THEORETICAL FRAMEWORKS IN NUCLEAR STRUCTURE: FROM SHELL MODEL TO ENERGY DENSITY FUNCTIONALS — A COMPREHENSIVE REVIEW

Abstract

This review examines the major theoretical frameworks used to describe atomic nuclear structure, emphasising their conceptual foundations, mathematical assumptions, applications, strengths, and limitations. Nuclear structure theory treats the nucleus as a strongly interacting quantum many-body system shaped by shell effects, pairing correlations, deformation, collective motion, and nucleon-nucleon interactions. The review discusses the nuclear shell model, collective models, self-consistent mean-field theory, nuclear energy density functional approaches, and modern ab initio many-body methods. It highlights how these frameworks operate at different levels of microscopic detail and computational complexity, making them suitable for different nuclear regions and observables. Attention is given to the complementarity between phenomenological models and microscopic approaches, especially in describing light, medium-mass, heavy, and exotic nuclei. The review also considers current efforts to connect shell-model interactions and energy density functionals with ab initio calculations based on realistic nuclear forces. Major challenges include deformation, shape coexistence, continuum coupling, drip-line nuclei, computational limitations, and theoretical uncertainty quantification. Although no single framework provides a complete description of all nuclei, each remains valuable within its domain. Future progress will depend on unified models, improved interactions, advanced computation, and systematic uncertainty analysis.