Nuclear density functional theory for collective excitation

The finite-amplitude method provides efficient solutions for the collective excitation within the nuclear density functional theory. It has been originally developed for the response functions of giant resonances, and the method recently covers low-energy collective levels, zero-energy Nambu-Goldstone modes, and sum rules, and variety of applications is open. I will overview these recent developments of the framework, and discuss physics of collective inertia for pairing rotation and shape coexistence, and efforts towards determination of next-generation nuclear energy density functional.