Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global optimization of harmonic oscillator basis in covariant density functional theory

B. Osei, A. V. Afanasjev, A. Dalbah

TL;DR

This study demonstrates that covariant density functional theory calculations based on a harmonic-oscillator basis can reach near-infinite-basis accuracy for moderate basis sizes ($N_F\approx20$) by globally optimizing the HO frequency through a scaling factor $f$ in $\hbar\omega_0 = f \times 41 A^{-1/3}$ MeV. The authors exploit the unique two-basis structure of meson-exchange functionals (fermionic and bosonic) to tune convergence, showing that fermionic energies converge from above while mesonic energies converge from below; by selecting optimal $f$, binding energies approach the infinite-basis solution with small $N_F$. They perform a global analysis across 882 even-even nuclei, demonstrating substantial reductions in the root-mean-square deviation $\delta B_{rms}$ from the infinite-basis results (down to ~0.025–0.031 MeV with mass-dependent $f_{opt-2}(A)$) and provide mass-aware parametrizations $f_{opt-2}(A)$. The work also contrasts meson-exchange with point-coupling functionals, shows the limited benefit of treating $\hbar\omega_0$ variationally at large $N_F$, and proposes practical guidelines to extend the approach to other EDF classes. Overall, the findings offer a path to high-precision global CDFT calculations using moderately sized bases, with clear prescriptions for $f$ and its mass dependence to minimize systematic basis biases.

Abstract

The present investigation focuses on the improvement of the accuracy of the description of binding energies within moderately sized fermionic basis. Using the solutions corresponding to infinite fermionic basis it was shown that in the case of meson exchange (ME) covariant energy density functionals (CEDFs) the global accuracy of the description of binding energies in the finite $N_F=16-20$ bases can be drastically (by a factor ranging from $\approx 3$ up to $\approx 9$ dependent on the functional and $N_F$) improved by a global optimization of oscillator frequency of the basis. This is a consequence of the unique feature of the ME functionals in which with increasing fermionic basis size fermionic and mesonic energies approach the exact (infinite basis) solution from above and below, respectively. As a consequence, an optimal oscillator frequency $\hbarω_0$ of the basis can be defined which provides an accurate reproduction of exact total binding energies by the ones calculated in truncated basis. This leads to a very high accuracy of the calculations in moderately sized $N_F=20$ basis when mass dependent oscillator frequency is used: global rms differences $δB_{rms}$ between the binding energies calculated in infinite and truncated bases are only 0.025 MeV and 0.031 MeV for the NL5(Z) and DD-MEZ functionals, respectively. Optimized values of the oscillator frequency $\hbarω_0$ are provided for three major classes of CEDFs, i.e. for density dependent meson exchange functionals, nonlinear meson exchange ones and point coupling functionals.

Global optimization of harmonic oscillator basis in covariant density functional theory

TL;DR

This study demonstrates that covariant density functional theory calculations based on a harmonic-oscillator basis can reach near-infinite-basis accuracy for moderate basis sizes () by globally optimizing the HO frequency through a scaling factor in MeV. The authors exploit the unique two-basis structure of meson-exchange functionals (fermionic and bosonic) to tune convergence, showing that fermionic energies converge from above while mesonic energies converge from below; by selecting optimal , binding energies approach the infinite-basis solution with small . They perform a global analysis across 882 even-even nuclei, demonstrating substantial reductions in the root-mean-square deviation from the infinite-basis results (down to ~0.025–0.031 MeV with mass-dependent ) and provide mass-aware parametrizations . The work also contrasts meson-exchange with point-coupling functionals, shows the limited benefit of treating variationally at large , and proposes practical guidelines to extend the approach to other EDF classes. Overall, the findings offer a path to high-precision global CDFT calculations using moderately sized bases, with clear prescriptions for and its mass dependence to minimize systematic basis biases.

Abstract

The present investigation focuses on the improvement of the accuracy of the description of binding energies within moderately sized fermionic basis. Using the solutions corresponding to infinite fermionic basis it was shown that in the case of meson exchange (ME) covariant energy density functionals (CEDFs) the global accuracy of the description of binding energies in the finite bases can be drastically (by a factor ranging from up to dependent on the functional and ) improved by a global optimization of oscillator frequency of the basis. This is a consequence of the unique feature of the ME functionals in which with increasing fermionic basis size fermionic and mesonic energies approach the exact (infinite basis) solution from above and below, respectively. As a consequence, an optimal oscillator frequency of the basis can be defined which provides an accurate reproduction of exact total binding energies by the ones calculated in truncated basis. This leads to a very high accuracy of the calculations in moderately sized basis when mass dependent oscillator frequency is used: global rms differences between the binding energies calculated in infinite and truncated bases are only 0.025 MeV and 0.031 MeV for the NL5(Z) and DD-MEZ functionals, respectively. Optimized values of the oscillator frequency are provided for three major classes of CEDFs, i.e. for density dependent meson exchange functionals, nonlinear meson exchange ones and point coupling functionals.

Paper Structure

This paper contains 13 sections, 5 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework and the details of the calculations
  3. The dependence of the results on the number of integration points in Gauss-Hermite and Gauss-Laguerre quadratures
  4. The convergence of the nuclear binding energies as a function of size of fermionic basis: the examples of spherical $^{208}$Pb and deformed $^{240}$Pu nuclei
  5. Meson exchange functionals
  6. Point coupling functionals
  7. The oscillator frequency $\hbar \omega_0$ of the basis as a variational parameter
  8. Impact of the coupling of fermionic and bosonic bases on convergence of binding energies
  9. Global analysis of the convergence errors for moderately sized fermionic bases
  10. General discussion on the optimization of the HO basis for meson exchange functionals
  11. Mass dependence of scaling factors $f$
  12. Conclusions
  13. ACKNOWLEDGMENTS

Figures (17)

  • Figure 1: Neutron density of $^{208}$Pb at the GH integration points for indicated calculational schemes. Lines and arrows indicate the position of the last GH integration point i.e. approximate extension of the nucleus covered by the calculations. The calculations are carried out with scaling factor $f=1.4$.
  • Figure 2: The dependence of binding energies on the number of the GH integration points $ngh$ in the ground state of $^{208}$Pb. The calculations are performed for indicated combinations of $N_F$ and scaling factor $f$. The right panel shows the results presented in the left one but in significantly reduced energy window. The $ngh=90$ solution corresponds to the exact one.
  • Figure 3: (a) Neutron densities at the GH integration points obtained in the calculations with $ngh=30$ and $N_F=10$, 20, 30, 40 and 50 compared with exact solution shown by red line. (b) The differences $\Delta \rho_{\nu}$ of neutron densities at the GH integration points obtained in the calculations with $ngh=30$ and indicated values of $N_F$. The calculations are carried out with scaling factor $f=1.4$.
  • Figure 4: The binding energies of the ground states of the $^{208}$Pb and $^{240}$Pu nuclei as a function of $N_F$ for different values of scaling factor $f$. Thin dashed line shows the exact value of binding energy corresponding to infinite basis.
  • Figure 5: The values of $N_F^{conv}$ at which the calculations converge as a function of scaling factor $f$ for indicated functionals. The convergence point $N_F^{conv}$ is reached when $|B(N_F^{conv}) - B(N_F=90)| \leq \varepsilon$ where $\varepsilon$ is numerical accuracy of the calculations of binding energy in variational calculations ($\varepsilon$ =1 keV in our case). The NL1, NL3, NL3*, NL5(E) and NL5(Z) CEDFs belong to the NLME class of the functionals. The DDME class of the functionals is represented by the DD-ME2, DD-MEX, DD-MEY and DD-MEZ CEDFs. The DD-PC1, PC-PK1, PC-Y and PC-Z ones are representatives of the PC class of the CEDFs.
  • ...and 12 more figures