The first 14 (3)Sigma(g)(+) and the first 15 (3)Sigma(u)(+) states of the H(2) molecule are computed with full configuration interaction both from Hartree-Fock molecular orbitals and Heitler-London atomic orbitals within the Born-Oppenheimer approximation, following recent studies for the (1)Sigma(g)(+) and (1)Sigma(u)(+) manifolds [Corongiu and Clementi, J. Chem. Phys. 131, 034301 (2009) and J. Phys. Chem. (in press)]. The basis sets utilized are extended and optimized Slater-type functions and spherical Gaussian functions. The states considered correspond to the configurations (1s(1)nl(1)) with n from 1 to 5; the internuclear separations sample the distances from 0.01 to 10,000 bohrs. For the first three (3)Sigma(g)(+) and (3)Sigma(u)(+) states and for the fourth and fifth (3)Sigma(g)(+) states, our computed energies at the equilibrium internuclear separation, when compared to the accurate values by Staszewska and Wolniewicz and by Kołos and Rychlewski, show deviations of about 0.006 kcal/mol, a test on the quality of our computations. Motivation for this work comes not only from obtaining potential energy curves for the high excited states of H(2) but also from characterizing the electronic density evolution from the united atom to dissociation to provide a detailed analysis of the energy contributions from selected basis subsets and to quantitatively decompose the state energies into covalent and ionic components. Furthermore, we discuss the origin of the seemingly irregular patterns in potential energy curves in the two manifolds, between 4 and 6-9 bohrs--there are two systems of states: the first, from the united atom to about 4 bohrs, is represented by functions with principal quantum number higher than the one needed at dissociation; this system interacts at around 4 bohrs with the second system, which is represented by functions with principal quantum number corresponding to one of the dissociation products.