are the simplest water clusters with shared protons. They have been widely studied experimentally and theoretically, and their proton transfer modes are of particularly high research interest. In H 5 O + 2 , the proton is shared by two water molecules, giving a symmetric structure. Through geometry optimization, this symmetric structure can . b From Ref. . c From Ref. , with basis set aug-cc-pVQZ. d These are bound OH stretch mode frequencies instead of proton transfer mode frequencies as a result of asymmetric proton placement predicted by the underlying theory.