This study examined the effect of the uptake of one and two electrons on the atomic structure of three isomers of $C_{20}$ clusters, namely the ring, bowl (corannulene like), and cage (the smallest fullerene). Geometry optimizations were performed using the hybrid density functional (B3LYP) methods for neutral, singly and doubly charged $C_{20},;{C_{20}}^-,and;{C_{20}}^{2-}$. These results show that the symmetry of the lowest energies for ring and bowl isomers were not changed, whereas the increasing order of energy for the cage (the smallest fullerene) isomers was changed from $D_{2h};<;C_{2h};{leq};C_2;of;C_{20};through;Ci;<;C_{2h};<;C_2;<;D_{2h};of;{C_{20}}^-;to;Ci;<;C_2;<;D_{2h};<;C_{2h};of;{C_{20}}^{2-}$. The reduced symmetry isomers of the cage have comparative energy and the ground state symmetry of the neutral and single and double charged $C_{20}$ decreased with increasing number of electrons taken up in the point of energetics. Interestingly, the difference in energy between the ground state and the next higher energy state of ${C_{20}}^{2-}$ was 3.5kcal/mol, which is the largest energy gap of the neutral, single anion and double anion of the cage isomers examined.