  We studied a multi-orbital Hubbard model at half-filling for two and three
orbitals per site on a two-site cluster via full exact diagonalization, in a
wide range for the onsite repulsion $U$, from weak to strong coupling, and
multiple ratios of the Hund coupling $J_H$ to $U$. The hopping matrix elements
among the orbitals were also varied extensively. At intermediate and large $U$,
we mapped the results into a Heisenberg model. For two orbitals per site, the
mapping is into a $S=1$ Heisenberg model where by symmetry both
nearest-neighbor $(\mathbf{S}_{i}\cdot\mathbf{S}_{j})$ and
$(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{2}$ are allowed, with respective
couplings $J_1$ and $J_2$. For the case of three orbitals per site, the
mappping is into a $S=3/2$ Heisenberg model with
$(\mathbf{S}_{i}\cdot\mathbf{S}_{j})$,
$(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{2}$, and
$(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{3}$ terms, and respective couplings
$J_1$, $J_2$, and $J_3$. The strength of these coupling constants in the
Heisenberg models depend on the $U$, $J_H$, and hopping amplitudes of the
underlying Hubbard model. Our study allows to establish bounds on how large the
ratios $J_2/J_1$ and $J_3/J_1$ can be. We show that those ratios are severely
limited and, as a crude guidance, we conclude that $J_2/J_1$ is less than 0.4
and $J_3/J_1$ is less than 0.2, establishing bounds on effective models for
strongly correlated Hubbard systems.
