Machine learning orbital-free density functional theory resolves shell effects in deformed nuclei

The accurate description of deformed atomic nuclei by orbital-free density functional theory has been a longstanding textbook challenge, due to the difficulty in accounting for quantum shell effects. Orbital-free density functional theory is, in principle, capable of describing all nuclear effects, as guaranteed by the Hohenberg-Kohn theorem. However, from a microscopic perspective, shell and deformation effects are intrinsically connected to single-orbital structures, posing a significant challenge for orbital-free approaches. Here, we develop a machine-learning-based orbital-free density functional theory, enabling the description of ground-state properties and potential energy curves for both spherical 16O and deformed 20Ne nuclei. To our knowledge, this is the inaugural instance where a fully orbital-free energy density functional has succeeded in taming the complex nuclear shell and deformation effects. It demonstrates that the orbital-free approach, rooted directly in the Hohenberg-Kohn theorem, is not only a theoretical concept but also a practical one for nuclear systems. The description of deformed nuclei within orbital-free density functional theory has been a longstanding challenge, due to the difficulty in capturing intricate quantum shell effects. Here the authors propose a machine-learning-based orbital-free method, enabling the description of nuclear quantum shell and deformation effects.