Language: Fortran 77, Fortran 90

Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP

POTHEA - a program for computing eigenvalues and surface eigenfunctions and their first derivatives with respect to the parameter of the parametric self-edjoined 2D elliptic partial differential equation

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

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

KANTBP - a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

ODPEVP - a program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem

POTHMF - a program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field


POTHEA - A PROGRAM FOR COMPUTING EIGENVALUES AND SURFACE EIGENFUNCTIONS
AND THEIR FIRST DERIVATIVES WITH RESPECT TO THE PARAMETER OF THE PARAMETRIC
SELF-ADJOINED 2D ELLIPTIC PARTIAL DIFFERENTIAL EQUATION

Authors: A.A.Gusev, O.Chuluunbaatar, S.I.Vinitsky, A.G.Abrashkevich

A FORTRAN 77 program POTHEA is presented for calculating with the given accuracy eigenvalues, surface eigenfunctions and their first derivatives with respect to a parameter of the parametric self-adjoined 2D elliptic partial differential equation with the Dirichlet and/or Neumann type boundary conditions on a finite two-dimensional region. The program calculates also potential matrix elements that are integrals of the products of the surface eigenfunctions and/or the first derivatives of the surface eigenfunctions with respect to a parameter. Eigenvalues and matrix elements computed by the POTHEA program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second order ordinary differential equations with the help of the KANTBP program [O. Chuluunbaatar et al, Comput. Phys. Commun. 177 (2007) 649–675].
Benchmark calculations of eigenvalues and eigenfunctions of the ground and first excited states of a Helium atom in the framework of a coupled-channel hyperspherical adiabatic approach are presented.
As a test desk, the program is applied to the calculation of the eigensolutions of a 2D boundary value problem, their first derivatives with respect to a parameter and potential matrix elements used in the benchmark calculations.

Details - Computer Physics Communications (2014), 185, pp.2636-2654 (2014):
http://cpc.cs.qub.ac.uk/summaries/AESX_v1_0.html
DOI: http://dx.doi.org/10.1016/j.cpc.2014.04.014


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

Authors: A.A.Gusev, O.Chuluunbaatar, S.I.Vinitsky, A.G.Abrashkevich

A FORTRAN 77 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, or a set of first, second and third type boundary conditions for discrete 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. Efficiency of the schemes proposed is demonstrated on an example of solution of quantum transmittance problem for a pair of coupled ions through the repulsive Coulomb barriers. 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.

Sources and detailed description (pdf) are submitted.

Details - Computer Physics Communications 185(2014)3341:
http://cpc.cs.qub.ac.uk/summaries/ADZH_v3_0.html
DOI: http://dx.doi.org/10.1016/j.cpc.2014.08.002

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

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.

External routines: GAULEG and GAUSSJ [2]

For details see: http://cpc.cs.qub.ac.uk/summaries/ADZH_v2_0.html


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.

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

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:
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 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:
1. the number of differential equations;
2. the number and order of finite elements;
3. the total number of hyperradial points on interval [0, ρmax]; and
4. 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.

External routines: GAULEG and GAUSSJ [1]

See also: http://cpc.cs.qub.ac.uk/summaries/ADZH_v1_0.html


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.

ODPEVP - A PROGRAM FOR COMPUTING EIGENVALUES AND EIGENFUNCTIONS AND
THEIR FIRST DERIVATIVES WITH RESPECT TO THE PARAMETER
OF THE PARAMETRIC SELF-ADJOINED STURM-LIOUVILLE PROBLEM

Authors: O.Chuluunbaatar, A.A.Gusev, S.I.Vinitsky, A.G.Abrashkevich

Nature of problem:

The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics physics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms, (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2].

Solution method:

The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF = EBF with respect to a pair of unknown (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 [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a threedimensional sphere [20].

Restrictions:

The computer memory requirements depend on:
1. the number and order of finite elements;
2. the number of points; and
3. the number of eigenfunctions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements. The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z) of Eq.(1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f1(z) and f2(z) of Eq.(1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions.

Running time:

The running time depends critically upon:
1. the number and order of finite elements;
2. the number of points on interval [zmin, zmax]; and
3. the number of eigenfunctions required.

RAM: depends on
1. the number and order of finite elements;
2. the number of points; and
3. the number of eigenfunctions required.

External routines: GAULEG [3].

See also: ttp://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.html


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.
  2. O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky and A.G. Abrashkevich. Comput. Phys. Commun. 179 (2008) 685-693.
  3. 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.
  4. O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov and V.V. Serov. Phys. Rev. A 77 (2008) 034702-1-4.
  5. E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430.
  6. Yu.N. Demkov and J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365.
  7. P.M. Krassovitskiy and N.Zh. Takibaev. Bulletin of the Russian Academy of Sciences. Physics, 70 (2006) 815-818.
  8. V.S. Melezhik, J.I. Kim and P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15.
  9. F.M. Pen kov, Phys. Rev. A 62 (2000) 044701-1-4.
  10. M. Born and X. Huang. Dynamical theory of crystal lattices, The Clarendon Press, Oxford, England, 1954.
  11. L.V. Kantorovich and V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964.
  12. U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace and G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640.
  13. C.V. Clark, K.T. Lu and A.F. Starace, Progress in Atomic Spectroscopy, Part C, eds. H.G. Beyer and H. Kleinpoppen (New-York: Plenum) (1984) 247-320.
  14. 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.
  15. A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40-64.
  16. K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982
  17. 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.
  18. Yu.A. Kuperin, P.B.Kurasov, Yu.B.Melnikov, S.P.Merkuriev, Ann.Phys. 205 (1991) 330-361.
  19. 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 (2008) 301-330.
  20. A.G. Abrashkevich, M.S. Kaschiev and S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.

POTHMF - A PROGRAM FOR COMPUTING POTENTIAL CURVES AND MATRIX ELEMENTS
OF THE COUPLED ADIABATIC RADIAL EQUATIONS FOR A HYDROGEN-LIKE ATOM
IN A HOMOGENEOUS MAGNETIC FIELD

Authors: O.Chuluunbaatar, A.A.Gusev, V.P.Gerdt, V.A.Rostovtsev, S.I.Vinitsky, A.G.Abrashkevich, M.S.Kaschiev, V.V.Serov

Nature of problem:

In the multi-channel adiabatic approach the Schrödinger equation for a hydrogen-like atom in a homogeneous magnetic field of strength γ (γ = B/B0, B0 ≅ 2.35 × 105T is a dimensionless parameter which determines the field strength B) is reduced by separating the radial coordinate, r, from the angular variables, (θ, φ), and using a basis of the angular oblate spheroidal functions [3] to a system of second-order ordinary differential equations which contain first-derivative coupling terms [4]. The purpose of this program is to calculate potential curves and matrix elements of radial coupling needed for calculating the low-lying bound and scattering states of hydrogen-like atoms in a homogeneous magnetic field of strength 0 < γ ≤ 1000 within the adiabatic approach [5]. The program evaluates also asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem needed to extract from the R-matrix a required symmetric shortrange open-channel reaction matrix K [6] independent from matching point [7]. In addition, the program computes the dipole transition matrix elements in the length form between the basis functions that are needed for calculating the dipole transitions between the low-lying bound and scattering states and photoionization cross sections [8].

Solution method:

The angular oblate spheroidal eigenvalue problem depending on the radial variable is solved using a series expansion in the Legendre polynomials [3]. The resulting tridiagonal symmetric algebraic eigenvalue problem for the evaluation of selected eigenvalues, i.e. the potential curves, is solved by the LDLT factorization using the DSTEVR program [2]. Derivatives of the eigenfunctions with respect to the radial variable which are contained in matrix elements of the coupled radial equations are obtained by solving the inhomogeneous algebraic equations. The corresponding algebraic problem is solved by using the LDLT factorization with the help of the DPTTRS program [2]. Asymptotics of the matrix elements at large values of radial variable are computed using a series expansion in the associated Laguerre polynomials [9]. The corresponding matching points between the numeric and asymptotic solutions are found automatically. These asymptotics are used for the evaluation of the asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem [7]. As a test desk, the program is applied to the calculation of the energy values of the ground and excited bound states and reaction matrix of multi-channel scattering problem for a hydrogen atom in a homogeneous magnetic field using the KANTBP program [10].

Restrictions:

The computer memory requirements depend on:

1. the number of radial differential equations;
2. the number and order of finite elements;
3. the total number of radial points.

Restrictions due to dimension sizes can be changed by resetting a small number of PARAMETER statements before recompiling (see Long Write Up and listing for details).

Running time:

The running time depends critically upon:

1. the number of radial differential equations;
2. the number and order of finite elements;
3. the total number of radial points on interval [rmin, rmax]

The test run which accompanies this paper took 7s required for calculating of potential curves, radial matrix elements, and dipole transition matrix elements on a finite-element grid on interval [rmin = 0, rmax = 100] used for solving discrete and continuous spectrum problems and obtaining asymptotic regular and irregular matrix radial solutions at rmax = 100 for continuous spectrum problem on the Intel Pentium IV 2.4 GHz. The number of radial differential equations was equal to 6. The accompanying test run using the KANTBP program took 2s for solving discrete and continuous spectrum problems using the above calculated potential curves, matrix elements and asymptotic regular and irregular matrix radial solutions. Note, that in the accompanied benchmark calculations of the photoionization cross-sections from the bound states of a hydrogen atom in a homogeneous magnetic field to continuum we have used interval [rmin = 0, rmax = 1000] for continuous spectrum problem. The total number of radial differential equations was varied from 10 to 18.

See also: http://cpc.cs.qub.ac.uk/summaries/AEAA_v1_0.html


References:

  1. 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.
  2. http://www.netlib.org/lapack/
  3. M. Abramovits and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965
  4. U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace and G.L. Webster, Phys. Rev. A 19 (1979) 1629 1640; C.V. Clark, K.T. Lu and A.F. Starace, Progress in Atomic Spectroscopy, eds. H.G. Beyer and H. Kleinpoppen (New-York: Plenum) Part C (1984) 247 320; U. Fano and A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986.
  5. M.G. Dimova, M.S. Kaschiev and S.I. Vinitsky, J. Phys. B 38 (2005) 2337 2352; O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, V.V. Serov, T.V. Tupikova and S.I. Vinitsky, Proc. SPIE 6537 (2007) 653706 1 18.
  6. M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167 257.
  7. M. Gailitis, J. Phys. B 9 (1976) 843 854; J. Macek, Phys. Rev. A 30 (1984) 1277 1278; S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova and O. Chuluunbaatar, Programming and Computer Software 33 (2007) 105 116.
  8. H. Friedrich, Theoretical Atomic Physics, New York, Springer, 1991
  9. R.J. Damburg and R.Kh. Propin, J. Phys. B 1 (1968) 681 691; J.D. Power, Phil. Trans. Roy. Soc. London A 274 (1973) 663 702.
  10. 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

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