Accelerating Molecular H${}_2^{+}$ Beam in AGS and RHIC

After the recent success of acceleration molecular H${}_2^{+}$ beam (from EBIS) in Booster to 1.0 GeV/u kinetic energy, we propose to carry out tests to accelerate H${}_2^{+}$ beam in AGS (up to 12 GeV/u) and to RHIC (up to 100 GeV/u). In this Note, we address the main effects that might hamper H${}_2^{+}$ beam acceleration, outline the impacts and benefits of high energy H${}_2^{+}$ beam to BNL.