  Understanding spin excitations and their connection to unconventional
superconductivity have remained a central issue since the discovery of the
cuprates. Direct measurement of the dynamical spin structure factor in the
parent compounds can provide key information on important interactions relevant
in the doped regime, and variations in the magnon dispersion have been linked
closely to differences in crystal structure between families of cuprate
compounds. Here, we elucidate the relationship between spin excitations and
various controlling factors thought to be significant in high-$T_c$ materials
by systematically evaluating the dynamical spin structure factor for the
three-orbital Hubbard model, revealing differences in the spin dispersion along
the Brillouin zone axis and the diagonal. Generally, we find that the absolute
energy scale and momentum dependence of the excitations primarily are sensitive
to the effective charge-transfer energy, while changes in the on-site Coulomb
interactions have little effect on the details of the dispersion. In
particular, our result highlights the splitting between spin excitations along
the axial and diagonal directions in the Brillouin zone. This splitting
decreases with increasing charge-transfer energy and correlates with changes in
the apical oxygen position, and general structural variations, for different
cuprate families.
