The Thomas-Reiche-Kuhn sum rule as a consequence of a non-singular optical susceptibility in semiconductors

The Thomas-Reiche-Kuhn optical (TRK) sum rules for bulk materials have customarily been obtained by combining the Kramers-Kronig relations with the high frequency limit of the optical susceptibility tensor $χ_{ij}$. Also, a non-singular expression for $χ_{ij}$ involve the reduction of some its parts to an effective mass tensor. In this paper we show that the latter procedure is intimately connected to the TRK sum rules, and in fact these sum rules can be obtained from it. In reaching this result, we present before a thorough description of the momentum matrix elements of Bloch eigenfunctions bypassing the so-called $\bf{k}-$representation.