  The Drude weight of the Hubbard model on the two-dimensional square lattice
is studied by the exact diagonalizations applied to clusters up to 20 sites. We
carefully examine finite-size effects by consideration of the appropriate
shapes of clusters and the appropriate boundary condition beyond the imitation
of employing only the simple periodic boundary condition. We successfully
capture the behavior of the Drude weight that is proportional to the squared
hole doping concentration. Our present result gives a consistent understanding
of the transition between the Mott insulator and doped metals. We also find, in
the frequency dependence of the optical conductivity, that the mid-gap
incoherent part emerges more quickly than the coherent part and rather
insensitive to the doping concentration in accordance with the scaling of the
Drude weight.
