On closer examination, some of the spectroscopic assignments for the fourth excited 1SIGMA(g)+ state of H-2 (4dsigma,P) and all of the ones for the fifth state (4s,O) prove to be wrong. To aid the re-evaluation of these experimental data we have performed (i) the first accurate ab initio calculation, over a wide range of internuclear distances, of the fifth and sixth electronic 1SIGMA(g)+ states, i.e., of their Born-Oppenheimer electronic energy curves and adiabatic corrections, and of their nonadiabatic couplings with the lower excited electronic 1SIGMA(g)+ states, (ii) an improved calculation of the first three excited electronic 1SIGMA(g)+ states and of their mutual couplings, and (iii) the first calculation of relativistic corrections for the first two of these excited states, EF and GK, at the minima of their double-well potential energy functions. We have found that the electronic energy curves of the pairs of singlet ndsigma and ns states (n = 3, 4), unlike the triplet 3dsigma and 3s states, cross each other at internuclear distances which are much smaller than their equilibrium R values and with off-diagonal electronic energies that are at least 2 orders of magnitude smaller than the diabatic coupling energies at the avoided crossings of the corresponding triplet states.