  The van der Waals-London's law, for a collection of atoms at large
separation, states that their interaction energy is pairwise attractive and
decays proportionally to one over their distance to the sixth. The first
rigorous result in this direction was obtained by Lieb and Thirring [LT], by
proving an upper bound which confirms this law. Recently the van der
Waals-London's law was proven under some assumptions by I.M. Sigal and the
author [AS]. Following the strategy of [AS] and reworking the approach
appropriately, we prove estimates on the remainder of the interaction energy.
Furthermore, using an appropriate test function, we prove an upper bound for
the interaction energy, which is sharp to leading order. For the upper bound,
our assumptions are weaker, the remainder estimates stronger and the proof is
simpler. The upper bound, for the cases it applies, improves considerably the
upper bound of Lieb and Thirring. However, their bound is much more general.
Here we consider only spinless Fermions.
