  We investigate the ground state properties of the two dimensional half-filled
one band Hubbard model in the strong (large-U) to intermediate coupling limit
({\it i.e.} away from the strict Heisenberg limit) using an effective spin-only
low-energy theory that includes nearest-neighbor exchange, ring exchange, and
all other spin interactions to order t(t/U)^3. We show that the operator for
the staggered magnetization, transformed for use in the effective theory,
differs from that for the order parameter of the spin model by a
renormalization factor accounting for the increased charge fluctuations as t/U
is increased from the t/U -> 0 Heisenberg limit. These charge fluctuations lead
to an increase of the quantum fluctuations over and above those for an S=1/2
antiferromagnet. The renormalization factor ensures that the zero temperature
staggered moment for the Hubbard model is a monotonously decreasing function of
t/U, despite the fact that the moment of the spin Hamiltonien, which depends on
transverse spin fluctuations only, in an increasing function of t/U. We also
comment on quantitative aspects of the t/U and 1/S expansions.
