Random Phase Approximation for Periodic Systems Employing Direct Coulomb Lattice Summation

A method to compute ground state correlation energies from the random phase approximation (RPA) is presented for molecular and periodic systems on an equal footing. The supermatrix representation of the Hartree kernel in canonical orbitals is translation-symmetry adapted and factorized by the resolution of the identity (RI) approximation. Orbital expansion and RI factorization employ atom-centered Gaussian-type basis functions. Long ranging Coulomb lattice sums are evaluated in direct space with a revised recursive multipole method that works also for irreducible representations different from Γ. The computational cost of this RI-RPA method scales as [Formula: see text](N⁴) with the system size in direct space, N, and as [Formula: see text](N_k²) with the number of sampled k-points in reciprocal space, N_k. For chain and film models, the exploration of translation symmetry with 10 k-points along each periodic direction reduces the computational cost by a factor of around 10-100 compared to equivalent Γ-point supercell calculations.