  Spin-only descriptions of the half-filled one-band Hubbard model are relevant
for a wide range of Mott insulators. In addition to the usual Heisenberg
exchange, many new types of interactions, including ring exchange, appear in
the effective Hamiltonian in the intermediate coupling regime. In order to
improve on the quantitative description of magnetic excitations in the
insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to
be consistent with band structure calculations and photoemission experiments on
these systems, we include second and third neighbor hopping parameters, t' and
t'', into the Hubbard Hamiltonian. A unitary transformation method is used to
find systematically the effective Hamiltonian and any operator in the spin-only
representation. The results include all closed, four hop electronic pathways in
the canonical transformation. The method generates many ring exchange terms
that play an important role in the comparison with experiments on La2CuO4.
Performing a spin wave analysis, we calculate the magnon dispersion as a
function of U,t,t' and t''. The four parameters are estimated by fitting the
magnon dispersion to the experimental results of Coldea et al. [Phys. Rev.
Lett. 86, 5377, {2001}] for La2CuO4. The ring exchange terms are found
essential, in particular to determine the relative sign of t' and t'', with the
values found in good agreement with independent theoretical and experimental
estimates for other members of the cuprate family. The zero temperature
sublattice magnetization is calculated using these parameters and also found to
be in good agreement with the experimental value estimated by Lee et al. [Phys.
Rev. B 60, 3643 (1999)]. We find a value of the interaction strength U~8t
consistent with Mott insulating behavior.
