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

 Sent to CCL by: Susi Lehtola []
 On 09/28/2015 01:39 PM, Thomas Manz tmanz],[ wrote:
 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
 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
 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)
   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   Lawrence Berkeley National Laboratory  USA