Since their discovery, DNA triplexes have received a large amount of interest because of their potential as viable strategy for control of gene expression and their formation with some specific drugs. For that reason, we have studied, by means of ab initio calculations, the structure of the bases of these DNA triplexes in terms of their interaction energies and electron densities for the different hydrogen bonds, which is one of the interactions that determine the structure and dynamics of nucleic acid molecules. A good agreement has been found with experimental results and previous calculations. In general, when a third base is added to a dimer, the strength of the hydrogen bonds of the dimer becomes modified provoking changes in their structures by increasing the distance in the major groove and decreasing the space in the minor groove.