  The so-called aromatic infrared bands are attributed to emission of
polycyclic aromatic hydrocarbons. The observed variations toward different
regions in space are believed to be caused by contributions of different
classes of PAH molecules, i.e. with respect to their size, structure, and
charge state. Laboratory spectra of members of these classes are needed to
compare them to observations and to benchmark quantum-chemically computed
spectra of these species. In this paper we present the experimental infrared
spectra of three different PAH dications, naphthalene$^{2+}$,
anthracene$^{2+}$, and phenanthrene$^{2+}$, in the vibrational fingerprint
region 500-1700~cm$^{-1}$. The dications were produced by electron impact
ionization of the vapors with 70 eV electrons, and they remained stable against
dissociation and Coulomb explosion. The vibrational spectra were obtained by IR
predissociation of the PAH$^{2+}$ complexed with neon in a 22-pole cryogenic
ion trap setup coupled to a free-electron infrared laser at the Free-Electron
Lasers for Infrared eXperiments (FELIX) Laboratory. We performed anharmonic
density-functional theory calculations for both singly and doubly charged
states of the three molecules. The experimental band positions showed excellent
agreement with the calculated band positions of the singlet electronic ground
state for all three doubly charged species, indicating its higher stability
over the triplet state. The presence of several strong combination bands and
additional weaker features in the recorded spectra, especially in the
10-15~$\mu$m region of the mid-IR spectrum, required anharmonic calculations to
understand their effects on the total integrated intensity for the different
charge states. These measurements, in tandem with theoretical calculations,
will help in the identification of this specific class of doubly-charged PAHs
as carriers of AIBs.
