The nuclear density functional theory (DFT) is a microscopic self-consistent framework suitable for describing heavy nuclei and performing large-scale studies. In this dissertation I discuss my research works on the development and application of the Skyrme nuclear-DFT framework, covering a broad range of topics, including the nucleon localization in rotating systems, the origin of reflection-asymmetric deformations, the parameter calibration for beta decays, and the development of a new coordinate-space DFT solver.The nucleon localization function (NLF), discussed in the first part, is a useful tool for the visualization of structure information. It has been utilized to characterize clustering and shell structure. How the NLF pattern evolves in rotating systems, how it visualizes internal nuclear structure, and how it is connected with single-particle (s.p.) orbits are discussed in this dissertation. The second part deals with nuclei having reflection-asymmetric shapes, which are important candidates for the search of permanent electric dipole moments. In this dissertation, the origin of pear-like deformation is investigated through both the multipole expansion of the energy density functional and the spectrum of canonical s.p. states. Theoretical predictions of beta-decay rates are discussed next; they are important for r-process simulations that involves nuclei whose experimental beta-decay data are unknown. To provide reliable predictions with quantified uncertainties, the χ^2 optimization is performed to constrain parameters that significantly affect beta-decay transitions in proton-neutron finite-amplitude-method calculations. Besides a well calibrated functional, a reliable and efficient DFT solver is also crucial. The Hartree-Fock-Bogoliubov (HFB) method in the coordinate space is preferred for deformed and weakly bound nuclei, as solvers based on basis expansions often have difficulty correctly describing continuum effects. A new HFB solver based on the canonical-basis HFB formalism in the three-dimensional coordinate space is developed in this dissertation. It is a well parallelized solver and has been carefully benchmarked against other established HFB solvers.
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