them generates a multiplicative factor of r + 1. We rename r back to q$_{k}$ and have We sum over one fewer { q$_{l}$ } , namely from q$_{1}$ to q$_{k}$ , where the prime indicates that q$_{1}$ + · · · + q$_{k}$ = j − k as before. The q$_{l}$ are not treated symmetrically in the above expression: q$_{k}$ is singled out by the extra factor q$_{k}$ + 1 because we chose to eliminate q$_{0}$ . However, the final result for I k j has the same value if instead we single out another q$_{l}$ to have the q$_{l}$ + 1 factor because we can rearrange the k multiplied factors and permute the q$_{l}$ via relabeling. Therefore, we symmetrize by summing over all k choices of q$_{l}$ (being singled out) and dividing by k . This amounts to averaging the q$_{l}$ + 1 factor: We are now ready to sum over j to eliminate the restriction over the { q$_{l}$ } , i.e. remove the prime. To do this, we reorder the j and k sums in the correlation energy where we added and subtracted I 1 1 = 1 2 $^{∑}$$_{t}$〈 t | V | t 〉 . The inner sum over j removes the constraint over the q$_{l}$ so Each factor of [ V P$_{Δ}$ )] $^{q$_{l}$}$V can be summed separately to yield an identical result, and in each case we are summing the geometric series for (1 − x ) − $^{1}$: We have defined the modified screened interaction W$_{Δ}$ ( ω ) for which the interband transition energy Δ is missing from the screening action. Thus the sum over j has yielded I k j = − 1 2 j ∮ dω 2 πi ∑ Δ (2Δ) k ( ω 2 − Δ $^{2}$) k ∑ q$_{1}$,...,q$_{k}$ ′ ∑ δ$_{1}$, ··· ,δ$_{k}$ 〈 δ$_{1}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V | δ$_{2}$ 〉〈 δ$_{2}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V | δ$_{3}$ 〉 · · · ( q$_{k}$ + 1) 〈 δ$_{k}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{k}$}$V | δ$_{1}$ 〉 . 1 k k ∑ l =1 ( q$_{l}$ + 1) = 1 k ( j − k + k ) = j k I k j = − 1 2 ∮ dω 2 πi ∑ Δ (2Δ) k k · ( ω 2 − Δ $^{2}$) k ∑ q$_{1}$,...,q$_{k}$ ′ ∑ δ$_{1}$, ··· ,δ$_{k}$ 〈 δ$_{1}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V | δ$_{2}$ 〉 · · · 〈 δ$_{k}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{k}$}$V | δ$_{1}$ 〉 , which happily cancels the 1 /j factor. So we now have the symmetric expression Φ GW c = ∞ ∑ j =2 j ∑ k =1 I k j = ∞ ∑ k =1 ∞ ∑ j = k I k j − I 1 1 ∞ ∑ j = k I k j = − 1 2 ∮ dω 2 πi ∑ Δ (2Δ) k k · ( ω 2 − Δ $^{2}$) k ∑ q$_{1}$,...,q$_{k}$ ∑ δ$_{1}$, ··· ,δ$_{k}$ 〈 δ$_{1}$ | [ V P$_{Δ}$ ] $^{q$_{1}$}$V | δ$_{2}$ 〉 · · · 〈 δ$_{k}$ | [ V P$_{Δ}$ ] $^{q$_{k}$}$V | δ$_{1}$ 〉 . W$_{Δ}$ ( ω ) ≡ ∞ ∑ q$_{l}$ =0 [ V P$_{Δ}$ ( ω )] $^{q$_{l}$}$V = ( I − V P$_{Δ}$ ( ω )) − $^{1}$V . (42) ∞ ∑ j = k I k j = − 1 2 ∮ dω 2 πi ∑ Δ (2Δ) k k · ( ω 2 − Δ $^{2}$) k ∑ δ$_{1}$, ··· ,δ$_{k}$ 〈 δ$_{1}$ | W$_{Δ}$ ( ω ) | δ$_{2}$ 〉 · · · 〈 δ$_{k}$ | W$_{Δ}$ ( ω ) | δ$_{1}$ 〉 . 22