I was reading about actinides and I need some help in understanding this paragraph from Concise Inorganic Chemistry by JD Lee -
The electronic structures of the actinides do not follow the simple pattern found in the lanthanides. Immediately after La the 4f orbitals become appreciably lower in energy than the 5d orbitals. Thus in the lanthanides the electrons fill the 4f orbitals in a regular way (apart from minor differences where it is possible to attain a half filled shell). It might have been expected that after Ac the 5f orbitals would become lower in energy than the 6d orbitals. However, for the first four actinide elements Th, Pa, U and Np the difference in energy between 5f and 6d orbitals is small. Thus in these elements (and their ions) electrons may occupy the 5f or the 6d levels, or sometimes both. Later in the actinide series the 5f orbitals do become appreciably lower in energy. Thus from Pt onwards the 5f shell fills in a regular way, and the elements become very similar.
How do we know when the energy of the orbitals of a subshell become lower or higher than the orbitals of another subshell? Can this just be found experimentally or is there some theoretical explanation too?