Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation

Two related methods to calculate the Kohn-Sham correlation energy within the framework of the adiabatic-connection fluctuation-dissipation theorem are presented. The required coupling-strength-dependent density-density response functions are calculated within exact-exchange time-dependent density-functional theory, i.e., within time-dependent density-functional response theory using the full frequency-dependent exchange kernel in addition to the Coulomb kernel. The resulting resolution-of-identity exact-exchange random-phase approximation (RI-EXXRPA) methods in contrast to previous EXXRPA methods employ an auxiliary basis set (RI basis set) to improve the computational efficiency, in particular, to reduce the formal scaling of the computational effort with respect to the system size N from N6 to N5. Moreover, the presented RI-EXXRPA methods, in contrast to previous ones, do not treat products of occupied times unoccupied orbitals as if they were linearly independent. Finally, terms neglected in previous E...