  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
considerd the entire region of internuclear distances 0.8 < R < 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 nineth 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.
