  Rhombohedral multilayer graphene (RnG) featuring partially flat bands has
emerged as an important platform to probe strong Coulomb correlation effects.
Theoretical consideration of local electron-electron interactions are of
particular importance for electronic eigenstates with a tendency to spatially
localize. We present a method to incorporate mean-field electron-electron
interaction corrections in the tight-binding hopping parameters of the band
Hamiltonian within the extended Hubbard model that incorporates ab initio
estimates of on-site ($U$) and inter-site ($V$) Hubbard interactions for the
$\pi$ bands of RnG. Our Coulomb-interaction renormalized band structures
feature electron-hole asymmetry, band flatness, band gap, and
anti-ferromagnetic ground states in excellent agreement with available
experiments for $n \geq 4$. We reinterpret the putative gaps proposed in $n=3$
systems in terms of shifting electron and hole density of states peaks
depending on the range of the Coulomb interaction models.
