Equilibrium structures, dissociation energies, and rovibrational energy levels of the electrostatic complexes formed by molecular hydrogen and first-row S-state transition metal cations Cr+, Mn+, Cu+, and Zn+ are investigated ab initio. Extensive testing of the CCSD(T)-based approaches for equilibrium structures provides an optimal scheme for the potential energy surface calculations. These surfaces are calculated in two dimensions by keeping the H-H internuclear distance fixed at its equilibrium value in the complex. Subsequent variational calculations of the rovibrational energy levels permits direct comparison with data obtained from equilibrium thermochemical and spectroscopic measurements. Overall accuracy within 2-3% is achieved. Theoretical results are used to examine trends in hydrogen activation, vibrational anharmonicity, and rotational structure along the sequence of four electrostatic complexes covering the range from a relatively floppy van der Waals system (Mn+center dot center dot center dot H-2) to an almost a rigid molecular ion (Cu+center dot center dot center dot H-2).