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This is inspired from an amazingly successful question on Operations Research Stack Exchange: What are the great unsolved problems in operations research?


Wikipedia has some huge lists of:

$\star$ But neither of them even mention the fact that the universal functonal in DFT is unknown! $\star$


Some great problems (not in the above lists!) were discussed in these answers:


Some great problems (not in either of the above lists, as far as I know!) are here:

  • Finding a multi-electron relativistic and quantum mechanical method:
    • the Schrödinger equation is non-relativistic,
    • the Klein-Gordon equation is relativistic but only works for spinless particles,
    • the Dirac equation is a 1-electron equation and only approximates QM to 1st order in $\alpha$,
    • the Dirac-Coulomb-Breit equation involves interacting electrons but is not invariant with respect to Lorentz transformations (it is no longer properly relativistic) and like the Dirac equation it is not properly quantum mechanical either since it is derive from first-order perturbation theory in the fine structure constant $\alpha$!
    • $\therefore$ There is no multi-electron, relativistic, QM equation like the above four for single e-.
  • High temperature superconductivity: For low-temperatures we have BCS theory, but for high-temperature superconductors we cannot even predict $T_c$ (the critical temperature).
  • How to get multi-reference coupled cluster working well like CCSD(T) for single-reference?
  • Can we come up with a black-box multi-reference method like CCSD(T) for single-reference?
  • Is there a robust way to automatically select active spaces?
  • How best to reach the CBS limit for post-SCF methods? How to solve the cusp problem?
  • How to go beyond Gaussian orbitals, and remain efficient?
  • Can a quantum computer demonstrate beating a classical computer in the modeling of matter?

Can you explain any of these, or perhaps discuss the most recent progress, in up to 3 paragraphs?


What are some other unsolved problems in the computational / theoretical study of matter, and can you explain them in up to 3 paragraphs?


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High-temperature superconductivity

High-Tc superconductor levitating above a magnet

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.

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    $\begingroup$ @NikeDattani Thanks. Regarding your edit, I think manifold is the correct word, not manyfold. At least according to the dictionary, only one of them means diverse. $\endgroup$
    – Anyon
    Jul 20 '20 at 22:21
  • $\begingroup$ You're right, I'm sorry about that! I got confused by "manifold" because I'm too used to seeing that word in the context of Reimannian manifolds in general relativity, or "manifold of electronic states" in quantum chemistry. How about multifold? $\endgroup$ Jul 20 '20 at 22:25
  • $\begingroup$ @NikeDattani On balance, I prefer manifold. But it does not matter much. If I saw the word in isolation I would probably first think of a topological space... It is a versatile word. $\endgroup$
    – Anyon
    Jul 21 '20 at 3:08
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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).

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