of the { Δ$_{1}$ , . . . , Δ$_{j}$ } happen to be coincident. This leads us to write I$_{j}$ as a sum over sets of contributions I k j labeled by the divergence index k , where The contour integral is along the real axis and closed over the lower complex ω half-plane, P$_{Δ}$ ( ω ) is the polarization with transition energy Δ missing, and { q$_{0}$, · · · , q$_{k}$ } are any nonnegative integers restricted to sum to j − k , The prime over the second sum denotes this restriction on { q$_{0}$, · · · , q$_{k}$ } . Eq. (41) is to be understood as follows: to get a contribution to I k j , we must have k of the j transition energies in Eq. (40) have the same value which we call Δ while all the remaining j − k transition energies must be different from Δ; most generally, the first q$_{0}$ factors in Eq. (40) have transitions differing from Δ, then there is a transition at Δ, then q$_{1}$ follow which differ, followed by another at Δ, etc. ; summing over all Δ and all possible { q$_{l}$ } includes all the possibilities. Using the cyclicity of the trace, we combine the first q$_{0}$ and last q$_{k}$ terms, where the transition { δ$_{1}$, · · · , δ$_{k}$ } sum only over those with energy Δ. In the above expression for I k j , there is no separate dependence on q$_{0}$ or q$_{k}$ but only on their sum r = q$_{0}$ + q$_{k}$ . There are r + 1 possibilities for q$_{0}$ and q$_{k}$ at fixed r . Summing over 21 I$_{j}$ = j ∑ k =1 I k j I k j = − 1 2 j$^{∮}$ dω 2 πi ∑ Δ ∑ q$_{0}$,...,q$_{k}$ $^{′}$tr $^{{}$[ V P$_{Δ}$ ( ω )] $^{q$_{0}$}$V Π$_{Δ}$( ω )[ V P$_{Δ}$ ( ω )] q$_{1}$ · · · V Π$_{Δ}$( ω )[ V P$_{Δ}$ ( ω )] q$_{k}$ $^{}}$. (41) P$_{Δ}$ ( ω ) = P ( ω ) − Π$_{Δ}$( ω ) = ∑ Δ ′ =Δ Π$_{Δ}$ ′ ( ω ) , k ∑ l =0 q$_{l}$ = j − k . I k j = − 1 2 j ∮ dω 2 πi ∑ Δ ∑ q$_{0}$,...,q$_{k}$ $^{′}$tr $^{{}$V Π$_{Δ}$( ω )[ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V Π$_{Δ}$( ω ) · · · V Π$_{Δ}$( ω )[ V P$_{Δ}$ ( ω )] q$_{0}$ + q$_{k}$ $^{}}$, I k j = − 1 2 j ∮ dω 2 πi ∑ Δ (2Δ) k ( ω 2 − Δ $^{2}$) k ∑ q$_{0}$,...,q$_{k}$ ′ ∑ δ$_{1}$, ··· ,δ$_{k}$ tr $^{{}$V | δ$_{1}$ 〉〈 δ$_{1}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V | δ$_{2}$ 〉〈 δ$_{2}$ | · · · V | δ$_{k}$ 〉〈 δ$_{k}$ | [ V P$_{Δ}$ ( ω )] q$_{0}$ + q$_{k}$ } = − 1 2 j ∮ dω 2 πi ∑ Δ (2Δ) k ( ω 2 − Δ $^{2}$) k ∑ q$_{0}$,...,q$_{k}$ ′ ∑ δ$_{1}$, ··· ,δ$_{k}$ 〈 δ$_{1}$ | [ V P$_{Δ}$ ( ω )] $^{q$_{1}$}$V | δ$_{2}$ 〉 · · · 〈 δ$_{k}$ | [ V P$_{Δ}$ ( ω )] q$_{0}$ + $^{q$_{k}$}$V | δ$_{1}$ 〉 , and then replace the Π$_{Δ}$ by explicit sums over transitions to get