BSkG3 Model: Unified Nuclear EDF

• BSkG3 model is a comprehensive energy density functional framework incorporating extended Skyrme interactions, pairing functionals, and full 3D Hartree–Fock–Bogoliubov solvers to describe nuclear structure and dynamics.
• It employs a global fitting protocol using weighted least squares across thousands of measured observables, achieving high accuracy for nuclear masses, charge radii, and fission barrier heights.
• The model unifies nuclear and astrophysical applications by providing an equation of state that meets neutron star constraints and supports r-process nucleosynthesis through detailed fission modeling.

The BSkG3 model is a unified, large-scale, non-relativistic energy density functional (EDF) framework designed to describe all known atomic nuclei, nuclear masses, fission properties, and neutron-star matter within a single, parameter-consistent approach. Developed by the Brussels-Montréal collaboration, BSkG3 integrates an extended Skyrme-type interaction, a microscopically motivated pairing functional, and full three-dimensional coordinate-space Hartree–Fock–Bogoliubov (HFB) solvers. Its parameterization is globally optimized to reproduce nuclear masses, charge radii, fission barriers, neutron resonances, and constraints from infinite nuclear matter, including the properties required for neutron star modeling. BSkG3 is notable for its systematic inclusion of triaxial and octupole deformations at the mean-field level and its ability to simultaneously achieve state-of-the-art accuracy for ground states, excited isomers, spontaneous fission half-lives, nuclear level densities, and the nuclear equation of state up to supranuclear densities (Grams et al., 2023, Grams et al., 2022, Sánchez-Fernández et al., 22 Aug 2025, Goriely et al., 7 Jan 2026, Sánchez-Fernández et al., 28 Apr 2026, Stryjczyk et al., 2024).

1. Energy Density Functional and Theoretical Framework

The BSkG3 model is built on an extended Skyrme energy density functional, incorporating density-dependent gradient terms and additional generalizations beyond the standard t₀–t₃ Skyrme parameters. The total energy for a nucleus is expressed as

$$E[\rho, \tau, \mathbf{J}, \tilde{\rho}] = E_\text{kin}[\tau] + E_\text{Sk}[\rho, \nabla\rho, \mathbf{J}] + E_\text{Coul}[\rho_p] + E_\text{pair}[\tilde{\rho}] + E_\text{corr}$$

where

• $E_\text{kin}$ is the kinetic energy,
• $E_\text{Sk}$ the central Skyrme interaction (including isoscalar/isovector and time-even/odd terms),
• $E_\text{Coul}$ the direct and exchange Coulomb energy,
• $E_\text{pair}$ a microscopically guided, density-dependent zero-range ($\delta$) pairing functional,
• $E_\text{corr}$ phenomenological corrections (center-of-mass, rotational, vibrational, Wigner).

Parametric density dependences (notably $t_4$, $t_5$, $\beta$, $E_\text{kin}$0) provide additional flexibility, enabling both the reproduction of empirical constraints at saturation and the required stiffness at supra-saturation densities. The EDF's coupling constants are linear combinations of the Skyrme parameters $E_\text{kin}$1, $E_\text{kin}$2 and optimized exponents (Grams et al., 2023, Grams et al., 2022).

The pairing functional is constructed to reproduce Brueckner–Hartree–Fock $E_\text{kin}$3 gaps in both symmetric and neutron matter, and includes gradient corrections to simulate induced pairing. Pairing strengths are fitted to odd-even mass staggering and ab initio gaps (Grams et al., 2023).

2. Parameter Optimization and Global Fitting Protocol

BSkG3 incorporates 28–30 free parameters encompassing the Skyrme interaction, density dependencies, spin–orbit and tensor terms, pairing strengths/gradients, and collective correction scales. The fit employs a weighted least-squares cost function covering:

• 2457 measured nuclear masses ($E_\text{kin}$4 MeV rms)
• 810 charge radii ($E_\text{kin}$5 fm rms)
• Primary and secondary fission barriers of 65 actinide nuclei (rms 0.33–0.51 MeV)
• Neutron resonance spacings, isomeric excitation energies, separation energies, $E_\text{kin}$6-decay Q-values (Grams et al., 2023, Grams et al., 2022, Goriely et al., 7 Jan 2026, Sánchez-Fernández et al., 22 Aug 2025)

Constraints from ab initio calculations of the equation of state (EOS) for both symmetric and neutron matter enforce realism up to densities $E_\text{kin}$7 fm$E_\text{kin}$8, necessary for neutron-star applications. The optimization uses neural network emulators to efficiently scan the high-dimensional parameter space, with final parameter sets re-validated by direct HFB Hamiltonian solver calculations on a 3D Cartesian grid (Grams et al., 2023, Grams et al., 2022).

3. Three-Dimensional Mean-Field Implementation and Symmetry Breaking

BSkG3 calculations are performed in a fully symmetry-unrestricted, three-dimensional mesh (cubic, with typical $E_\text{kin}$9 fm on $E_\text{Sk}$0 grids). The HFB equations,

$E_\text{Sk}$1

are solved to self-consistency, with time-odd fields, triaxial, octupole, and higher deformations, and explicit blocking for odd and odd-odd systems (Grams et al., 2023, Stryjczyk et al., 2024, Goriely et al., 7 Jan 2026). Constraints on multipole moments ($E_\text{Sk}$2, $E_\text{Sk}$3, $E_\text{Sk}$4) are imposed via Lagrange multipliers to explore the nuclear deformation energy landscape. The code permits the spontaneous breaking of rotational, axial, reflection, and time-reversal symmetries, and is the first mass model to implement such general freedom for global nuclear modeling (Grams et al., 2023, Sánchez-Fernández et al., 28 Apr 2026, Stryjczyk et al., 2024).

4. Applications to Nuclear Structure, Fission, and Level Densities

Masses, Deformations, and Low-Energy Spectra

BSkG3 achieves high accuracy in global masses, charge radii, and separation energies across the full nuclear chart. It systematically describes triaxial and octupole ground states, resolving puzzles such as anomalous fission yields in neutron-rich Rh isotopes and enabling correct spin-parity assignments in odd–odd nuclei (e.g., $E_\text{Sk}$5Rh, where triaxiality is essential to reproduce the observed isomer ordering) (Stryjczyk et al., 2024).

Fission Properties and Spontaneous Fission Half-Lives

Fission barriers, potential energy surfaces (PES), and spontaneous fission lifetimes are calculated using multidimensional constrained HFB (MOCCa) and least-action pathfinding methods, including full triaxial and octupole degrees of freedom. The collective inertia tensor is obtained via the ATDHFB cranking approximation. BSkG3 reproduces fission barrier heights to within 0.33 MeV rms and ground-state SF half-lives within $E_\text{Sk}$6 orders of magnitude of experiment, matching or surpassing mic-mac approaches and providing a unique first-principles tool for r-process nucleosynthesis and transuranic nuclei (Sánchez-Fernández et al., 22 Aug 2025, Sánchez-Fernández et al., 28 Apr 2026).

Nuclear Level Densities

In combination with a combinatorial approach, the BSkG3 mean-field model underpins energy-, spin-, and parity-dependent nuclear level density (NLD) tables for $E_\text{Sk}$78500 nuclei. The spontaneous breaking of triaxial and reflection symmetry in the mean-field solution is reflected in NLD, improving agreement with neutron resonance spacings ($E_\text{Sk}$8) and cumulative discrete levels, and providing reliable inputs for statistical reaction models (Goriely et al., 7 Jan 2026).

5. Infinite Matter Equation of State and Neutron Stars

BSkG3 is tuned to satisfy empirical saturation and symmetry energy constraints at $E_\text{Sk}$9 fm$E_\text{Coul}$0 (e.g., $E_\text{Coul}$1 MeV, $E_\text{Coul}$2 MeV), as well as high-density neutron EOS requirements. It yields an equation of state that is compatible with observations of $E_\text{Coul}$3 pulsars, resulting in maximum neutron star masses $E_\text{Coul}$4 and radii $E_\text{Coul}$5 km (Grams et al., 2023, Grams et al., 2022). This unifies nuclear structure and astrophysical applications within a single EDF platform.

6. Computational Aspects and Model Performance

Large-scale HFB calculations with BSkG3 require advanced HPC resources but are tractable for thousands of nuclei. Key computational strategies include:

• Full 3D coordinate-space HFB solvers (MOCCa code)
• Multidimensional constraint handling for collective variables
• Efficient least-action path algorithms (e.g., PyNEB) for fission observables
• Neural network surrogates for objective function acceleration during global optimization
• Systematic tabulation and release of model outputs (mass tables, PES, NLD, fission rates) for practical use (Goriely et al., 7 Jan 2026, Sánchez-Fernández et al., 28 Apr 2026).

For application to (n,γ) and other cross-section calculations, renormalization schemes enable matching of calculated level densities to experimental resonance spacings and discrete spectra.

7. Broader Implications and Future Directions

By resolving key structural, spectroscopic, and astrophysical observables with a single non-phenomenological Skyrme–HFB EDF, BSkG3 establishes a benchmark for nuclear modeling. Its consistent inclusion of triaxiality, octupole degrees of freedom, and high-density matter constraints directly impact modeling of fission recycling, r-process nucleosynthesis, and neutron star composition and cooling. Ongoing development targets improved treatment of collective inertia (full Thouless–Valatin) and the integration of SF half-lives and NLDs directly into the fitting protocol. The BSkG3 framework and parameter set underpin the current best-practice mass and fission data generation for large-scale theoretical and applied nuclear science (Grams et al., 2023, Sánchez-Fernández et al., 28 Apr 2026, Goriely et al., 7 Jan 2026, Stryjczyk et al., 2024).