  The energy-weighted sum rule for an electric dipole transition operator of a
Schiff type differs from the Thomas-Reiche-Kuhn sum rule by several corrective
terms which depend on the number of system components, ${\cal N}$. The
deviations are evaluated by two distinct approaches. One of them is based on
the charge density expansion around an uniform distribution and provides an
analytical ${\cal N}$ dependence for the sum rule while, the other one hinges
of the RPA approach and yields compact expressions for corrections as function
of the RPA amplitudes. Although the formalism might be considered for any many
body system, we applied it for illustration, to the case of $Na$ clusters. One
concludes that the RPA results for $Na$ clusters obey the modified TRK sum
rule.
