How can free of self-interaction error easily be concluded in DFT?

[bsmile]

I saw in a reference "For a system consisting just of one electron, DFT predicts an non-physical self-interaction energy." (Theor Chem Acc (2009) 123:171–182), and thus wrong single electron energy. Does this mean I can quickly conclude whether a specific exchange-correlation functional has been made self-interaction error free by doing a single hydrogen calculation to see whether it is able to reach the correct hydrogen atomic energy?

 

[cgk]

No. The self-interaction error in DFT cannot be fixed---at least not without deteriorating DFT's accuracy into nothingness. The reason for this is that there is a balance of errors---the self-interaction error actually reduces DFT's susceptibility to making errors with strong correlation. And everything you can do is shift the errors from one side to the other... (essentially, between local DFT and full Hartree-Fock).

In short: in 2014 there is *not a single functional in existence* which can treat both the H2^+ cation and the H2 molecule correctly (e.g., compute a correct dissociation curve for both). And it is very unlikely that this will ever change.

If you are interested in these topics, I would recommend you to read up some of the recent work of Aron Cohen, who has done a fantastic job of pointing out the relationships between different problems of DFT.

 

[bsmile]

Thanks for the message. Does this mean the full Hartree-Fock does not have self-interaction error while local DFT has?

 

[bsmile]

Meanwhile, in the paper I cited above, it specifically mentioned " A thorough analysis in the framework of
DFT was performed by Perdew and Zunger [43], who suggested an explicit correction term that makes any XC-functional SIE-free. An alternative way of avoiding the SIE of approximate XC-functionals was suggested by Colle and Salvetti already in 1975 who used wave function theory (WFT) to develop the first SIE-free C-functional [23]. The Colle–Salvetti (CS) functional laid the basis for the work of Lee et al. to develop the LYP correlation functional, [9] which is also SIE-free." How to possibly understand that?

 

[bsmile]

I know of Aron Cohen's work. I enter the DFT field quite recently. Too much information flushes to me which sometimes is quite confusing.

 

[cgk]

Thanks for the message. Does this mean the full Hartree-Fock does not have self-interaction error while local DFT has?

That is right. While Hartree-Fock does have problems, electron self-interaction is not one of them. For example, Hartree-Fock is exact for the H_2^+ cation, unlike all real world density functionals.

Meanwhile, in the paper I cited above, it specifically mentioned " A thorough analysis in the framework of
DFT was performed by Perdew and Zunger [43], who suggested an explicit correction term that makes any XC-functional SIE-free.(...)

This self-interaction correction does indeed reduce a major part of the self-interaction error of KS (but not all of it--only a part of the 1e parts). The problem with this correction is two-fold: (1) It massively deteriorates the practical performance of DFT. For example, thermochemical results obtained with DFT protocols modified as such are even worse than HF's. (2) Whether the resulting method should still be called "DFT" is debatable. They claim it is a theory. I would call it a hack.

(...)An alternative way of avoiding the SIE of approximate XC-functionals was suggested by Colle and Salvetti already in 1975 who used wave function theory (WFT) to develop the first SIE-free C-functional [23]. The Colle–Salvetti (CS) functional laid the basis for the work of Lee et al. to develop the LYP correlation functional, [9] which is also SIE-free." How to possibly understand that?

I do not quite get what they mean with this. The CS functional is indeed parameterized on correlation energies which are self-interaction free[1] (it is fitted to correlation energies of noble atoms). This also applies for LYP, which is maybe best described as a reparameterization of CS to make it easier to apply in actual calculations. However, this SIE-freeness of the correlation functional does not actually say much: First, it is highly debatable whether in strict DFT you can separate XC contributions into C and X contributions (and n-representability contributions, which are often omitted). This is often done for practical reasons, but there is no strict basis for this in theory---in theory there is only one XC functional. Second, one would always expect the dominant self-interaction error contribution to come from the X functional, not the C functional. So even if CS/LYP, which are pure C-functionals, would be strictly self-interaction free, this would not help at all, as you still need a X from somewhere. As a result, for example, BLYP has massive self-interaction errors---even if they come from B instead of LYP. (and if you add CS or LYP to Hartree-Fock, the results are bad again)

I know of Aron Cohen's work. I enter the DFT field quite recently. Too much information flushes to me which sometimes is quite confusing.

I was in the same boat :). The foundations of DFT can be very confusing, and they are often misrepresented. To be honest, I have since adopted the more practical approach of completely ignoring them and treating DFT as some kind of hacked up Hartree-Fock---which indeed is how it is used in practice in >99% of the cases, even if DFT evangelists would tell you otherwise.

As far as the "in principle exact"-ness is concerned: It also may help to consider that with the same argumentation as in, say, Levy's constrained search formalism of DFT, you can show that classical force fields are in principle exact. You just have to vary over the full-CI wave function for a given set of atomic coordinates... and, voila: You have an exact energy for each configuration of atomic positions. And how does that help? Well... not very much, actually. It helps to check arguments in DFT if they really go beyond this kind of "exactness".



[1] Whether these are the correlation energies a DFT functional should be parameterized on is a different question: In some sense they implicitly assume that the KS determinant describes a *real* electronic wave function, not some kind of fake auxiliary system which just produces the same density (the latter is the "official" interpretation of DFT's KS system).