  The Kubo single band sum rule is used to determine the optical spectral
weight of a tight binding band with further than nearest neighbour hopping. We
find for a wide range of parameters and doping concentrations that the change
due to superconductivity at low temperature can be either negative or positive.
In contrast, the kinetic energy change is always negative. We use an ARPES
determined tight binding parametrization of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
to investigate whether this can account for recent observations of a positive
change in the spectral weight due to the onset of superconductivity. With this
band structure we find that in the relevant doping regime a straightforward BCS
calculation of the optical spectral weight cannot account for the experimental
observations.
