# CCL:G: two-electron range-separated Coulomb and exchange integrals

*From*: Susi Lehtola <susi.lehtola|alumni.helsinki.fi>
*Subject*: CCL:G: two-electron range-separated Coulomb and exchange
integrals
*Date*: Mon, 28 Sep 2015 15:15:18 -0700

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.
` --
-----------------------------------------------------------------------
Mr. Susi Lehtola, PhD Chemist Postdoctoral Fellow
susi.lehtola#alumni.helsinki.fi Lawrence Berkeley National Laboratory
http://www.helsinki.fi/~jzlehtol USA
-----------------------------------------------------------------------