content_image/368883.jpg content_image/368884.jpg content_image/368885.jpg content_image/368886.jpg content_image/368887.jpg content_image/368888.jpg content_image/368889.jpg content_image/368890.jpg
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 

<img src='content_image/368883.jpg'>

<img src='content_image/368884.jpg'>

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, 

<img src='content_image/368885.jpg'>

and { q$_{0}$, · · · , q$_{k}$ } are any nonnegative integers restricted to sum to j − k , 

<img src='content_image/368886.jpg'>

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, 

<img src='content_image/368887.jpg'>

and then replace the Π$_{Δ}$ by explicit sums over transitions to get 

<img src='content_image/368888.jpg'>

<img src='content_image/368889.jpg'>

<img src='content_image/368890.jpg'>

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 

