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Triatomic butterfly molecules

We detail the rich electronic and vibrational structure of triatomic "butterfly" molecules, ultra-long-range Rydberg molecules bound by resonant $p$-wave scattering. We divide these molecules into two sub-classes depending on their parity under reflection of the electronic wave function through the molecular plane. The trimers with odd reflection parity have topographically smooth potential energy surfaces except near the collinear configuration. Here, the vibrational wave function is confined tightly in the symmetric-stretch and bending modes, but only loosely in the asymmetric stretch mode. The trimers with even reflection parity exhibit far richer potential surfaces with abundant minima, but only a few of these are deep enough to localize the vibrational states. These minima are correlated with the electronic wave functions of the butterfly dimer, contributing to a building principle for trimers.