  In this work, the required algebra to employ the resolution of the identity
approximation within Piris Natural Orbital Functional (PNOF) is developed,
leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced
from fifth-order to fourth-order, and the memory scaling is reduced from
fourth-order to third-order, allowing significant computational time savings.
After the DoNOF-RI calculation has fully converged, a restart with four-center
electron repulsion integrals can be performed to remove the effect of the
auxiliary basis set incompleteness, quickly converging to the exact result. The
proposed approach has been tested on cycloalkanes and other molecules of
general interest to study the numerical results as well as the speed-ups
achieved by PNOF7-RI when compared with PNOF7.
