Why 1-RDM doesn't matter?

Usually molecules have even number of electrons, the ground state is not degenrated, and occupy same orbital but different spin.
In this case, Darwin Smith's theorem applies, thus problem can be very simplified.
http://link.aps.org/doi/10.1103/PhysRev.147.896
This is very important theorem that we do not obtain 1-RDM from fake wave function.

However, in doublet state, we don't have such symmetry, thus we have degenerated ground state. The total energy can be calculated very accurately, but fails to calculate the dipole moment which should be non-zero but zero.