CCL: Proper scaling of HF exchange for hybrid functionals

 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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 Mr. Susi Lehtola, PhD             Junior Fellow, Adjunct Professor
 susi.lehtola===alumni.helsinki.fi   University of Helsinki
 http://susilehtola.github.io/     Finland
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 Susi Lehtola, dosentti, FT        tutkijatohtori
 susi.lehtola===alumni.helsinki.fi   Helsingin yliopisto
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