Convergence of Density-Matrix Expansions for Nuclear Interactions

We extend density-matrix expansions in nuclei to higher orders in derivatives of densities and test their convergence properties. The expansions allow for converting the interaction energies characteristic to finite- and short-range nuclear effective forces into quasilocal density functionals. We also propose a new type of expansion that has excellent convergence properties when benchmarked against the binding energies obtained for the Gogny interaction.

Figure 1

Precision of the gradient expansions (4) of the direct interaction energies calculated for the finite-range Gogny D1S interaction [15]. The nine nuclei used for the test are ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, ${}^{40,48}\mathrm{Ca}$, ${}^{56,78}\mathrm{Ni}$, ${}^{100,132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$. Positive (fourth order) and negative (second and sixth orders) errors are shown with open and closed symbols, respectively.

Figure 2

Comparison of a Taylor expansion and the DT expansion (with ${\overline{\rho }}_v^t=0$ and $a=4/k_F$), applied to the SNM nonlocal density in second, fourth, and sixth orders. The figure is drawn for $k_F^0=1.35{\mathrm{fm}}^{-1}$ corresponding to the equilibrium density of SNM [9, 15]. The shaded regions indicate the integration ranges needed to obtain 1%, 0.1%, 0.01%, and 0.001% accuracy for the SNM exchange energy with the Gogny D1S interaction.

Figure 3

The rms deviations between the exact and approximate exchange energies of Eq. (7), calculated for the nine nuclei listed in the caption of Fig. 1. The four panels show results obtained in the four spin-isospin channels labeled by $V_{vt}$. Results obtained in the present work (DT) are compared with those corresponding to the NV [2] and PSA [7] expansions.