Kohn-Sham-DFT always uses approximations for exchange and correlation energy. Due to these approximations, it may happen that these energies involve a contribution in which an electron formally interacts with its own field in an unphysical way. This error is found in basically every functional, but in some it's negligibly small while for other functionals it spoils the results - these are the ones for which you read the comment you cited.

Jürgen Weippert

Too many thanks for your nice and informative comments.

Saeed

Self interaction arises from including all occupied orbital densities in the Hartree potential of any single-particle approach to many-body quantum mechanics. This can allow the computational cost of the single-particle method to be reduced significantly, but introduces a spurious contribution to the Hartree potential for every occupied single-particle orbital, where each orbital feels the Hartree potential of its own density.

Advanced single-particle methods can remove this spurious 'self-interaction' by including the exchange interaction, such as Hartree-Fock. Within Kohn-Sham theory, as Jürgen Weippert explains, the exact exchange-correlation (xc) potential removes self-interaction completely. However, in practice, owing to the need to approximate the xc part of the Kohn-Sham potential, the full self-interaction is not removed, leading to the infamous self-interaction error. Hence, the size of this error corresponds to the reliability of the approximation to the xc potential.

Matthew J. P. Hodgson

Many thanks for your kind attention to clarifying this issue perfectly and completely.

Sincerely,

Saeed

Self interaction error is an error present for density functional approximations within density functional theory (DFT). When we approximate the electronic electrostatic potential as a functional of the electron density (known as the Hartree potential), we consider a single particle (an electron) interacting with a classical charge distribution which, unfortunately, includes the very electron in question.

Matthew J. P. Hodgson 's answer is not quite correct as Hartree-Fock theory (the quintessential single particle theory, arguably) is completely free of self interaction error as we consider explicit interactions between single particle orbitals: the evaluation of the two-electron integrals. Such a computation is less efficient than the calculation of the Hartree potential in DFT, however.

Self interaction error actually brings about the most important discrepancy between DFT and Hartree-Fock theory. If we choose to work without an approximate exchange and correlation functional in DFT, the missing contribution to the exact ground state total energy of the many-body system comes from the missing exchange and correlation energy but also the self interaction error. Hartree-Fock theory, on the other hand, exactly calculates the exchange energy and features no self interaction error. The only missing energy is the correlation energy.

This is a source of some frustration for me as it means that, in the language of many-body wavefunction theorists (i.e, those using Hartree-Fock and post-Hartree-Fock methods), the exact "exchange and correlation" functional in DFT is actually an "exchange, correlation and self interaction correcting" functional.

@Jack S. Baker

Many thanks for your detailed answer.

Jack S. Baker Not sure what you mean when you say my answer is "not quite correct". I wrote about Hartree-Fock theory: "Advanced single-particle methods can remove this spurious 'self-interaction' by including the exchange interaction, such as Hartree-Fock."

Hi Matthew J. P. Hodgson . Let me clarify. I do not think the statement "Advanced single-particle methods can remove this spurious 'self-interaction' by including the exchange interaction, such as Hartree-Fock" is correct.

Perhaps you find this pedantic, but Hartree-Fock theory doesn't "remove" self interaction error. It just was never present in the theory to begin with. Because the electrostatic potential is not approximated using the Hartree potential (the coulomb integral is used instead), and the exchange integrals exactly cancel any would-be self interaction, it simply doesn't appear.

Perhaps you are referring to Hybrid DFT and not Hartree-Fock theory in isolation? Here, there is some approximate removal of self interaction error.

Jack S. Baker I take your point. However, for the purposes of addressing the original question, my answer stresses that it is the inclusion of all occupied orbital densities in the Hartree potential which gives rise to self interaction.

The Hartree-Fock single-particle Hamiltonian contains the Hartree potential, so the self-interaction error would be present if it were not removed by the Fock exchange term. While it is accurate to state that there is no self-interaction error in the Hartree-Fock method, such a statement alone does not offer the same depth of insight.

This cancellation of the Hartree self interaction with the Fock exchange term is the reason why the self-interaction correction is considered part of the exchange potential in Kohn-Sham theory. Kohn and Sham disassembled their effective potential into the external potential, the Hartree potential (derived from Hartree theory) and the exchange-correlation potential owing to the historical progression of single-particle approaches. As the Fock exchange term removes self interaction in Hartree-Fock theory, the self-interaction correction is classed as 'exchange' in the Kohn-Sham approach.