Accurate 2d finite element calculations for hydrogen in magnetic fields of arbitrary strength

Abstract

Recent observations of hundreds of hydrogen-rich magnetic white dwarf stars with magnetic fields up to 105 T (103 MG) have called for more comprehensive and accurate databases for wavelengths and oscillator strengths of the H atom in strong magnetic fields for all states evolving from the field-free levels with principal quantum numbers n≤10. We present a code to calculate the energy eigenvalues and wave functions of such states which is capable of covering the entire regime of field strengths B=0T to B∼109T. We achieve this high flexibility by using a two-dimensional finite element expansion of the wave functions in terms of B-splines in the directions parallel and perpendicular to the magnetic field, instead of using asymptotically valid basis expansions in terms of spherical harmonics or Landau orbitals. We have paid special attention to the automation of the program such that the data points for the magnetic field strengths at which the energy of a given state are calculated can be selected automatically. Furthermore, an elaborate method for varying the basis parameters is applied to ensure that the results reach a pre-selected precision, which also can be adjusted freely. Energies and wave functions are stored in a convenient format for further analysis, e.g. for the calculation of transition energies and oscillator strengths. The code has been tested to work for 300 states with an accuracy of better than 10−6 Rydberg across several symmetry subspaces over the entire regime of magnetic field strengths. Program summaryProgram title: H2dbCatalogue identifier: AERB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 158867No. of bytes in distributed program, including test data, etc.: 322671Distribution format: tar.gzProgramming language: Fortran95.Computer: 1 HP Compaq dc5750.Operating system: Linux.RAM: At least 2 GBClassification: 2.1.External routines: Lapack (http://www.netlib.org/lapack/)Nature of problem:The hydrogen problem in the presence of a magnetic field of arbitrary strength shall be solved for all states up to a principal quantum number of n=10. We obtain the full energy vs. field strength function within a certain precision.Solution method:We expand the wave functions in a 2d B-spline basis, vary the corresponding energy functional for the B-spline coefficients and solve the resulting generalised eigenvalue problem. The B-spline basis parameters are adapted iteratively to ensure the overall precision of our results.Restrictions:Non-relativistic Hamiltonian, non-moving atom.Unusual features:Automated analysis of the states at magnetic field strengths from B=0T to B=109T.Running time:Seconds to minutes per single result; hours to days for a full analysis.