The statistical phase space theory (PST) in its orbital transition state version has been used to calculate the dissociation rate associated with the loss of atomic hydrogen from a polycyclic aromatic hydrocarbon molecule. The PST model has been applied to the naphthalene cation with input data obtained exclusively from first-principle calculations using density functional theory. Without any fitting parameters, the calculated dissociation rates are found to agree well with available measurements. The effects of vibrational anharmonicities are investigated and are shown to lower the dissociation rates by a factor of about five.