We have performed DFT calculations to devise some possible fullerene dimers (from $C_{60}$ and $C_{80}$) connected through $C_{24}$ and $C_{36}$ bridge cages with the face-to-face linking model. The fullerene dimers with $C_{36}$ bridges have lower binding energies and greater HOMO-LUMO gaps than those of the fullerene dimers with $C_{24}$ bridges. Also, the replacement of $C_{60}$ cages with $C_{80}$ ones always leads to an increase in binding energies and HOMO-LUMO gaps in these systems. Dimerization of $C_{60}$ and $C_{80}$ fullerenes with $C_{24}$ and $C_{36}$ results in a significant decrease in antiaromaticity of the antiaromatic cages $C_{24}$ and $C_{80}$, and an increase in the aromaticity of the aromatic cages $C_{36}$ and $C_{60}$. Therefore, DFT results indicate that those fullerene dimers involving the initially harshly antiaromatic $C_{24}$ or $C_{80}$ cages are more energetically favorable configuration than the fullerene dimers involving the aromatic $C_{36}$ and $C_{60}$ cages.