Precision of finite-difference representation in 3D coordinate-space Hartree-Fock-Bogoliubov calculations

Background: The precision of nuclear Hartree-Fock-Bogoliubov (HFB) calculations in coordinate space is limited by box discretization schemes. In particular, for the finite-difference (FD) discretization method, both the resolution and box size determine the calculation error. Purpose: The current work studies the accuracy of the FD approximation as applied to the three-dimensional (3D) nuclear HFB problem. Methods: By (1) taking the wave functions solved in the harmonic oscillator (HO) basis and (2) representing the HFB problem in coordinate space using the FD method, the current work carefully evaluates the error due to box discretization by examining the deviation of the resulting HFB matrix, the total energies in the coordinate space, from those calculated with the HO method, the latter of which is free from numerical error within its model configuration. To estimate how the box discretization error accumulates with self-consistent iterations, HF and HFB calculations (with the two-basis method) were carried out for Ca40, Mo110, and Sn132. The resulting energies were compared with those from the HO basis, and 3D coordinate space calculations in the literature. Results: The analysis shows that for a grid spacing of ≤0.6 fm, the off-diagonal elements of the HFB matrix are extremely small (<1keV). The calculated quasiparticle spectra differ from those of HO calculations by a few keV. Self-consistent HF and HFB calculations within the current FD method, with the box discretization schemes suggested in the above analysis, give results similar to those using the HO basis. For the HFB calculations, the FD and HO methods predict different single-particle spectra, making an exact comparison difficult. This makes the analysis of precision at a certain iteration useful. Conclusions: With the described FD approximation to the differential operators, together with the way various densities and the Hamiltonian are constructed, it can be concluded that for a box grid spacing of ≤0.7 fm, the accuracy of each calculated energy contribution is on the order of a few tens of keV. The box size needs to be large enough for the normal densities to be smaller than 10-8fm-3 at the edge of the box. With this discretization scheme, the number of the calculated matrix elements differs from that of the HO calculation by ≤10% of the total number of the HFB matrix elements. For the single- and quasiparticle energies, the accuracy is on the order of a few keV. The above conclusions have been verified by performing self-consistent HF and HFB calculations.