We propose a novel type of Rydberg dimer, consisting of a Rydberg-state atom bound to a distant positive ion. The molecule is formed through long-range electric-multipole interaction between the Rydberg atom and the point-like ion. We present potential energy curves (PECs) that are asymptotically connected with Rydberg $nP$- or $nD$-states of rubidium or cesium. The PECs exhibit deep, long-range wells which support many vibrational states of Rydberg-atom-ion molecules (RAIMs). We consider photo-association of RAIMs in both the weak and the strong optical-coupling regimes between initial and Rydberg states of the neutral atom. Experimental considerations for the realization of RAIMs are discussed.
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