Second quantisation formula (electron-phonon coupling)

[peterprp]

Hello,
I can't understand where this formula ((8.5), Rössler, Solid State Theory, Springer) comes from (see attached file).
The label l stands for the electrons positions and c and c dagger are the free electron creation and annihilation operator. This expression is used to write the linear term in the perturbative expansion of the electron-phonon coupling in second quantization.

Thank you in advance.

 

[peterprp]

Sorry, there is a mistake in the formula. The right-hand side is
$\sum_\vec{k} c^\dagger_{k+q}c_k$

 

[DrDu]

In the sum on the left hand side, it would be more precise to include $c^\dagger_{r_l}c_{r_l}$. Then use the usual representation of the $c_r$ in terms of the $c_k$.