The structural, electronic, energy, and vibrational characteristics of the Al13X- and Al13X2- clusters, with an aluminum-centered (Al-c) icosahedral cage Al-13 and with one or two outer-sphere ligands X = H, F, Cl, Br, OH, NH2, CH3, C-6 H-5, have been calculated within the B3LYP approximation of the density functional theory using the 6-31G* and 6-311+G* basis sets. In all Al13X- radicals, the unpaired electron is localized at the cage atom Al* located opposite the Al-X bond. This Al* atom is the most favorable site for attaching the second X ligand of any nature (trans-addition rule). According to the previously suggested molecular model of the valence state of the [Al-13(-)] "superatom," the calculated energies D-1(Al-13(-)- X) of addition of the first ligand to the anion are about 1 eV lower than the corresponding energies of addition of the second ligand D-2(X Al-13(-)- X). The structure of the Al-13 cage depends on the nature of the nature of the substituent X and can radically change in going from anions to their neutral congeners. In the lowest-lying Al-13 isomer with electronegative substituents X (Hal, OH, NH2, CH3, etc.), the aluminum cage has a marquee structure ( 1, symmetry C-s) with a hexagonal base and a pentagonal "roof." For Al-13 analogues with electropositive ligands X (Al, Li, Na), a tridentate isomer ( T, C-3v) with the X substituent coordinated to a face of the Al-13 icosahedron is preferable. In the case of moderately electronegative X ligands (of the H type), the marquee (1) and icosahedral (T) isomers are close in energy. The stretching vibration frequencies of isomers 1 and T differ significantly in magnitude and intensity so that vibrational spectroscopy methods can be especially applicable to their experimental identification.