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This question came to mind while writing another question here Extended Hybrid Methods, but I felt it was distinct enough to ask separately.

In double hybrids DFT methods, you essentially perform a hybrid DFT calculation (DFT plus Hartree-Fock exchange) and then perform an MP2 calculation using the orbitals from the hybrid calculation. The energies from both these calculations are weighted and combined to give a final energy.

I'm curious about this last step; beyond fitting, are there heuristics that can be used to set the correlation coefficient for a particular type of problem? Obviously there is no exact answer for picking how much correlation comes the DFT part and how much from MP2, but what sort of considerations go into that decision? I know for regular hybrids different ratios of exchange are chosen depending on the problem they hope to address with that functional. For example, the original PBE0 paper makes an argument based on perturbation theory that ~1/4 HF exchange should be the best for atomization energies. I was wondering if there were similar physical rationale for setting the correlation ratio for double hybrids.

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It's determined by fitting. You optimize the functional (i.e. the coefficients therein) to yield the lowest errors possible.

See e.g. the wB97M(2) functional for an example of a recent double hybrid.

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    $\begingroup$ Thank you the answer! I know in a lot of cases these weights are optimized, but these parameters are occasionally chosen with some physical motivation. For example, the hybrid M06-HF, PBE0, and HSE06 all set the ratio without additional fitting, citing a desire to give the functional some property (e.g. the original PBE0 paper makes an argument based on perturbation theory that ~1/4 HF exchange should be the best for atomization energies). I was wondering if there were similar rationale for setting the correlation ratio for double hybrids. $\endgroup$
    – Tyberius
    May 17 '20 at 20:25
  • $\begingroup$ @Tyberius in the 2006 Grimme paper (the first double-hybrid paper) that I gave a link to in my "DFT milestones" answer, the coefficient $c$ is determined by a fit, as Susi says here. Perhaps your question could be: "Is there any double-hybrid where a fixed ratio is used rather than a fitted parameter?" I know for single-hybrids this is the case for PBE0, which is 3/4 PBE and 1/4 HF. I don't know anything about double-hybrids other than what I learned while writing that Milestones answer, so I only know B2PLYP which uses fitting rather than a fixed ratio. $\endgroup$ Jun 4 '20 at 3:51
  • $\begingroup$ @NikeDattani I tried to edit it to make my intention a little clearer. $\endgroup$
    – Tyberius
    Jun 4 '20 at 4:04
  • $\begingroup$ @Tyberius I had just finished sending Stefan Grimme an email asking for an answer to your other question (extended hybrids) and the Milestones question. Now that you've changed the question a bit, I could send him this one too: I didn't include this in the email because I believe the answer really was as basic as "It's determined by fitting", but now it's actually a very interesting question that I'm curious to know the answer to! Something like the 3:1 ratio in PBE0... is there such a thing for any double hybrids? Good question. $\endgroup$ Jun 4 '20 at 4:10
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Let's define the energy obtained by a double-hybrid as $$ E^\text{DHDF}_\text{xc} = a \cdot E^\text{exact}_\text{x} + b \cdot E^\text{(m)GGA}_\text{x} + c \cdot E^\text{(m)GGA}_\text{c} + d \cdot E^{\text{PT2}}_\text{c} $$ where the weighted summands are the energy contribution of exact/Fock exchange, the (meta)-GGA DFT exchange energy, the (meta)-GGA DFT correlation energy, and the second-order perturbative correction$^1$.

Many researchers are of the opinion that the coefficients for exchange and correlation should sum to unity, respectively: $$ a + b = 1 \\ c + d = 1 $$ which seems valid as long as the functionals reproduce some exactly known values, such as for the uniform electron gas.

Some researchers have suggested that there is a relationship between $a$ and $d$, as well: $$ a^n = d $$ where it is a matter of debate whether the exponent $n$ should be $2$ or $3$. For proponents of $3$, see work by Brémond and Adamo) wherein they claim to have found a parameter-free double-hybrid by using $a = 0.5$, $d = 0.125$ and the PBE functional. Sharkas, Toulouse, Savin suggested $2$ instead. Note that the first double-hybrid functional, B2PLYP, leans towards $2$ with $a = 0.53$ and $d = 0.27$ (though it was largely fitted).

So while there is a large amount of fitting going on, and most of the most successful DHDFs have been found this way, attempts to find a physical motivation exist.


$^1$ Some people say that it's not exactly MP2 because that would start from HF orbitals.

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  • $\begingroup$ ... but since there's many ways in which one can argue the exponent could be chosen, you end up with a fitting procedure albeit with a constraint to integer $n$.. $\endgroup$ Jun 4 '20 at 7:00
  • $\begingroup$ Another reason why it's not the traditionally known MP2 is because single excitations are ignored (at least in the case of Grimme's 2006 paper). It is still a 2nd order perturbation thery based on the idea of Møller & Plesset though. $\endgroup$ Jun 4 '20 at 14:39
  • $\begingroup$ So the answer is yes: 1/2 , 1/2 , 1/8 , 7/8, but the physical motivation is not given (except maybe inside the referenced paper, but not in this answer). $\endgroup$ Jun 4 '20 at 14:43
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In most of double hybrid KS-DFT approaches, there is a disconnect between xc potential obtained from xc functional. As correlation corresponding to the MP2 part is not included in the self-consistent procedure. Thus one did not improve the KS density due to correlation functional of MP2. Parameters used in double-hybrid has no physical motivation except fitting to certain data. Physically motived way to construct double hybrid is described in paper titled "Ionization potential optimized double-hybrid density functional approximations" https://aip.scitation.org/doi/abs/10.1063/1.4962354

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