Physically motivated double hybrid DFT?

Question

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.

Answer 1

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.

Answer 2

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.

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

Answer 3

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