One of the biggest problems facing DFT is that of self-interaction: each electron effectively interacts with itself, because the potential derives from the total charge density of the system. This is not an issue for the exact (unknown) density functional, or for Hartree-Fock, but is the cause of significant error in many DFT functionals. Approaches such as DFT+U[1],[2],[3] and hybrid functionals (far too many to reference !) are aimed in part at fixing this problem.

Probably the earliest attempt to remove this error is the self-interaction correction of Perdew and Zunger[4] which corrects the potential for each Kohn-Sham orbital, complicating the calculation considerably over a standard DFT calculation. (Ironically, this paper, which has over 11,000 citations, is best known for its appendix C, where a parameterisation of the LDA XC energy is given.) However, this process is notoriously slow to converge and is not widely used.

A recent paper[5] showed that, even for isolated molecules, complex orbitals were required to achieve convergence, and this approach has now been tested for atomisation energies of a standard set of 140 molecules[6]. The tests compare the new complex SIC implementation against the standard, real implementation, as well as various GGAs, hybrid functionals and meta-GGAs. The complex SIC, when coupled with the PBEsol functional[7], gives good results (though ironically the PBEsol functional was developed to improve PBE for solids). Not surprisingly, the best results are from hybrids, but meta-GGA improves the energies almost as well.

This study highlights the problem with DFT at the moment: there are many different approaches, which often work well for specific problems. SIC is cheaper than hybrid calculations, and can be important for charge transfer problems (and Rydberg states). The results for convergence and complex orbitals are interesting, but based on these results, I would use meta-GGA for atomisation energies, as a good compromise between accuracy and cost (almost the same as GGA).

[1] Phys. Rev. B 52, R5467 (1995) DOI:10.1103/PhysRevB.52.R5467

[2] Phys. Rev. B 57, 1505 (1998) DOI:10.1103/PhysRevB.57.1505

[3] Int. J. Quantum Chemistry 114, 14 (2014) DOI:10.1002/qua.24521

[4] Phys. Rev. B. 23, 5048 (1981) DOI:10.1103/PhysRevB.23.5048

[5] J. Chem. Theory Comput. 12, 3195 (2016) DOI:10.1021/acs.jctc.6b00347

[6] J. Chem. Theory Comput. in press (2016) DOI:10.1021/acs.jctc.6b00622

[7] Phys. Rev. Lett. 100, 136406 (2008) DOI:10.1103/PhysRevLett.100.136406