expression we would obtain if we had taken the exact expression of Eq. (38) and computed ω$_{p}$ by only solving the block diagonal parts of the RPA/Casida Eq. (36). This provides a different viewpoint on why the screened interaction W$_{Δ}$ appearing in Eq. (44) must be based on a polarizability that does not include any transitions at energy Δ: interactions of transitions with energy Δ among themselves are already included exactly by this formula with no screening ( i.e. when W$_{Δ}$ = V ). So the screened interaction in Eq. (44) must be responsible for capturing the couplings among transitions of differing energies which is precisely why transitions at Δ do not contribute to W$_{Δ}$ . At this point, a great deal of simplification is achieved if the only degeneracies present are normal ones ( i.e. those due to a symmetry such as spin degeneracy for an spin unpolarized system or molecular or crystalline symmetry in real space). Generically, we expect this to be the case as any weak perturbation will remove an accidental degeneracy. For the case of normal degeneracy, the degenerate subspace spanned by the transitions { δ$_{p}$ } with energy Δ transforms as an irreducible representation of the symmetry group. On the other hand, the screening matrix W$_{Δ}$ ( ω ) must transform as the identical representation ( i.e. , as a scalar). Thus the only non-zero entries in the matrix W$_{Δ}$ ( ω ) will be the diagonal entries, and the diagonals will all be equal by symmetry. The matrix square root is trivial in this diagonal basis and Eq. (44) becomes This relation can be viewed as a scalar version of the more general matrix Eq. (44). In the unusual case of accidental degeneracies, the matrix W$_{Δ}$ ( ω ) will have off-diagonal entries and it is necessary to evaluate the matrix square root of Eq. (44). Since these degeneracies are rare, the matrices in question will be small and using any method (such as diagonalization) will not impact the computational load. We note that these approximate forms of the GW -RPA correlation energy also suffer from the unboundedness problem discussed in Section 8 that plagues the exact GW -RPA correlation energy of Eq. (38). For example, consider scaling all interband transitions by a factor λ , so Δ$_{t}$ → λ Δ$_{t}$, and then sending λ to zero. As long as matrix elements of W$_{Δ}$ remain finite, the correlation energies of Eqs. (44,45) both diverge to negative infinity for the same reason that Eq. (38) diverges. (In fact, when all interband transition energies are scaled to zero, the system will show very effective metallic screening and the matrix elements of W$_{Δ}$ will actually go to zero.) We present numerical evidence of this divergence in Section 11. We end this section with some comments about the actual usage of Eqs. (44,45). Com- putationally, using these expressions requires re-computation of the screening for each tran- sition energy. In a na¨ıve implementation, the full matrix is recomputed, and since each computation of P scales as O ( N $^{4}$), the overall computational load for evaluating Eq. (44) is O ( N $^{6}$) and thus no better than the exact diagonalization approach of Eqs. (38) and (36). Furthermore, the sums in Eqs. (44,45) are very difficult to converge numerically because transition energies in the continuum are very closely spaced and thus W$_{Δ}$ ( ω ) will have con- tributions from many poles near ω = Δ. Converging such a sum requires a very dense representation of the continuum together with some regularization procedure to control the very large contributions from nearby poles. In Section 11, we discuss this problem for atomic systems. Thus, the utility of Eqs. (44,45) is not for numerical computation per se but rather 25 Φ GW c ≈ 1 2 ∑ t $^{√}$Δ 2 t + 2Δ$_{t}$ 〈 t | W$_{Δ}$$_{t}$ (Δ$_{t}$) | t 〉 − Δ$_{t}$ − 〈 t | V | t 〉 . (45)