We used the hybrid density-functional (B3LYP/6-31G(d,p)) method to analyze the substitution patterns of $C_{20}H_{18}X_2$ derivatives (X = F, Cl, Br, or OH) obtained as disubstituted $C_{20}H_{20}$ cages. Our results suggest that the cis-1 regioisomers (1,2-dihalo derivatives) are less stable than the trans-1 regioisomers (1,20-dihalo derivatives), whereas in the case of the dihydroxy derivatives, the cis-1 regioisomer is more stable than the trans-1 regioisomer. This implies that in the dihalo-induced strain cages of $C_{20}H_{18}X_2$, the strain effect would affect the relative energies, while in the dihydroxide, the hydrogen bonds have a stronger effect on the relative energies in cis-1 regioisomer than the strain effect do. Thus this supports the experimental result in which the bisvicinal tetrol was of particular preparative-synthetic interest as a substitute for the lacking bisvicinal tetrabromide. Further, the topologies of the HOMO and LUMO characteristics of all $C_{20}H_{18}Cl_2$ and $C_{20}H_{18}Br_2$ regioisomers with the same symmetry are same, but they are different from those of $C_{20}H_{18}F_2$ and $C_{20}H_{18}(OH)_2$. This indicates that the five regioisomers of each $C_{20}H_{20}$ disubstituted derivative will have an entirely different set of characteristic chemical reactions.