From owner-chemistry _-at-_)ccl.net Mon Sep 28 19:46:01 2015
From: "Susi Lehtola susi.lehtola]=[alumni.helsinki.fi" <owner-chemistry^-^server.ccl.net>
To: CCL
Subject: CCL:G: two-electron range-separated Coulomb and exchange integrals
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X-Original-From: Susi Lehtola <susi.lehtola|alumni.helsinki.fi>
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Date: Mon, 28 Sep 2015 15:15:18 -0700
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Sent to CCL by: Susi Lehtola [susi.lehtola%alumni.helsinki.fi]
On 09/28/2015 01:39 PM, Thomas Manz tmanz],[nmsu.edu wrote:
> Hi,
>
> The range-separated Coulomb integral
>
> 1/r = erf(k*r)/r + (1-erf(k*r))/r
>
> arises in range-separated hybrid functionals.
> There are many papers about functionals that use this kind of
> range-separation.
> My question is regarding the 2-electron integral terms (i.e., exchange
> and Coulomb integrals)
> using this type of range separation. Does anyone know where the analytic
> forms for this have been published?

Range-separation is not used for Coulomb integrals. It's only used for 
the exchange integrals.

> They should be of the form
> <g1(r1)*g2(r1)*(erf(k*r)/r)*g3(r2)*g4(r1)>
> where <> means integrate over positions r1 and r2 and g1, g2, g3, g4 are
> Gaussian basis sets.
>
> We are actually trying to do the analogous Coulomb/exchange integrals
> for electrons confined to a plane (i.e., two-dimensional Gaussians), but
> we maybe could figure out how to do this by looking at the
> three-dimensional case.
>
> I would be grateful if anyone could provide references on the
> mathematical derivations of the analytic integrals for the
> three-dimensional case.

Implementations in Q-Chem and Gaussian are documented AFAIK in
   Adamson, Dombroski, Gill; JCC 9, 921 (1999)
and
   Heyd, Scuseria, Ernzerhof; JCP 118, 8207 (2003)

Both are based on a trivial modification of the PRISM algorithm for 
range-separated functionals. The only change is that the recursion 
relations are modified a tiny bit.

Obara-Saika recursion relations can also be modified in the same manner: 
just by a small change to the (00|00)^m auxiliary integrals, where you 
get a second Boys function term with a slightly different prefactor.
-- 
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Mr. Susi Lehtola, PhD             Chemist Postdoctoral Fellow
susi.lehtola#alumni.helsinki.fi   Lawrence Berkeley National Laboratory
http://www.helsinki.fi/~jzlehtol  USA
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