Application of the gradient method to Hartree-Fock-Bogoliubov theory

A computer code is presented for solving the equations of the Hartree-Fock-Bogoliubov (HFB) theory by the gradient method, motivated by the need for efficient and robust codes to calculate the configurations required by extensions of the HFB theory, such as the generator coordinate method. The code is organized with a separation between the parts that are specific to the details of the Hamiltonian and the parts that are generic to the gradient method. This permits total flexibility in choosing the symmetries to be imposed on the HFB solutions. The code solves for both even and odd particle-number ground states, with the choice determined by the input data stream. Application is made to the nuclei in the $\mathit{sd}$ shell using the universal $\mathit{sd}$-shell interaction B (USDB) shell-model Hamiltonian.

Figure 1

Number of iterations required for convergence using Eq. (19) and fixed $\eta$. At the point $\eta =0.12$ MeV${}^{-1}$ and beyond, the iteration process is unstable. The converged solutions and their energies are the same for all values of $\eta$ shown in the plot. All values produce converged solutions. The system is ${}^{24}\mathrm{Mg}$ with three constraints: $N$, $Z$, and $\langle Q_Q\rangle =10$ $\hslash /m{\omega }_0$. The convergence criterion is $\left|H_c^{20}\right|<1.0\times {10}^{-2}$ MeV. See Sec. 6b for further details.

Figure 2

Single-step energy change as a function of $\eta$ in Eq. (19). The configuration that was updated is the tenth iteration step of the system in Fig. 1.

Figure 3

Error in constrained quantities as a function of iteration number for the $\eta =0.1$ run of the ${}^{24}\mathrm{Mg}$ iterations in Fig. 1. Quantities constrained are $N$ (open circles), $Z$ (filled squares), and $Q_Q$ (filled circles).

Figure 4

Number of iterations required to convergence for the calculated configurations on the deformation energy curve, Fig. 5.

Figure 5

HFB energies as a function of deformation, using the $Q_Q$ quadrupole constraint. The nucleus is ${}^{24}\mathrm{Mg}$, $N=Z=4$ in the $\mathit{sd}$ shell.

Figure 6

Local minima for 50 runs of ${}^{24}\mathrm{Mg}$ with different random starting configurations. Shown are the number of cases as a function of energy.