From owner-chemistry@ccl.net Thu Jun 27 04:01:00 2019
From: "Susi Lehtola susi.lehtola#%#alumni.helsinki.fi" <owner-chemistry,,server.ccl.net>
To: CCL
Subject: CCL: Proper scaling of HF exchange for hybrid functionals
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X-Original-From: Susi Lehtola <susi.lehtola~!~alumni.helsinki.fi>
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Date: Thu, 27 Jun 2019 10:49:24 +0300
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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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