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Although H$_{2}$ is the simplest and the most abundant molecule in the Universe, its formation in the interstellar medium, especially in the photodissociation regions is far from being fully understood. According to suggestions, the formation of H$_{2}$ is catalyzed by polyaromatic hydrocarbons (PAHs) on the surface of interstellar grains. In the present study, we have investigated the catalytic effect of small PAHs with an imperfect aromatic system. Quantum chemical computations were performed for the H-atom-abstraction and H-atom-addition reactions of benzene, cyclopentadiene, cycloheptatriene, indene, and 1H-phenalene. Heights of reaction barriers and tunneling reaction rate constants were computed with density functional theory using the MPWB1K functional. For each molecule, the reaction path and the \warn{rate constants} were determined at 50 K using ring-polymer instanton theory, and the temperature dependence of the \warn{rate constants} was investigated for cyclopentadiene and cycloheptatriene. The computational results reveal that defects in the aromatic system compared to benzene can increase the rate of the catalytic H$_{2}$ formation at 50 K.