What ab initio methods are better suited to modelling disorder in materials?

Coherent Potential Approximation (CPA) Method.

This method is implemented successfully in the first-principles matter modeling packages Questaal and EMTO.

You can find a general introduction about CPA on Wikipedia.

A very neat recent development for the modelling of disorder is the so-called localization landscape theory, first introduced here.

Consider a general Hamiltonian $hat{H}=-frac{hbar^2}{2m}nabla^2+V$. The landscape function $u$ is defined as the solution to this equation:

$$ hat{H}u=1. $$

This innocent-looking landscape function has been shown to encode a great deal of information for studying the complex energy landscapes associated with disorder. For example, it can be shown that (i) the eigenstates $psi(mathbf{r})$ and eigenenergies $E$ of the Hamiltonian obey $|psi(mathbf{r})|leq Eu(mathbf{r})$, so that the localization landscape function $u$ delimits localization regions in space, or that (ii) the function $W(mathbf{r})=1/u(mathbf{r})$ acts as a confining potential that controls quantities like the exponential decay of Anderson localized states.

The localization landscape theory has recently been adopted in materials simulations of disorder. This series of three papers provides a very comprehensive introduction to the theory, and a demonstration of what is possible:

1. Basic theory and modelling: paper.
2. Urbach tails of disordered InGaN alloy quantum well layers: paper.
3. Carrier transport and recombination in light emitting diodes: paper.

Better late than never for a partial answer.

I have worked with modeling the disorder of coverage on surfaces of materials. Recently we published a paper where we give a very general approach to modeling adsorbate-adsorbate interactions which we have seen is a problem in current literature. By using non-ideal buildups of coverage (where each adsorbate is not equally spaced), we have been able to identify better models of NO coverage on Pt₃Sn. You can see below that at accessible high coverages, the structure is heavily disordered and would be incredibly difficult to find.

A large problem with modeling disorder is getting any useful information out of it. This problem needs to be solved before very disordered models will give better results than ideal models. In contrast to the comment by Camps, I don't feel making a good disordered model is difficult, the hard part is understanding what part of the disorder is actually relevant to the result.