Quantum chemistry was used to investigate the kinetics of the reactions of 1,3-C4H6 and 1,3-butadien-1-yl with phenyl and benzene, respectively, with the aim of elucidating mechanisms that might lead to the formation of naphthalene and indene. Kinetic constants for each elementary reaction involved in the reactive processes were calculated with density functional theory and a modified G2MP2 method. Small vibrational frequencies, when necessary, were treated as hindered rotors to calculate their rotational potential energies quantum mechanically and the corresponding partition functions. Global rate constants for the formation of the different products were determined with QRRK theory. The main result of this study is that if 1,3-C4H6 and 1,3-butadien-1-yl are formed in significant amounts then they can contribute significantly to the formation of naphthalene in a flame. It was also found that activation energies and reaction enthalpy changes can be influenced significantly by the level of theory adopted in the calculations, with B3LYP differing from the more accurate G2MP2 calculations by up to 7 kcal/mol. This was attributed to the known problems of DFT in describing radicals having multiple resonance structures. The calculated rates of formation of indene for the investigated reaction channels were all too slow to compete with alternative mechanisms proposed in the literature.