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We develop and implement a formalism which enables calculating the analyticalgradients of particle-hole random-phase approximation (RPA) ground-state energywith respect to the atomic positions within the atomic orbital basis setframework. Our approach is based on a localized resolution of identity (LRI)approximation for evaluating the two-electron Coulomb integrals and theirderivatives, and the density functional perturbation theory for computing thefirst-order derivatives of the Kohn-Sham (KS) orbitals and orbital energies.Our implementation allows one to relax molecular structures at the RPA levelusing both Gaussian-type orbitals (GTOs) and numerical atomic orbitals (NAOs).Benchmark calculations show that our approach delivers high numerical precisioncompared to previous implementations. A careful assessment of the quality ofRPA geometries for small molecules reveals that post-KS RPA systematicallyoverestimates the bond lengths. We furthermore optimized the geometries of thefour low-lying water hexamers -- cage, prism, cyclic and book isomers, anddetermined the energy hierarchy of these four isomers using RPA. The obtainedRPA energy ordering is in good agreement with that yielded by the coupledcluster method with single, double and perturbative triple excitations, despitethat the dissociation energies themselves are appreciably underestimated. Theunderestimation of the dissociation energies by RPA is well corrected by therenormalized single excitation correction.

Localized resolution of identity approach to the analytical gradients of random-phase approximation ground-state energy: algorithm and benchmarks