  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.
