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Manuscript Title: A highly optimized code for calculating atomic data at neutron star magnetic field strengths using a doubly self-consistent Hartree-Fock-Roothaan method
Authors: C. Schimeczek, D. Engel, G. Wunner
Program title: HFFERII
Catalogue identifier: AECC_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 183(2012)1502
Programming language: Fortran 95.
Computer: Cluster of 1-13 HP Compaq dc5750.
Operating system: Linux.
Has the code been vectorised or parallelized?: Yes, parallelized using MPI directives.
RAM: 1 GByte per node
Keywords: Neutron star magnetic fields, Atomic data, B-splines, Hartree-Fock- Roothaan.
PACS: 31.15.xr, 32.60.+i, 95.30.Ky, 97.60.Jd.
Classification: 2.1.

External routines: MPI/GFortran, LAPACK, BLAS, FMlib (included in the package).

Does the new version supersede the previous version?: Yes

Nature of problem:
Quantitative modellings of features observed in the X-ray spectra of isolated magnetic neutron stars are hampered by the lack of sufficiently large and accurate databases for atoms and ions up to the last fusion product, iron, at strong magnetic field strengths. Our code is intended to provide a powerful tool for calculating energies and oscillator strengths of medium-Z atoms and ions at neutron star magnetic field strengths with sufficient accuracy in a routine way to create such databases.

Solution method:
The Slater determinants of the atomic wave functions are constructed from single-particle orbitals ψi which are products of a wave function in the z direction (the direction of the magnetic field) and an expansion of the wave function perpendicular to the direction of the magnetic field in terms of Landau states, ψi(ρ, varphi, z) = Pi(zNLn=0tinφni(ρ,varphi). The tin are expansion coefficients, and the expansion is cut off at some maximum Landau level quantum number n = NL. In the previous version of the code only the lowest Landau level was included (NL = 0), in the new version NL can take values of up to 7. As in the previous version of the code, the longitudinal wave functions are expanded in terms of sixth-order B-splines on finite elements on the z axis, with a combination of equidistant and quadratically widening element borders. Both the B-spline expansion coefficients and the Landau weights tin of all orbitals have to be determined in a doubly self-consistent way: For a given set of Landau weights tin, the system of linear equations for the B-spline expansion coefficients, which is equivalent to the Hartree-Fock equations for the longitudinal wave functions, is solved numerically. In the second step, for frozen B-spline coefficients new Landau weights are determined by minimizing the total energy with respect to the Landau expansion coefficients. Both steps require solving non-linear eigenvalue problems of Roothaan type. The procedure is repeated until convergence of both the B-spline coefficients and the Landau weights is achieved.

Reasons for new version:
The former version of the code was restricted to the adiabatic approximation, which assumes the quantum dynamics of the electrons in the plane perpendicular to the magnetic field to be fixed in the the lowest Landau level, n = 0. This approximation is valid only if the magnetic field strengths are large compared to the reference magnetic field BZ, for a nuclear charge Z, BZ = Z24.70108 ×105 T.

Summary of revisions:
In the new version, the transverse parts of the orbitals are expanded in terms of Landau states up to n = 7, and the expansion coefficients are determined, together with the longitudinal wave functions, in a doubly self-consistent way. Thus the back-reaction of the quantum dynamics along the magnetic field direction on the quantum dynamics in the plane perpendicular to it is taken into account. The new ansatz not only increases the accuracy of the results for energy values and transition strengths obtained so far, but also allows their calculation for magnetic field strengths down to B >≅ BZ, where the adiabatic approximation fails.

Intense magnetic field strengths are required, since the expansion of the transverse single-particle wave functions using 8 Landau levels will no longer produce accurate results if the scaled magnetic field strength parameter βZ = B/BZ becomes much smaller than unity.

Unusual features:
A huge program speed-up is achieved by making use of pre-calculated binary files. These can be calculated with additional programs provided with this package.

Running time:
1-30 minutes