A highly optimized code for calculating atomic data at neutron star magnetic field strengths using a doubly self-consistent HartreeFockRoothaan method ?
Our previously published code for calculating energies and boundbound transitions of medium-Z elements at neutron star magnetic field strengths [D. Engel, M. Klews, G. Wunner, Comput. Phys. Comm. 180 (2009) 302311] was based on the adiabatic approximation. It assumes a complete decoupling of the (fast) gyration of the electrons under the action of the magnetic field and the (slow) bound motion along the field under the action of the Coulomb forces. For the single-particle orbitals this implied that each is a product of a Landau state and an (unknown) longitudinal wave function whose B-spline coefficients were determined self-consistently by solving the HartreeFock equations for the many-electron problem on a finite-element grid. In the present code we go beyond the adiabatic approximation, by allowing the transverse part of each orbital to be a superposition of Landau states, while assuming that the longitudinal part can be approximated by the same wave function in each Landau level. Inserting this ansatz into the energy variational principle leads to a system of coupled equations in which the B-spline coefficients depend on the weights of the individual Landau states, and vice versa, and which therefore has to be solved in a doubly self-consistent manner. The extended ansatz takes into account the back-reaction of the Coulomb motion of the electrons along the field direction on their motion in the plane perpendicular to the field, an effect which cannot be captured by the adiabatic approximation. The new code allows for the inclusion of up to 8 Landau levels. This reduces the relative error of energy values as compared to the adiabatic approximation results by typically a factor of three (1/3 of the original error), and yields accurate results also in regions of lower neutron star magnetic field strengths where the adiabatic approximation fails. Further improvements in the code are a more sophisticated choice of the initial wave functions, which takes into account the shielding of the core potential for outer electrons by inner electrons, and an optimal finite-element decomposition of each individual longitudinal wave function. These measures largely enhance the convergence properties compared to the previous code, and lead to speed-ups by factors up to two orders of magnitude compared with the implementation of the HartreeFockRoothaan method used by Engel and Wunner in [D. Engel, G. Wunner, Phys. Rev. A 78 (2008) 032515]. Program title: HFFER II Catalogue identifier: AECC_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECC_v2_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.: v 55?130 No. of bytes in distributed program, including test data, etc.: 293?700 Distribution format: tar.gz Programming language: Fortran 95 Computer: Cluster of 113 HP Compaq dc5750 Operating system: Linux Has the code been vectorized or parallelized?: Yes, parallelized using MPI directives. RAM: 1 GByte per node Classification: 2.1 External routines: MPI/GFortran, LAPACK, BLAS, FMlib (included in the package) Catalogue identifier of previous version: AECC_v1_0 Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 302 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 ?i ? i 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, ? i ( ? , ? , z ) = P i ( z ) ? n = 0 N L t i n ? n i ( ? , ? ) . The tin t in are expansion coefficients, and the expansion is cut off at some maximum Landau level quantum number n=NL n = N L . In the previous version of the code only the lowest Landau level was included (NL=0) ( N L = 0 ) , in the new version NL N L 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 t in of all orbitals have to be determined in a doubly self-consistent way: For a given set of Landau weights tin t in , the system of linear equations for the B-spline expansion coefficients, which is equivalent to the HartreeFock 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 lowest Landau level, n=0 n = 0 . This approximation is valid only if the magnetic field strengths are large compared to the reference magnetic field BZ B Z , for a nuclear charge Z,BZ=Z24.70108105 T Z , B Z = Z 2 4.70108  10 5 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 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 B ? B Z , where the adiabatic approximation fails. Restrictions:  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 ?Z=B/BZ ? Z = B / B Z 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: 130 min.
atomic data neutron star magnetic fields hartreefockroothaan