Broyden's method in nuclear structure calculations

Broyden's method, widely used in quantum chemistry electronic-structure calculations for the numerical solution of nonlinear equations in many variables, is applied in the context of the nuclear many-body problem. Examples include the unitary gas problem, the nuclear density functional theory with Skyrme functionals, and the nuclear coupled-cluster theory. The stability of the method, its ease of use, and its rapid convergence rates make Broyden's method a tool of choice for large-scale nuclear structure calculations.

Figure 1

Comparison between the linear mixing algorithm and the standard Broyden's method in SLDA for systems of $(N_{\uparrow },N_{\downarrow })=(30,30)$ (top), $(N_{\uparrow },N_{\downarrow })=(30,29)$ (middle), and $(N_{\uparrow },N_{\downarrow })=(30,15)$ (bottom). The basis size is $N_r=40$ and the convergence parameter of Eq. (13) is $\eta =1$. For the asymmetric systems, the equations exhibit a discontinuity responsible for small plateaus in the plots. Introducing a small temperature $T=0.02{\epsilon }_F$ restores the continuity. Once the approximate solution is found, the temperature may be set back to zero.

Figure 2

Comparison between linear mixing ($\alpha =0.5$, black symbols) and the modified Broyden's method ($\alpha =0.7,M=7$) in HFODD for a high-spin superdeformed band in ${}^{151}\mathrm{Tb}$ at two values of rotational frequencies $\hslash {\omega }_y=0.5$ and 1 MeV.

Figure 3

Comparison between linear mixing ($\alpha =0.5$, black symbols) and Broyden's method ($\alpha =0.7,M=7$) in HFODD for the superdeformed minimum in ${}^{152}\mathrm{Dy}$ at the quadrupole moment of $Q_2=20$ eb (SLy4 Skyrme functional, $N_{\mathrm{osc}}=14$) and the spherical ground state of ${}^{208}\mathrm{Pb}$ (SkP Skyrme functional, $N_{\mathrm{osc}}=12$).

Figure 4

Comparison between linear mixing (dotted line) and Broyden's method in HFBTHO for ${}^{120}\mathrm{Sn}$ (top) and ${}^{194}\mathrm{Rn}$ (bottom). The largest element of ${\mathit{F}}^{(m)}$ is shown as a function of $m$. The modified Broyden mixing was carried out with $\alpha =0.7,w_0=0.01$, and $w_n=1$ for $M=3$ (dashed line) and $M=7$ (solid line). The inset shows the slow convergence of the linear mixing for the pathological case of ${}^{194}\mathrm{Rn}$. Here, the tolerance ${10}^{-5}$ is reached after 4,345 iterations.

Figure 5

Energy curve of ${}^{212}\mathrm{Ra}$ versus quadrupole deformation obtained in the constrained HFBTHO calculations. The local minima (maxima) are marked by dots (squares). While in the unconstrained calculations the linear mixing converges at the local minima only, Broyden's method yields both local minima and maxima, as indicated by arrows.

Figure 6

Convergence of CCSD ground-state energy of ${}^{28}\mathrm{Si}$ (top) and ${}^{56}\mathrm{Ni}$ (bottom) using the modified Broyden's method with $M=4$, 8, and 12 compared to the simple mixing with $\alpha =0.7$. The calculations were carried out in a model space of $N_{\mathrm{osc}}=10$ major oscillator shells.