was intended for calculate of the energy levels, reflection and transmission amplitude matrices and corresponding wave functions of the bound state problem and scattering problem for the system of CCSODEs on a whole axis. Moreover, the scattering problem is solved under the condition that potential matrix elements in left and right asymptotic regions have only a “almost” diagonal form, and the left and right thresholds are the same.
A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at left- and right-boundary points for the discrete spectrum and scattering problems. The resulting system of such equations, containing potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. The scattering problem is solved with non-diagonal potential matrix elements in the left and/or right asymptotic regions and different left and right threshold values. Benchmark calculations for the fusion cross sections of ³⁶S+⁴⁸Ca, ⁶⁴Ni+¹⁰⁰Mo reactions are presented. As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions of the exact solvable wave-guide model, and also the fusion cross sections and mean angular momenta of the ¹⁶O+¹⁴⁴Sm reaction.
Program Title: KANTBP
CPC Library link to program files: https://doi.org/10.17632/4vm9fhyvh3.1
Licensing provisions: CC BY NC 3.0
Programming language: FORTRAN
Nature of problem: In the adiabatic approach [1], a multidimensional Schrödinger equation for quantum reflection [2], the photoionization and recombination of a hydrogen atom in a homogeneous magnetic field [3–6], the three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [7], wave-guide models [8], the fusion model of the collision of heavy ions [9–11], and low-energy fusion reactions of light- and medium mass nuclei [12] is reduced by separating the longitudinal coordinate, labeled as z, from transversal variables to a system of second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present a program based on the use of high-order accuracy approximations of the finite element method (FEM) for calculating energy levels, reflection and transmission matrices and wave functions for such systems of coupled-channel second order differential equations (CCSODEs) on finite intervals of the variable $z \in [z_{\min}, z_{\max}]$ with homogeneous boundary conditions of the third-type at the left- and right-boundary points, which follow from the discrete spectrum and scattering problems.
Solution method: The boundary-value problems for the system of CCSODEs are solved by the FEM using high-order accuracy approximations [13,14]. The generalized algebraic eigenvalue problem $A F = E B F$ with respect to pair unknowns $(E, F)$ , arising after the replacement of the differential eigenvalue problem by the finite-element approximation, is solved by the subspace iteration method [14]. The generalized algebraic eigenvalue problem of a special form $(A - E B) F = D F$ with respect to pair unknowns $(D, F)$ arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the $L D L^{T}$ factorization of the symmetric matrix and back-substitution methods [14].
Additional comments including restrictions and unusual features: The user must supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSL and ASYMSR (when solving the scattering problem) which evaluate asymptotics of the wave functions at boundary points in the case of a boundary conditions of the third-type for the above problems.
- [1]
  M. Born, Festschrift Goett. Nach. Math. Phys. K1 (1951) 1–6.
- [2]
  H. Friedrich, Theoretical Atomic Physics, 3rd ed., Springer, Berlin, 2006.
- [3]
  A. Alijah, J. Hinze, J.T. Broad, J. Phys. B 23 (1990) 45–60.
- [4]
  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.
- [5]
  O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov, V.V. Serov, Phys. Rev. A 77 (2008) 034702.
- [6]
  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 (2008) 301–330.
- [7]
  G.L. Goodvin, M.R.A. Shegelski, Phys. Rev. A 72 (2005) 042713.
- [8]
  G. Chuluunbaatar, A.A. Gusev, O. Chuluunbaatar, S.I. Vinitsky, L.L. Hai, EPJ Web Conf. 226 (2020) 02008.
- [9]
  H.J. Krappe, K. Moehring, M.C. Nemes, H. Rossner, Z. Phys. A. 314 (1983) 23–31.
- [10]
  T. Ichikawa, K. Hagino, and A. Iwamoto, Phys. Rev. C 75 (2007) 064612.
- [11]
  C.L. Jiang, B.B. Back, K.E. Rehm, K. Hagino, G. Montagnoli, A.M. Stefanini, Eur. Phys. J. A 57 (2021) 235.
- [12]
  V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, H. Lenske, Eur. Phys. J. A 56 (2020) 19.
- [13]
  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.
- [14]
  K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982.
Spectrum of beryllium dimer in ground X1Σ<inf>g</inf>+ state
2021, Journal of Quantitative Spectroscopy and Radiative Transfer
Citation Excerpt :
It is also important for modeling of a near-surface diffusion of the beryllium dimers [12] in connection with the well-known multifunctional use of beryllium alloys in modern technologies of the electronic, space and nuclear industries [13], and, in particular, the ITER project [14]. The adaptation of technique for solving the above class of eigenvalue, metastable and scattering problems for the second order ordinary differential equations using the programs ODPEVP [15], KANTBP[16–18], and KANTBP 5M[19], i.e., the upgraded version of KANTBP 4M [20], implementing the finite element method (FEM) [21,22] in Fortran and Maple, respectively, is also a subject of the present study. The paper has the following structure.
The calculations of the spectrum of vibrational-rotational bound states and new metastable states of the beryllium dimer in ground $X^{1} Σ_{g}^{+}$ state important for laser spectroscopy are presented. The problem is solved using the potential energy curves from [A.V. Mitin, Chem. Phys. Lett. 682, 30 (2017)] and [M. Lesiuk et al, Chem. Theory Comput. 15, 2470 (2019)], and the authors’ software package that implements the iteration Newton method and the high-accuracy finite element method. The efficiency of the proposed approach is demonstrated by the upper and lower estimates of the spectrum of vibrational-rotational bound states and, for the first time, rotational-vibrational metastable states with complex-valued energy eigenvalues (with negative imaginary parts of the order of ( $10^{- 20} \div 6$ ) cm $^{- 1}$ ) in the beryllium dimer. The existence of these metastable states is confirmed by calculating the corresponding scattering states with real-values resonance energies.
Schemes of Finite Element Method for Solving Multidimensional Boundary Value Problems
2024, Journal of Mathematical Sciences (United States)
Unusual chemical bond and spectrum of beryllium dimer in ground X1Σ+<inf>g</inf> state
2023, arXiv
Symbolic-Numerical Algorithm for Solving the Problem of Heavy Ion Collisions in an Optical Model with a Complex Potential
2023, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Bayesian analysis on non-resonant behavior of 12C + 12C fusion reaction at sub-barrier energies
2022, Chinese Physics C

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^☆: This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

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References (0)

Cited by (22)

KANTBP 3.1: A program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel and adiabatic approaches

Spectrum of beryllium dimer in ground X<sup>1</sup>Σ<inf>g</inf><sup>+</sup> state

Schemes of Finite Element Method for Solving Multidimensional Boundary Value Problems

Unusual chemical bond and spectrum of beryllium dimer in ground X<sup>1</sup>Σ<sup>+</sup><inf>g</inf> state

Symbolic-Numerical Algorithm for Solving the Problem of Heavy Ion Collisions in an Optical Model with a Complex Potential

Bayesian analysis on non-resonant behavior of <sup>12</sup>C + <sup>12</sup>C fusion reaction at sub-barrier energies

KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach☆

Abstract

Program summary