Resolution of the identity approximation applied to PNOF family of correlation calculations

PNOFi (i = 1-7) family of functionals deal with electronic structure correlation calculations by means of the reduced electron density matrix approach (1-RDM). The major advantage of a 1-RDM formulation is that the unknown functional only needs to incorporate electron correlation. PNOFi family of functionals provide an efficient way of including dynamic and static correlation as compared to wave function methods. However, a transformation from atomic orbital integrals to molecular orbital integrals has to be performed in order to computate the Coulomb and Exchange matrices in molecular orbital representation J^OM and K^OM respectively, which involves an overall fifth power time scaling factor.
Resolution of the identity (RI) approximation reduces the scaling factor of both, J^OM and K^OM matrices, and lowers the global scaling to fourth power. As a consequence, less memory is required to perform PNOFi calculations with an overall speed up in the calculation. RI is applied directly in the evaluation of J^OM and K^OM matrices, thus, improvement would be valid for all PNOFi family, however, the PNOF5 functional has been used in this work. Impact of RI on the convergence, and the obtained error in the correlation energy is analyzed for selected large size systems.