CCL: Proper scaling of HF exchange for hybrid functionals

[Kjell Jorner <kjell.jorner*o*gmail.com>]

Thanks for the feedback! What I am looking for is a good way to scale the HF exchange applicable to the range 0-100 rather rather than developing a new functional. I agree that the way forward in functional development is not modifications of B3LYP.

The approach I am looking for would mainly be used for assessing sensitivity of a certain property to HF exchange. What I am trying to get at is that there are two approaches for doing this in the literature, and as far as I have seen in the original papers and in the reviews that Marcel lists, there is no motivation for choosing one over the other. My conclusion so far is that "percentage of HF exchange" is defined in relation to either (1) Slater exchange or (2) total DFT exchange, including corrections from GGA etc. No distinction between these two definitions seems to be made in the literature, and they are freely mixed e.g. in tables when comparing the % HF exchange of different functionals.

Maybe there is no definitive theoretical justification for choosing one of these definitions over the other as we are dealing with density functional approximations. Personally, I think that approach 2 seems more reasonable as it gives no DFT exchange component at all in the limit of 100% HF exchange. This is also in line with Becke's original implementation of the adiabatic connection method in his half-and-half functional (where B88 is the combination of Slater and GGA exchange correction):

E_XC = 0.5*E_X(HF) + 0.5*E_X(B88) + E_C(LYP)

Best,

Kjell

Den tors 27 juni 2019 kl 17:37 skrev Martin Kaupp martin.kaupp() tu-berlin.de <owner-chemistry ~ ccl.net>:

Hi all,

In addition to the mentioned constant HF exchange admixture ("global hybrids") and range-separated functionals ("RS hybrids"), where the admixture depends on the interelectronic distance, let me mention a third possibility that has so far received less attention: in "local hybrid functionals" the HF exchange admixture is different at each point in (real) space, governed by a so-called "local mixing function" (LMF). Most of the additional implementation challenges for such local hybrids have been overcome, and a set of "first-generation" local hybrids are available in Turbomole since version 7.3  (in the upcoming version 7.4, parallelization and some more possibilities have been enabled), which for example provide outstanding performance for triplet excitations in TDDFT but also give remarkably good thermochemistry and barriers. We are continuing to work on improved LMFs and functionals, on excited-state gradients, NMR shifts and spin-spin coupling constants, on a relativistic two-component implementation of many properties, and a comprehensive review has appeared recently: 

Local Hybrid Functionals: Theory, Implementation, and Performance of an Emerging New Tool in Quantum Chemistry and Beyond T. M. Maier, A. V. Arbuznikov, M. Kaupp WIREs Comp. Mol. Sci. 2019, 9, e1378. DOI: 10.1002/wcms.1378.

Many regards,

Martin Kaupp

Am 27.06.2019 um 09:49 schrieb Susi Lehtola susi.lehtola#%#alumni.helsinki.fi:

Sent to CCL by: Susi Lehtola [susi.lehtola|,|alumni.helsinki.fi]
On 6/26/19 1:10 PM, Kjell Jorner kjell.jorner/agmail.com wrote:

Hello,

I have a question about the best way to scale HF exchange in a hybrid functional. For example, B3LYP features three sources of exchange:
1. Exact HF exchange
2. Slater exchange
3. GGA correction to Slater exchange

The approach taken by Becke in his original B3-paper from 1993 is to have one parameter that scales HF and Slater exchange so that the total is unity. A second parameter controls the amount of GGA exchange correction. My interpretation is that in this way, the GGA correction is optimized in a semiempirical manner together with the admixture of HF exchange. He writes "Clearly, the coefficient a_x has value less than unity, since the presence of the E_x_exact term reduces the need for the gradient correction Delta_E_X_B88."

In the literature, there are two approaches two scaling the HF exchange in B3LYP:
1. Adjusting only the balance between HF and Slater exchange, keeping the GGA exchange correction fixed. This is exemplified by the B3LYP* functional which uses 15% HF exchange with an unchanged 72% GGA correction (Hess, 2002).
2. Adjusting the balance between HF and Slater exchange, as well as scaling the GGA exchange correction accordingly (Kulik, 2015).

 From my intuition, it does not make sense to have a GGA correction in the limit 100% HF exchange. Method 2 would therefore be preferred when one wants to assess the effect of HF exchange over a large range. Does anyone have any comments or are aware of any literature on this topic?

B3LYP is old, as has been established many times on this list. Instead of fixing the functional form beforehand (what you are repeating above), the proper way to optimize is to adjust everything simultaneously - including the funtional form - see e.g. the papers on combinatorially optimized functionals (wB97X-V, B97M-V, wB97M-V) by Mardirossian and Head-Gordon.

For a more usual, limited use case, one just scales between full DFT exchange and exact exchange, possibly in a range-separated manner (e.g. long-range only); this may give you information on e.g. self-interaction errors.

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