We review the role of self-consistency in density functional theory. We apply a recent analysis to both Kohn-Sham and orbital-free DFT, as well as to Partition-DFT, which generalizes all aspects of standard DFT. In each case, the analysis distinguishes between errors in approximate functionals versus errors in the self-consistent density. This yields insights into the origins of many errors in DFT calculations, especially those often attributed to self-interaction or delocalization error. In many classes of problems, errors can be substantially reduced by using 'better' densities. We review the history of these approaches, many of their applications, and give simple pedagogical examples.

With the growing complexity of systems that can be treated with modern electronic-structure methods, it is critical to develop accurate and efficient strategies to partition the systems into smaller, more tractable fragments. We review some of the various recent formalisms that have been proposed to achieve this goal using fragment (ground-state) electron densities as the main variables, with an emphasis on partition density-functional theory (PDFT), which the authors have been developing. To expose the subtle but important differences between alternative approaches and to highlight the challenges involved with density partitioning, we focus on the simplest possible systems where the various methods can be transparently compared. We provide benchmark PDFT calculations on homonuclear diatomic molecules and analyze the associated partition potentials. We derive a new exact condition determining the strength of the singularities of the partition potentials at the nuclei, establish the connection between charge-transfer and electronegativity equalization between fragments, test different ways of dealing with fractional fragment charges and spins, and finally outline a general strategy for overcoming delocalization and static-correlation errors in density-functional calculations.

Approximate molecular calculations via standard Kohn-Sham density functional theory are exactly reproduced by performing self-consistent calculations on isolated fragments via partition density functional theory [P. Elliott, K. Burke, M. H. Cohen, and A. Wasserman, Phys. Rev. A 82, 024501 (2010)]. We illustrate this with the binding curves of small diatomic molecules. We find that partition energies are in all cases qualitatively similar and numerically close to actual binding energies. We discuss qualitative features of the associated partition potentials.

The non-additive non-interacting kinetic energy is calculated exactly for fragments of H2, Li2, Be2, C2, N2, F2, and Na2 within partition density-functional theory. The resulting fragments are uniquely determined and their sum reproduces the Kohn-Sham molecular density of the corresponding XC functional. We compare the use of fractional orbital occupation to the usual PDFT ensemble method for treating the fragment energies and densities. We also compare Thomas-Fermi and von Weizäcker approximate kinetic energy functionals to the numerically exact solution and find significant regions where the von Weizäcker solution is nearly exact.

We demonstrate how these errors, which plague density-functional calculations of bond-stretching processes, can be avoided by employing the alternative framework of partition density-functional theory (PDFT) even with simple local and semi-local functionals for the fragments. Our method is illustrated with explicit calculations on simple systems exhibiting delocalization and static-correlation errors, stretched H + 2 , H 2 , He + 2 , Li + 2 , and Li 2 . In all these cases, our method leads to greatly improved dissociation-energy curves. The effective KS potential corresponding to our selfconsistent solutions display key features around the bond midpoint; these are known to be present in the exact KS potential, but are absent from most approximate KS potentials and are essential for the correct description of electron dynamics.