Grounding the nuclear energy density functional in ab initio calculations: deriving the surface, spin-orbit, and effective-mass terms, a local-density validity criterion, and quantified uncertainties

Nuclear density functional theory (DFT) predicts masses, radii, and other ground-state properties across the whole nuclide chart through an energy density functional (EDF) whose coupling constants are traditionally fitted to experimental data. Marino et al. (Phys. Rev. C 104, 024315 (2021)) showed that the bulk part of a nuclear EDF can instead be derived from ab initio equations of state of symmetric and pure neutron matter, but found that the quality of the resulting local-density-approximation (LDA) functional depends strongly and still unexplainedly on the underlying microscopic Hamiltonian (a chiral interaction worked fairly well; a hard phenomenological potential overbound nuclei by about 10 MeV per nucleon), and that surface and spin-orbit terms could only be added by phenomenological grid search, while the effective-mass terms were omitted altogether. The open problem is to complete this program: derive the non-bulk terms of the EDF — isoscalar and isovector density-gradient (surface) terms, the spin-orbit term, and the effective-mass ($\rho\tau$) terms — systematically from ab initio calculations of inhomogeneous model systems such as the static response of perturbed nuclear matter or neutron drops; establish an operational, quantitative criterion that predicts when the local-density mapping from a given Hamiltonian to an EDF is reliable; and calibrate the functional statistically so that ab initio uncertainties propagate into error bars on finite-nucleus predictions. As of 2026, the machinery is under active development (density-functional treatment of perturbed nuclear matter, Bayesian/emulator tools, exact Monte Carlo benchmarks of nuclear-matter correlations), but the first attempts to constrain the gradient terms from ab initio response calculations were reported unsatisfactory, no hardness criterion exists, and no ab initio-derived EDF with quantified uncertainties has been produced: the problem remains open.

Background

Nuclear density functional theory (DFT) is the standard framework for predicting ground-state properties — masses, radii, deformations — of nuclei across the entire nuclide chart, including exotic nuclei unreachable by experiment. Its central object is the energy density functional (EDF), $E[\rho_n, \rho_p]$, a functional of the neutron and proton densities whose variational minimization yields the single-particle orbitals and the total binding energy. In electronic DFT the functional is built non-empirically from calculations of the uniform electron gas; in nuclear physics, by contrast, the EDFs in everyday use (Skyrme, Gogny, covariant) are phenomenological: their coupling constants are adjusted to measured masses and radii, so their extrapolations away from known nuclei carry unquantified model errors.

"Ab initio" nuclear theory starts instead from microscopic Hamiltonians — two- and three-nucleon forces, today mostly from chiral effective field theory (chiral EFT, the low-energy expansion of QCD in pion and nucleon degrees of freedom) — and solves the many-body problem with systematically improvable methods. Two such methods are central here: self-consistent Green's functions (SCGF), which iterates the Dyson equations for the nucleon propagator in a diagrammatic truncation such as the algebraic diagrammatic construction (ADC), and auxiliary-field diffusion Monte Carlo (AFDMC), which projects out the ground state by imaginary-time evolution with Monte Carlo sampling of coordinates and spin-isospin amplitudes. Applied to uniform matter, these methods deliver the equation of state (EoS), the energy per particle of symmetric nuclear matter (SNM, equal neutrons and protons) and of pure neutron matter (PNM).

Marino et al. (2021) took the first steps toward grounding a nuclear EDF in such ab initio inputs. They computed the SNM and PNM EoS with the chiral interaction NNLO$_{\rm sat}$ (a chiral EFT potential adjusted to nuclear radii as well as binding energies) via SCGF, and with the phenomenological Argonne AV4$^\prime$+UIX$_c$ Hamiltonian via AFDMC; parameterized the EoS as a polynomial in the Fermi momentum $k_F \propto \rho^{1/3}$ (the momentum of the highest occupied state); and defined the functional's bulk part through the local density approximation (LDA), i.e. the energy density of uniform matter evaluated at the local density $\rho(\mathbf{r})$. Tested on magic (closed-shell) nuclei, the NNLO$_{\rm sat}$-based LDA EDF reproduced binding energies and charge radii fairly well (especially for heavy nuclei), whereas the AV4$^\prime$+UIX$_c$-based one overbound nuclei by roughly 10 MeV per nucleon and gave too-small radii. The authors attributed this contrast to LDA's difficulty in capturing the strong short-range correlations induced by the hard Argonne core, while noting that a quantitative measure of interaction "hardness" is missing. To go beyond LDA they added, in the spirit of the "Jacob's ladder" hierarchy of electronic DFT, a gradient approximation (GA) rung — isoscalar and isovector density-gradient terms $C_t^{\Delta\rho}\,\rho_t\,\Delta\rho_t$ (surface terms; $t=0,1$ denote isoscalar and isovector density combinations) and a spin-orbit term coupling the density to the spin-current density $\mathbf{J}$ — but the three couplings could only be fixed by a grid search against experimental data, not derived from the ab initio input. These GA EDFs also omitted the $\rho\tau$ terms (with $\tau$ the kinetic-energy density) that generate an effective mass $m^* \neq m$, which is crucial for nuclear spectroscopy, and carried no statistical calibration or error bars.

The literature already points to model systems that could in principle constrain such non-bulk terms microscopically: neutrons confined in an external harmonic trap ("neutron drops", computable by quantum Monte Carlo; Gandolfi, Carlson, Pieper, Phys. Rev. Lett. 106, 012501 (2011)) and nuclear matter weakly perturbed by an external field, whose static response function — the coefficient of the quadratic energy change under a sinusoidal density perturbation — is directly sensitive to gradient terms (Buraczynski and Gezerlis, Phys. Rev. Lett. 116, 152501 (2016); Phys. Rev. C 95, 044309 (2017)). A quantitative comparison of Hamiltonians in terms of spectral-function sum rules, the natural seed for a "hardness" measure, was proposed by Rios, Carbone, Polls (Phys. Rev. C 96, 014003 (2017)). What remained — and remains — open is the systematic mapping from such ab initio data to EDF couplings, a criterion telling in advance for which Hamiltonians the local-density mapping is reliable, and uncertainty quantification throughout the construction.

Problem Statement

Complete the construction of a nuclear energy density functional grounded in ab initio calculations, beyond the bulk rung already established. Given microscopic Hamiltonians (chiral EFT interactions and phenomenological potentials) and controlled ab initio many-body methods for homogeneous and weakly inhomogeneous nuclear matter (for example SCGF/ADC or AFDMC), the task is to:

1. Derive the EDF's surface (density-gradient) terms systematically from ab initio calculations of inhomogeneous model systems — for example the static response of perturbed symmetric and pure neutron matter, or neutron drops — rather than by grid search against experimental data, with an inversion that controls the finite-size effects and statistical errors of the ab initio pseudo-data.

2. Likewise constrain from ab initio input the spin-orbit term, which was present in the 2021 gradient-approximation EDFs but tuned by grid search against experimental data, and include the $\rho\tau$ (effective-mass) terms, which those EDFs omitted.

3. Establish an operational, quantitative criterion — for instance a measure of the short-range-correlation or high-momentum content ("hardness") of the underlying Hamiltonian — that predicts, before finite-nucleus testing, whether the local-density or gradient-approximation mapping from that Hamiltonian to an EDF will be reliable; the criterion should at minimum explain the observed contrast between the NNLO$_{\rm sat}$-based and AV4$^\prime$+UIX$_c$-based functionals.

4. Calibrate the resulting functional statistically, propagating the uncertainties of the ab initio input (many-body truncation, statistical and systematic errors) into error bars on predicted finite-nucleus observables.

The problem is solved when all coupling constants of the functional are traceable to ab initio input and its predictions for closed-shell nuclei — binding energies and charge radii, together with at least one observable sensitive to the spin-orbit and effective-mass terms — are competitive with modern empirical EDFs on a broad benchmark set of nuclei, with quantified uncertainties and an explicitly stated domain of validity over microscopic Hamiltonians. The extension to time-dependent DFT applications, which the source authors also envisioned, presupposes this static functional and constitutes a subsequent application phase of the program rather than part of the completion boundary stated here.

References

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