We define a square matrix W$_{Δ}$ ( ω ) that contains the matrix elements of W$_{Δ}$ ( ω ) among all the degenerate transitions { δ$_{1}$, · · · , δ$_{k}$ } of energy Δ: which allows us to compactify the above expression as The result of Eq. (43) is exact and summing it over k will recover the same answer as the exact result in Eq. (38) albeit expressed in a completely different manner. We can, in principle, perform the contour integral. On the lower complex ω half-plane, the integrand has a k th order pole at ω = Δ as well as a large number of other poles coming from W$_{Δ}$ ( ω ) which physically correspond to the screening modes or plasma frequencies. We would have to sum over all residues to obtain the integral exactly. However, the resulting expression is unwieldy and can not be simplified in any meaningful manner known to us. Therefore, we make a physical approximation to make the integral tractable. The basic approximation is in the spirit of the COHSEX approximation: we assume that the physically important plasma frequencies are at much higher energies than the dominant interband transition energies. This is generally reasonable for solids and extended systems where plasmons are strongly collective modes with high frequencies due to the long range of the Coulomb interaction. In other words, we assume that the screening dynamics is much faster than the interband dynamics. We return to this point at the end of Section 10 where we discuss what type of physics is and is not included in this type of approximation. Separately, we provide some evidence of the relatively good quality of this type of approximation for atomic systems in Section 11. Mathematically, this approximation means that we ignore the contributions of the residues coming from W$_{Δ}$ ( ω ) itself and instead include only the residue of the low-energy pole at Δ. If ˜ ω is the energy of a typical pole of W$_{Δ}$ ( ω ), then the neglected terms are proportional to powers of the dimensionless ratio Δ / ˜ ω . Therefore, if we write the integral as a power series in Δ / ˜ ω , our approximation amounts to keep only the leading terms of order (Δ / ˜ ω ) $^{0}$. Equivalently, we pretend that W$_{Δ}$ ( ω ) is smooth and analytic so that the only poles in the integral of Eq. (43) come from the denominator at ω = Δ. Since W$_{Δ}$ ( ω ) is built from the polarization P$_{Δ}$ ( ω ), both are missing transitions at energy Δ and are well behaved at and about ω = Δ. We use Eq. (12) to find Again, due to the assumption of the smoothness of W$_{Δ}$ ( ω ) at low frequencies, its derivatives at ω = Δ are also assumed negligible compared to the derivatives of ( ω +Δ) − $^{k}$: this amounts to discarding terms with positive powers of Δ / ˜ ω . So we arrive at W$_{Δ}$ ( ω )$_{p,q}$ ≡ 〈 δ$_{p}$ | W$_{Δ}$ ( ω ) | δ$_{q}$ 〉 , ∞ ∑ j = k I k j = − 1 2 ∮ dω 2 πi ∑ Δ (2Δ) k k · ( ω 2 − Δ $^{2}$) $_{k}$tr $^{{ (}$W$_{Δ}$ ( ω ) k $^{}}$. (43) ∞ ∑ j = k I k j ≈ 1 2 ∑ Δ (2Δ) k k ! d k − 1 dω k − 1 [ 1 ( ω + Δ) k · tr $^{{ (}$W$_{Δ}$ ( ω ) k } ] ∣ ∣ ω =Δ . ∞ ∑ j = k I k j ≈ 1 2 ∑ Δ (2Δ) k k ! · tr $^{{ (}$W$_{Δ}$ (Δ) ) k } · d k − 1 dω k − 1 [ 1 ( ω + Δ) k ] ∣ ∣ ω =Δ . 23