on-site repulsion
Picture renting a tiny studio apartment. One person lives there comfortably. But if a second person moves in, things get cramped, tense, and expensive — sharing such a small space carries a real cost. On-site repulsion is the electron version of that crowding penalty: the energy price two electrons pay for crowding onto the same atom.
In a lattice of atoms, an electron can sometimes sit on an atom that already holds another electron. Because both are negatively charged and now confined to the same tiny region, their Coulomb repulsion spikes. Physicists capture this single number — often written U — as the cost of double occupancy on one site. It is the central 'interaction' ingredient of the Hubbard model and the knob that, when large enough, turns a would-be metal into a Mott insulator.
It matters because on-site repulsion distills the messy, long-range Coulomb interaction down to one local, manageable number, making strongly correlated physics tractable. The honest caveat: it is an idealization. Real repulsion also acts between electrons on neighboring atoms, and the true value of U inside a material is hard to pin down because screening by other electrons reshapes it.
In hydrogen, pulling atoms far apart leaves one electron per atom and no conduction. Forcing a second electron onto an atom — making a negative hydrogen ion — costs a definite chunk of energy you can look up in tables. That energy is essentially the on-site repulsion U, the real-world price of double occupancy.
Forcing a second electron onto a hydrogen atom costs a fixed energy — that's on-site repulsion U.
The famous symbol U for on-site repulsion is, confusingly, the same letter often used for energy in general — but here it has a precise meaning: the extra energy to put a second electron on an already-occupied site. Bigger U means more strongly correlated.