Based on emph{ab initio} theoretical calculations of the optical spectra of vertical heterostructures of MoSe$_2$ (or MoS$_2$) and WSe$_2$ sheets, we reveal two spin-orbit-split Rydberg series of excitonic states below the textsl{A} excitons of MoSe$_2$ and WSe$_2$ with a significant binding energy on the order of 250,meV for the first excitons in the series. At the same time, we predict crystalographically aligned MoSe$_2$/WSe$_2$ heterostructures to exhibit an indirect fundamental band gap. Due to the type-II nature of the MoSe$_2$/WSe$_2$ heterostructure, the indirect transition and the exciton Rydberg series corresponding to a direct transition exhibit a distinct interlayer nature with spatial charge separation of the coupled electrons and holes. The experimentally observed long-lived states in photoluminescence spectra of MoX$_2$/WY$_2$ heterostructure are attributed to such interlayer exciton states. Our calculations further suggest an effect of stacking order on the peak energy of the interlayer excitons and their oscillation strengths.
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