What are the great unsolved questions in Matter Modeling?

Relativistic correlation methods are another interesting topic: usually one employs the no-pair approximation, which doesn't correlate the negative energy states. However, there's really no reason why the negative energy states shouldn't experience correlation effects, as well...

I think there's been pretty good effort recently for the automatic selection of active spaces with the DMRG method, see J. Comput. Chem. 40, 2216 (2019). Somewhat similar approaches have also been used in earlier works, e.g. J. Chem. Phys. 140, 241103 (2014) ran large-active-space calculations to figure out a smaller active space in which the production-level calculations were run.

As to the beyond Gaussian orbitals question, numerical atomic orbitals (NAOs) are pretty good for this when combined with density fitting approaches; e.g. here's a RI-CCSD(T) study with NAOs: J. Chem. Theory Comput. 15, 4721 (2019).

High-temperature superconductivity

Superconductivity is a fascinating macroscopic quantum phenomenon in which, as some material is cooled below a critical temperature, its electrical resistance abruptly vanishes. A superconductor can also expel magnetic flux, which allows levitation effects as shown in the picture above. The conventional form of superconductivity was first discovered in Mercury in 1911 by Heike Kamerlingh Onnes, but it took until 1957 for the microscopic Bardeen-Cooper-Schrieffer (BCS) theory to explain its origin. In short, electrons form bound states called Cooper pairs, due to an effective attractive interaction mediated by phonons. However, there is a less conventional, less understood cousin known as high-temperature superconductivity, or high-$T_c$ superconductivity.

It is mentioned both on Wikipedia's unsolved problems in physics page and on the unsolved problems in chemistry page, but it equally applies to the study of matter. Since the 1986 discovery by Bednorz and Müller of superconductivity in a copper oxide, with a transition temperature of $35$ K (high for superconductors!), there's been an immense amount of experimental, computational and theoretical activity in the field. The goals are manifold, including finding a room temperature superconductor, and to understand the mechanism. Often these systems are very complex, formed from multi-layered crystals, and involve some degree of doping and electron-electron interactions, making their modeling a complex task indeed.

Promising computational avenues include accurate simulations of model Hamiltonians (e.g. Hubbard Hamiltonians) in an effort to find the mechanism, and the ongoing development of suitable ab initio methods to model these systems. At this point, I personally think that such approaches represent the most likely path to understanding these materials, barring some breakthrough. However, that doesn't mean progress has stopped elsewhere. For example, additional clues keep coming in from experiments establishing new classes of superconducting materials, and surprising transport properties.