Describing a nucleus with quantum many-body theory is a fascinating and challenging problem with a long history. Previously I wrote about the highly successful shell model which treats the nucleons as weakly interacting with one another.
A complementary approach for nuclei with an equal even number of protons and of neutrons is to view the basic "quasi-particle" as an alpha particle (a bound set of 2 neutrons and 2 protons). This takes into account some aspect of the strong short-range interactin between nucleons. A really nice accessible review by Martin Freer is worth looking at (especially figures 1, 4, and 5). This gives a natural description of the Hoyle state, "The importance of this state is arguably unparalleled in nuclear science." [In order to explain stellar nucleosynthesis of carbon, Sir Fred Hoyle successfully predicted the existence, energy, and quantum numbers of this state in carbon.]
There is a nice Physics article by Michael Carpenter which considers Do alpha particles cluster inside heavy nuclei? and gives a viewpoint on a recent PRL showing evidence for a alpha + 208Pb state in 212Po.