A basis set of generalized nonspherical Gaussian functions (GGTOs) is presented and discussed. As a first example we report on Born-Oppenheimer energies of the hydrogen molecule. Although accurate results have been obtained, we conclude that H-2 is too ＇＇simple＇＇ to allow for a substantial gain by using nonspherical functions. We rather expect that these functions may be particularly useful in calculations on large systems. A single basis set of GGTOs was used to simultaneously calculate the potential energy curves of several states within each subspace of (1.3)Sigma(g,u) symmetry. We hereby considered the entire region of internuclear distances 0.8 less than or equal to R less than or equal to 1000 a.u. In particular the results for the fourth up to sixth electronic states show a high accuracy compared to calculations which invoke explicitely correlated functions, e.g., the relative accuracy is at least of the order of magnitude of 10(-5) a.u. Energies for the 4 (1)Sigma(u)(+) and 4-6 (3)Sigma(u)(+) were improved and accurate data for the 6 (3)Sigma(g)(+), 5 (1)Sigma(u)(+) and 6 (1)Sigma(u)(+) state are, to the best of the authors＇ knowledge, presented for the first time. Energy data for the seventh up to the ninth electronic state within each subspace were obtained with an estimated error of the order of magnitude of 10(-4) a.u. The 7 (1)Sigma(g)(+) and the 6 (1)Sigma(u)(+) state were found to exhibit a very broad deep outer well at large internuclear distances.