Hartree-Fock-Roothaan calculations for many-electron atoms and ions in neutron-star magnetic fields

The quantitative analysis of the electromagnetic spectra of isolated neutron stars by means of model atmosphere calculations requires extensive data sets of atomic energy values and transition probabilities in intense magnetic fields. We present a method for the fast computation of wave functions, energies, and oscillator strengths of medium-$Z$ atoms and ions at neutron star magnetic field strengths $B\gtrsim {10}^7\mathrm{T}$ which strikes a balance between numerical accuracy and computing times. We use a Hartree-Fock ansatz in which each single-electron orbital is expanded in terms of Landau states with one longitudinal expansion function, and each Landau level contributes with a different weight to the orbital. Both the longitudinal expansion functions and the Landau weights are determined in a doubly self-consistent way. Hartree-Fock equations are solved by decomposing the $z$ axis in finite elements and expanding the longitudinal wave functions in terms of sixth-order $B$-splines. The contributions of the eight lowest Landau levels are taken into account. The procedure can be efficiently parallelized. Results are presented for the ground states and different excited states of atoms and ions for nuclear charges $Z=2,\dots ,26$ and $N=2,\dots ,26$ electrons, and for oscillator strengths. Wherever possible, a comparison with the results of previous calculations is made.

Figure 1

(Color online) Complete quadratic $(k=3)$ $B$-spline basis sets for interpolation in the interval [0, 36] with quadratically widening nodes. The vertical dashed lines indicate the positions of the nodes. The borders of the complete interpolation interval are triple nodes.

Figure 2

(Color online) Simplified flow diagram of the HFFER algorithm. In the HFFE stage the Hartree-Fock equations are solved to determine the $B$-spline expansion coefficient vectors ${\alpha }_i$ of the orbitals. The converged values ${\alpha }_i^{\mathrm{conv}}$ are handed over to the HFR stage, where the Hartree-Fock-Roothaan equations are solved to obtain new converged Landau weights ${\mathbf{t}}_i^{\mathrm{conv}}$, which in turn serve as “frozen” input for the next HFFE stage of the iteration. Both the HFFE and the HFR stage are MPI parallelized. Adiabatic approximation wave functions coefficients ${\alpha }_i^{\mathrm{ini}}$ enter the initial HFFE iteration, with ${\mathbf{t}}_i^{\mathrm{ad}}=(1,0,0,\dots )$.

Figure 3

(Color online) Energy values obtained by the HFFER procedure as a function of the inverse number of Landau levels considered, for the ground state of neutral iron at $B=5\times {10}^8\mathrm{T}$. The value at $1∕(N_L+1)=1$ corresponds to that of the adiabatic approximation. For comparison the very accurate result of a DQMC calculation [9] is shown by the horizontal line. It is seen that the HFFER values saturate at $E\approx -108.0\mathrm{keV}$. The remaining error with regard to the DQMC result is the consequence of our approximation (3).

Figure 4

(Color online) Comparison between the longitudinal orbitals obtained in adiabatic approximation, and with eight Landau channels for the ground state of neutral silicon at $B=1.0\times {10}^8\mathrm{T}$ (four innermost electrons). Note the increase of the wave function amplitude close to the nucleus for $N_L=7$.