  Hubbard-like Hamiltonians are widely used to describe on-site Coulomb
interactions in magnetic and strongly-correlated solids, but there is much
confusion in the literature about the form these Hamiltonians should take for
shells of p and d orbitals. This paper derives the most general s, p and d
orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and
presents them in ways convenient for practical calculations. We use the full
configuration interaction method to study p and d orbital dimers and compare
results obtained using the correct Hamiltonian and the collinear and vector
Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the
nature of the ground state, the time evolution of excited states, and the
electronic heat capacity.
