content_image/363968.jpg content_image/363969.jpg content_image/363970.jpg content_image/363971.jpg content_image/363973.jpg content_image/363974.jpg content_image/363975.jpg content_image/363976.jpg
them generates a multiplicative factor of r + 1. We rename r back to q$_{k}$ and have 

<img src='content_image/363968.jpg'>

<img src='content_image/363969.jpg'>

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: 

<img src='content_image/363970.jpg'>

which happily cancels the 1 /j factor. So we now have the symmetric expression 

<img src='content_image/363971.jpg'>

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 

<img src='content_image/363973.jpg'>

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 

<img src='content_image/363974.jpg'>

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 ) − $^{1}$: 

<img src='content_image/363975.jpg'>

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 

<img src='content_image/363976.jpg'>

