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Accurately described excitonic properties of transition metal dichalcogenideheterobilayers (HBLs) are crucial to comprehend the optical response and thecharge carrier dynamics of them. Excitons in multilayer systems posses inter orintralayer character whose spectral positions depend on their binding energyand the band alignment of the constituent single-layers. In this study, wereport the electronic structure and the absorption spectra of MoS$_2$/WS$_2$and MoSe$_2$/WSe$_2$ HBLs from first-principles calculations. We explore thespectral positions, binding energies and the origins of inter and intralayerexcitons and compare our results with experimental observations. The absorptionspectra of the systems are obtained by solving the Bethe-Salpeter equation ontop of a G$_0$W$_0$ calculation which corrects the independent particleeigenvalues obtained from density functional theory calculations. Ourcalculations reveal that the lowest energy exciton in both HBLs possessesinterlayer character which is decisive regarding their possible deviceapplications. Due to the spatially separated nature of the charge carriers, thebinding energy of inter-layer excitons might be expected to be considerablysmaller than that of intra-layer ones. However, according to our calculationsthe binding energy of lowest energy interlayer excitons is only $\sim$ 20\%lower due to the weaker screening of the Coulomb interaction between layers ofthe HBLs. Therefore, it can be deduced that the spectral positions of theinterlayer excitons with respect to intralayer ones are mostly determined bythe band offset of the constituent single-layers. By comparing oscillatorstrengths and thermal occupation factors, we show that in luminescence at lowtemperature, the interlayer exciton peak becomes dominant, while in absorptionit is almost invisible.

Inter- and intra-layer excitons in MoS$_2$/WS$_2$ and MoSe$_2$/WSe$_2$ heterobilayers