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Combining the congruence check and the first-principles calculations, we have systematically investigated the structural stabilities and gap distributions of possible diamondoids (CnHm) with the carbon numbers (n) from 10 to 41. A simple method for the nomenclature is proposed, which can be used to distinguish and screen the candidates with high efficiency. Different from previous theoretical studies, the possible diamondoids can be enumerated according to our nomenclature, without any predetermination from experiments. The structural stabilities and electronic properties have been studied by density functional based tight binding and first-principles methods, where a nearly linear correlation is found between the energy gaps obtained by these two methods. According to the formation energy of structures, we have determined the stable configurations as a function of chemical potential. The maximum and minimum energy gaps are found to be dominated by the shape of diamondoids for clusters with a given number of carbon atoms, while the gap decreases in general as the size increases due to the quantum confinement. Published by AIP Publishing. [ABSTRACT FROM AUTHOR]