Why do computational models of electron-electron collisions avoid using cross-sections? Consider the Boltzmann equation in the case of a homogeneous position distribution.
$\frac{\partial f(p, t)}{\partial t} = \mathcal{Q}^{ee}[f](p)$
where $\mathcal{Q}$ represents the collision integral. I would expect this to take the form
$\int dw(p, p' | q, q') f(q)f(q') d^3q d^3q' - \int dw(q, q'|p, p') f(p) f(p')d^3pd^3p'$
where $dw(p, p' | q, q') = |q-q'| \frac{d\sigma(p, p' | q, q')}{dp dp'} d^3p d^3 p' \delta^3(p + p', q + q')$, with a collision cross-section $\sigma$ coresponding to electron-electron collisions. However, reference numerical schemes such as ELENDIF seem to go to great lengths to avoid this, preferring solutions to a Fokker-Planck equation, e.g.

Why is this done?
 A: An explanation is available from chapter 2.2 of Oxenius, "Theoretical foundations of non-LTE plasma spectroscopy":

Elastic collisions of electrically charged particles with other charged particles differ significantly from the collisions discussed in the preceding section. Indeed, in contrast to binary collisions involving at least one neutral particle, a charged particle in a plasma interacts simultaneously with a great number of other charged particles owing to the long range of the Coulomb interaction.

i.e. localised, "strong" collisions in the style of scattering are a much weaker effect than the cumulative many-body long-range repulsive interactions of the electrons. For this, a Fokker-Planck collision theory is needed. Explicitly, this takes the form
$$ \frac{\partial f}{\partial t} = - \partial_i \left(\left\langle \frac{\Delta v_i}{\Delta t} \right\rangle f(v) \right) + \frac{1}{2} \partial_i \partial_j \left(\left\langle \frac{\Delta v_i \Delta v_j}{\Delta t} \right\rangle f(v) \right)$$
where
$$\left\langle \frac{\Delta v_i}{\Delta t} \right\rangle = \frac{4\pi q^4}{m^2} \ln \Lambda \frac{\partial}{\partial v_i} 2 \int \frac{f(v')}{|v-v'|}dv'$$
$$\left\langle \frac{\Delta v_i \Delta v_j}{\Delta t} \right\rangle = \frac{4\pi q^4}{m^2} \ln \Lambda \frac{\partial^2}{\partial v_i \partial v_j} \int f(v')|v-v'|dv'$$
