  The derivation of effective spin models describing the low energy magnetic
properties of undoped CuO_2-planes is reinvestigated. Our study aims at a
quantitative determination of the parameters of effective spin models from
those of a multi-band model and is supposed to be relevant to the analysis of
recent improved experimental data on the spin wave spectrum of La_2CuO_4.
Starting from a conventional three-band model we determine the exchange
couplings for the nearest and next-nearest neighbor Heisenberg exchange as well
as for 4- and 6-spin exchange terms via a direct perturbation expansion up to
12th (14th for the 4-spin term) order with respect to the copper-oxygen hopping
t_pd. Our results demonstrate that this perturbation expansion does not
converge for hopping parameters of the relevant size. Well behaved
extrapolations of the couplings are derived, however, in terms of Pade
approximants. In order to check the significance of these results from the
direct perturbation expansion we employ the Zhang-Rice reformulation of the
three band model in terms of hybridizing oxygen Wannier orbitals centered at
copper ion sites. In the Wannier notation the perturbation expansion is
reorganized by an exact treatment of the strong site-diagonal hybridization.
The perturbation expansion with respect to the weak intersite hybridizations is
calculated up to 4th order for the Heisenberg coupling and up to 6th order for
the 4-spin coupling. It shows excellent convergence and the results are in
agreement with the Pade approximants of the direct expansion. The relevance of
the 4-spin coupling as the leading correction to the nearest neighbor
Heisenberg model is emphasized.
