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Manuscript Title: 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
Program title: POTHMF
Catalogue identifier: AEAA_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 178(2008)301
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
  1. the number of radial differential equations;
  2. the number and order of finite elements;
  3. the number of radial points.
Test run requires 4 MB.
Keywords: eigenvalue and multi-channel scattering problems, Kantorovich method, finite element method, R-matrix calculation, multi-channel adiabatic approximation, ordinary differential equations, high-order accuracy approximation.
PACS: 02.60.Lj, 03.65.Nk, 31.15.Ja, 32.80.Fb, 33.55.Be.
Classification: 2.5.

External routines: POTHMF uses some Lapack routines, copies of which are included in the distribution (see README file for details).

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.

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