Taylor expand the Lindhard function
\(\)\chi_0(\q,i\omega) = -\frac{1}{N_s}\sum_{\k} \frac{n_F(\xi_{\k}) - n_F(\xi_{\k+\q})}{\xi_{\k} - \xi_{\k+\q} + i\omega}\(\)
for small \(q = |\q|\) and \(\omega/qv_F\), and assuming a spherically symmetric dispersion relation.
We will expand \(\xi_{\k+\q}\) to second order in \(\q\), and the Fermi function to linear order. We thus write
\(\)\xi_{\k+\q} = \xi_{\k} + q_\alpha\partial_\alpha\xi_{\k} + \frac{1}{2} q_\alpha q_\beta \partial_\alpha\partial_\beta \xi_{\k} + \cdots \(\)