  We study the long-time behavior of the 3-dimensional repulsive nonlinear
Hartree equation with an external attractive Coulomb potential $-Z/|x|$, which
is a nonlinear model for the quantum dynamics of an atom. We show that, after a
sufficiently long time, the average number of electrons in any finite ball is
always smaller than 4Z (respectively 2Z in the radial case). This is a
time-dependent generalization of a celebrated result by E.H. Lieb on the
maximum negative ionization of atoms in the stationary case. Our proof involves
a novel positive commutator argument (based on the cubic weight $|x|^3$) and
our findings are reminiscent of the RAGE theorem. In addition, we prove a
similar universal bound on the local kinetic energy. In particular, our main
result means that, in a weak sense, any solution is attracted to a bounded set
in the energy space, whatever the size of the initial datum. Moreover, we
extend our main result to Hartree--Fock theory and to the linear many-body
Schr\"odinger equation for atoms.
