  The variational two-electron reduced density matrix (v2RDM) method is
generalized for the description of total angular momentum ($J$) and projection
of total angular momentum ($M_{J}$) states in atomic systems described by
non-relativistic Hamiltonians, and it is shown that the approach exhibits
serious deficiencies. Under ensemble $N$-representability constraints, v2RDM
theory fails to retain the appropriate degeneracies among various $J$ states
for fixed spin ($S$) and orbital angular momentum ($L$), and, for fixed $L$,
$S$, and $J$, the manifold of $M_{J}$ states are not necessarily degenerate.
Moreover, a substantial energy error is observed for a system for which the
two-electron reduced density matrix is exactly ensemble $N$-representable; in
this case, the error stems from violations in pure-state $N$-representability
conditions. Unfortunately, such violations do not appear to be good indicators
of the reliability of energies from v2RDM theory in general. Several states are
identified for which energy errors are near zero and yet pure-state conditions
are clearly violated.
