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:
- One of the first questions ever asked on this site was: "What is the closest thing we have to "the" universal density functional?"
- We can't yet theoretically predict the ground state hyperfine splitting of the H atom as accurately as we can measure it.
- What is the "engineering complexity" for building a universal quantum computer?
- Have anyons been confirmed to exist? (this one's for you Anyon!)
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?