Abstract Dissipative dark matter predicts rich observable phenomena that can be tested with future large-scale structure surveys. As a specific example, we study atomic dark matter, consisting of a heavy particle and a light particle charged under a dark electromagnetism. In particular, we calculate the cosmological evolution of atomic dark matter focusing on dark recombination and dark molecule formation. We have obtained the relevant interaction rate coefficients by rescaling the rates for normal hydrogen, and evolved the abundances for ionized, atomic, and molecular states using a modified version of Recfast++ (which we have released publicly at ^{a a} https://github.com/jamesgurian/RecfastJulia ). We also provide an analytical approximation for the final abundances. We then calculate the effects of atomic dark matter on the linear power spectrum, which enter through a dark photon diffusion and dark acoustic oscillations. At formation time, the atomic dark matter model suppresses halo abundances on scales smaller than the diffusion scale, just as warm dark matter models suppress the abundance below the free-streaming scale. The subsequent evolution with radiative cooling, however, will alter the halo mass function further.