I am trying to follow the derivation of the Born-Oppenheimer approximation proposed on Wikipedia.

I know this is not the sexiest source, but i thought it could give a nice first glimpse. But actually, I have troubles understanding their point.

My first problem is at the very beginning: why do they include the $$\sum_{A>B}\frac{Z_AZ_B}{R_{AB}}$$ term in the electronic hamiltonian $H_e$? I thought the whole point of BOE was to separate electronic and nuclear equations, but here they seem to integrate the nuclei-nuclei interaction in the electronic term (they only leave the nuclear kinetic energy out). It doesn't sound like the thing to do... What did I get wrong ?

Another problem comes later down the line. I really Don't understand how they get the relation:

$$(H_n(\boldsymbol{R}))_{k'k} = \delta_{k,k'}T_n-\sum_{A,\alpha}\frac{1}{M_A}< \chi_k'|P_{A\alpha}|\chi_k>_{(\boldsymbol{r})}P_{A\alpha}+< \chi_k'|T_n|\chi_k>_{(\boldsymbol{r})}$$

In my understanding of what is above in the derivation, I thought that the very definition of $(H_n(\boldsymbol{R}))_{k'k}$ was: $(H_n(\boldsymbol{R}))_{k'k} = < \chi_k'|T_n|\chi_k>_{(\boldsymbol{r})}$ But clearly I'm wrong. What is the definition of the matrix elements of $H_n$ then ?

Sorry in advance if those misunderstandings are basic mistakes !