# Hybrid functional calculations using different approach

In hybrid functional such as B3LYP, PBE0, HSE etc, some part of HF exchange and/or correlation is added to some part of DFT X/C part in order to get better results. My question is if I am adding 1/4 th of HF exchange part that means I already calculated HF exchange part and HF is undoubtedly better than DFT, then why don't we just use the full HF solution that we already have in our hand?

• I guess the theoretical reason is, the artificial system of KS is not the mean field. Hence, the exchange effect is not exactly the same as wave-function based schemes. – Rodriguez Jun 9 '16 at 2:32
• HF is usually less accurate than DFT because it lacks electron correlation – Jan Jensen Jun 9 '16 at 5:45

In hybrid functional such as B3LYP, PBE0, HSE etc, some part of HF exchange and/or correlation is added to some part of DFT X/C part in order to get better results.

First, there is no correlation in HF, only exchange, so you can not add part of the HF correlation into DFT, as you said. In principle, of course, if you want to add not only exchange from the wave function theory (WFT) methods but also some correlation, you could use other that the HF sources, but the conventional hybrid DFT corresponds to mixing in just (a part of) the HF exchange.

Now, you very question of

why don't we just use the full HF solution that we already have in our hand?

is addressed in almost any textbook on DFT. I suggest that you read, for instance, Section 6.6 Hybrid Functionals in Koch & Holthausen, specifically, pages 78-80. The authors provide a relatively simple argument for for the well-known failure of the exact exchange/density functional correlation combination approach.

In short, the source of the problem is that the separation of an exchange-correlation entity (would it be energy, or hole, or something else) into exchange and correlation contributions is artificial. Only the whole exchange-correlation entity has a physical meaning, while the above mentioned individual contribution are introduced just for convenience. Usually, it means that exchange and correlation contributions has to be chosen consistently so that their sum, an exchange-correlation entity, has some desired properties.

For instance, the exchange-correlation hole in molecular systems is usually relatively localized, so that the exchange hole and the correlation hole must both be either localized or delocalized (in a very special way so that combined they result in a localized hole). In LDA- and GGA-DFT both the exchange and the correlation hole are localized, but when we mix HF exchange and LDA/GGA-DFT correlation in the most straightforward way, i.e., $$E_{\mathrm{XC}} = E_{\mathrm{X}}^{\mathrm{HF}} + E_{\mathrm{C}}^{\mathrm{DFT}} \, ,$$ then, I quote:

We combine the exact, delocalized exchange hole with a localized model hole for correlation. Because the cancellation between the two individual holes cannot take place, the resulting total hole has the wrong characteristics.

1) Koch, W., & Holthausen, M. C. (2001). A Chemist's Guide to Density Functional Theory, Second Edition, Wiley-VCH Verlag GmbH.

As already noticed by Jan, Hartree-Fock is a poor theory, since it completely neglects correlation effects. DFT is in principle an exact theory but in practice its accuracy depends on the approximations and the self-interaction is not exactly cancelled as it is in Hartree-Fock theory. Both approximations have strengths and shortcomings and it is by combining them to some extent that we can obtain better results.

A striking example of why we don't use the whole Fock exact exchange in hybrid functionals is found in solid-state physics. Semilocal approximations (LDA and GGA) largely underestimate the band gap of semiconductors and insulators: this is known as the band gap problem of DFT. Hybrid functionals can be used to reproduce the experimental band gap by tuning the portion of exact exchange taken into account (1/4 is absolutely not a magic number and higher or lower mixing coefficients can and should be used). The portion of exact exchange needed to reproduce the experimental band gap depends on the system, but it is never 100%: this is because Hartree-Fock theory tends to overestimate the band gap.

[1] A. Alkauskas, P. Broqvist and A. Pasquarello, Phys. Status Solidi B, 248, No. 4, 775–789 (2011).