Derivation from Second Order Term

We have

$$V_{\rm eff}^{(2)} \equiv \sum_{kk'} p_{k}p_{k'} \langle kk'\vert VG_0(E_{kk'}) V \vert kk'\rangle$$

$$= \sum_{kk'nn'} p_{k}p_{k'} \frac{\langle kk'\vert V\vert nn'\rangle \langle nn' \vert V \vert kk'\rangle} {E_{k}+E_{k'} -E_{n}-E_{n'}},$$ (2.26)

where we have inserted $\hat{1}= \sum_{nn'} \vert nn'\rangle \langle nn'\vert$ twice and used

$$\langle nn'\vert G_0(z)\vert mm'\rangle = \frac{\delta_{nm}\delta_{mm'}}{z-E_{nn'}}.$$ (2.27)

Exercise: verify these expressions.