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The interactions between five amino acid based anions ([AA](-) (AA = Gly, Phe, His, Try, and Tyr)) and N7,N9-dimethylguaninium cation ([dMG](+)) have been investigated by the hybrid density functional theory method B3LYP together with the basis set 6-311++G(d,p). The calculated interaction energy was found to decrease in magnitude with increasing side-chain length in the amino acid anion. The interaction between the [dMG](+) cation and [AA](-) anion in the most stable configurations of ion pairs is a hydrogen bonding interaction. These hydrogen bonds (H bonds) were analyzed by the quantum theory of atoms in molecules (QTAIM) and natural bond orbital (NBO) analysis. Finally, several correlations between electron densities in bond critical points of hydrogen bonds and interaction energy as well as vibrational frequencies in the most stable configurations of ion pairs have been checked.