Introduction

Although DFT is very successful for a wide set of problems, it suffers from a number of key issues. One of the more significant problems is the derivative discontinuity error. The derivative of the electronic energy of a quantum system can be discontinuous with respect to different degrees of freedom. This is a consequence of the integer nature of electrons. If the calculated energy of the system does not fulfill this physical discontinuity requirement, the error manifests itself in various manners (even when $N$, the number of electrons in the system stays constant). For example:

• The complete failure to give the total energy of H₂ and H₂⁺,
• The missing gap in Mott insulators such as stretched H₂
• The thermodynamic limit of the 1D Hubbard Model
• A qualitatively incorrect density in the HZ molecule with two electrons and incorrect electron transfer processes.
• etc.

Currently, all approximate functionals in DFT, including hybrids, miss the derivative discontinuity. There are some workarounds. The reference in [1] is a good review on the matter.

Situation in GW

GW, due to how it is constructed, should not suffer from this issue. As so DFT. However, in practice, both suffer from it. The problem in current GW implementations is the incorrect handling of the correlation [2-3]. This is similar to $V_xc(\mathbf r)$ being not correct in DFT. The advantage of GW over DFT is that, if you can afford it (like, you have unlimited access to Tier0 resources afford it, or working on Hydrogen and nothing else), you can invest in 4 point vertex correction terms. 

Hence, the problem goes unsolved, unless a more clever method for calculating the corresponding correlation diagrams is found.

References:

1.  Paula Mori-Sánchez  and  Aron J. Cohen Phys. Chem. Chem. Phys., 2014,16, 14378-14387
2. Nils Erik Dahlen, Robert van Leeuwen, and Ulf von Barth Phys. Rev. A 73, 012511
3. Adrian Stan, Nils Erik Dahlen, and Robert van Leeuwen J. Chem. Phys. 130, 114105 (2009)
4. J. M. Tomczak, Ph.D. thesis, Ecole Polytechnique, France (2007).
5. C. Verdozzi, R. W. Godby, and S. Holloway Phys. Rev. Lett. 74, 2327