I list some suggestions of possible answers:
* Size extensitivity and/or Size consistency of DFT method
* Perturbation approaches for DFT methods
* DFT methods that scale linearly with system size
* Expressing properties as functionals of the electron density and/or develop such functionals
* Can one include geminals in some way into DFT?
* Is it possible to express the electron density as explicit functional of the external potential?
* Is a DFT possible that uses more than one density, e.g. besides the electron density also the proton density?
* Second quantisation approach to DFT: How can one use second quantisation methods for DFT? Can one formulate or analyze density functionals in terms of the second quantised Hamilton operator, especially for the ab initio Hamiltonian in a finite basis set?
* Can one express the two-particle density as a functional of the one-particle density (exactly and/or approximately)?
* Do we need for parametrising new density functionals new approaches, e.g.,
* Monte-Carlo calculations of the electrons in a -Z/r potential in addition to a uniform background needed for charge neutrality
* results of correlated quantum chemistry methods like MPn or Coupled Cluster, Full CI or MCSCF in large basis sets? (instead of/in addition to Monte Carlo)
* special functionals for specific basis sets that are used in state-of-the-art quantum chemical calculations
* a better understanding of the relation of DFT to chemical concepts like Valence Bond, bonds as electron pairs, van der Waals interactions, vibronic coupling, dissociation and ionization of molecules
* a better understanding of the relation of DFT to the Born-Oppenheimer approximation and its breakdown
I add the possible answer
* Constrained DFT (an example is DFT+U), see for instance the link provided below.
https://arxiv.org/abs/1309.6981
Another possible answer might be
* DFT with magnetic fields:
* magnetic field density functional theory (BDFT) (see the first link)
* current density functional theory (CDFT/Current-DFT) and Time-dependent Current-DFT (see e.g. the second link)
https://dx.doi.org/10.1103%2FPhysRevA.50.3089
http://dx.doi.org/10.1093/acprof:oso/9780199563029.003.0010
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