  We employ a first-principles lattice-gas Hamiltonian (LGH) approach to
determine the lateral interactions between O atoms adsorbed on the Pd(100)
surface. With these interactions we obtain an ordering behavior at low coverage
that is in quantitative agreement with experimental data. Uncertainties in the
approach arise from the finite LGH expansion and from the approximate
exchange-correlation (xc) functional underlying the employed density-functional
theory energetics. We carefully scrutinize these uncertainties and conclude
that they primarily affect the on-site energy, which rationalizes the agreement
with the experimental critical temperatures for the order-disorder transition.
We also investigate the validity of the frequently applied assumption that the
ordering energies can be represented by a sum of pair terms. Restricting our
LGH expansion to just pairwise lateral interactions, we find that this results
in effective interactions which contain spurious contributions that are of
equal size, if not larger than any of the uncertainties e.g. due to the
approximate xc functional.
