There's two facets to this question:
- What methods can be used for excited states in crystals? (the title, and final sentence)
- What methods can be used for excited states with plane waves? (paragraphs 1 and 3)
I will begin by answering the second question, which is in some sense grounded by this assumption:
"it is not possible, as far as I know, to do plane-wave based
"configuration interaction" calculations."
While CI calculations on plane-wave basis sets are rare, there are in fact examples, and while this Nature paper might not be the only example, it is the one with which I'm most familiar. Here is a quote from the paper (emphasis on the plane wave part was added by me):
"The determinants in this work are composed from antisymmetrized
products of one-electron orbitals obtained from a prior Hartree–Fock
calculation in a large basis of periodic plane waves within the
framework of the projector-augmented wave method, as implemented in
VASP."
Since that quote was talking about FCIQMC, I will provide a reference for how FCIQMC can treat excited states quite straightforwardly, but you might agree that if FCIQMC can be used with plane waves, then other CI approaches (along with excited-state extensions) can too.
The Nature paper also shows results for solids, using MP2 and coupled-cluster, which have excited-state extensions such as EOM-CC (a.k.a. LR-CC), STEOM-CC and Fock-Space CC. On the topic of coupled-cluster, I will also say that two of the same authors from the Nature paper, also describe a "pseudized Gaussian" approach for treating periodic systems, in this paper. Pseudized Gaussians were used in numerous papers which treat periodic systems with coupled cluster, many of them references in this recent review article on coupled cluster for materials science.
This conveniently allows me to transition into answering the first of the two questions listed at the top of this answer, which is the one which doesn't mention plane wave sets explicitly. Indeed excited state coupled cluster calculations have been done using Gaussians, for example in this paper in which the opening line of the abstract is:
We present the results of Gaussian-based ground-state and
excited-state equation-of-motion coupled-cluster theory with single
and double excitations for three-dimensional solids.
One of the authors of that paper also published an earlier paper in Science on crystalline benzene, which uses not only coupled-cluster but also DMRG with Dunning basis sets.
Conclusion: In addition to TDDFT or other approaches to DFT involving excited states, and the methods based on the GW approximation and the Bethe-Salpeter equation, which Geoff said in the comment he might elaborate more on, DMRG, CI, CC, and their excited-state extensions can be used for plane waves, pseudized Gaussians, or even pure Gaussians, even for periodic systems.
