The use of complete orthonormal sets of Psi(alpha)-exponential type orbitals (Psi(alpha)-ETOs) is investigated for electronic structure calculations of one-electron diatomic molecular systems. The linear combinations of atomic orbitals (LCAOs) method is used to determine the electronic energy levels of one-electron homonuclear (He-2(3+), Li-2(5+), Be-2(7+) and B-2(9+)) and heteronuclear (HeH2+, LiH3+, BeH4+ and BH5+) molecules by employing the same screening constant for a given set of atomic orbitals of each molecular function. For each molecule, the calculations are carried out for the lowest two energy terms of sigma, pi and delta states using the values of nuclear separations 0.01 <= R <= 20, and Psi(alpha)-ETOs basis sets with alpha = 1, 0 and -1. The results are in good agreement with those found in the literature. The strategies developed in this work can be useful in the study of multielectronic molecules with the help of 9 Psi(alpha)-ETOs basis sets.