The book integrates computer-assisted calculations developed since Wybourne’s classic publication on the topic. It contains useful Maple™ routines, discussions, and new aspects of the theory of f-electron systems. Establishing a unified basis for understanding state-of-the-art applications and techniques used in the field, the book reviews fundamentals based on Wybourne’s graduate lectures, which include the theory of nuclei, the theory of angular momentum, Racah algebra, and effective tensor operators. It then describes magnetic and hyperfine interactions and their impact on the energy structure and transition amplitudes of the lanthanide ions. The text culminates with a relativistic description of ↔ electric and magnetic dipole transitions, covering sensitized luminescence and a new parametrization scheme of f-spectra. enables scientists to construct accurate and reliable theoretical models to elucidate lanthanides and their properties. This text is ideal for exploring a range of lanthanide applications including electronic data storage, lasers, superconductors, medicine, nuclear engineering, and nanomaterials.