Several structures of small CdiXi clusters, i less than or equal to 16, have been characterized. Ringlike structures have been found to be the lowest lying structures for i less than or equal to 5 clusters and three-dimensional spheroid structures for larger ones, i = 6-16. This trend has been ascribed to the stability of obtuse X-Cd-X angles in the first case and to the stability gained from higher coordination in the second case. The three-dimensional structures may be envisioned as being built from Cd2X2 squares and Cd3X3 hexagons, as it was the case for ZniOi, X = O, S, Se, Te, and CdiOi three-dimensional structures, studied previously in our group. Second-order free-energy differences reveal that the most stable spheroids are the i = 12 ones. The highest occupied molecular orbital-lowest unoccupied molecular orbital gaps are found to fit in the experimental trend observed for nanoparticles, which might be an indication of a possible reorganization of the nanoparticle surfaces. Vibrational spectra are provided for their use in experimental characterization.