The hybrid density-functional (B3LYP/6-31G(d,p)) method was used to analyze the substitution effect on the $C_{20}H_{20}$ cage based on calculation of the frontier orbitals of fifteen $C_{20}H_{17}(OH)_3$ derivatives. All substitution products were geometrically optimized without constraints and confirmed by frequency analysis. The results suggest that the cis-1 cis-1 cis-2 regioisomer is the most stable isomer, which implies that hydrogen bonding exerts a stronger effect on the relative energies of the trihydroxide than long-range interactions. Thus, this supports the experimental result in which the bisvicinal tetrol was of particular preparative-synthetic interest. While the LUMO of each of the $C_{20}H_{17}(OH)_3$ regioisomers was equivalently delocalized over the void within the cage, the HOMO was limitedly delocalized on substituents and carbons in close proximity to the substituents. The characteristics of the HOMO of each of the regioisomers vary based on the substitution sites. This indicates that the 15 regioisomers of each $C_{20}H_{20}$ trisubstituted derivative might undergo an entirely different set of characteristic chemical reactions with electrophilic reagents. The results further suggest that the penta-substituted OH groups on the surface of the fullerene cage are more likely to be localized on a pentagon than to be homogeneously delocalized.