13 Random Phase Approximation and Beyond: Energy and First-Order Properties

Ground state energies and analytic first-order properties (e.g., ’gradients’ for structure optimizations) can be computed within the random phase approximation (RPA) using the rirpa module. Theory and development of the rirpa module is published in references [298, 299] for the energy and reference [300, 301] for the first-order properties. See also [302] for a survey of applications of RIRPA. For two-component relativistic RPA energy calculations, see reference [303]. For perturbative beyond-RPA calculations, see references [304, 305]. For orbital-self-consistent approach called the generalized Kohn–Sham semi-canonical projected RPA (GKS-spRPA), see reference [306]. \(\sigma\)-functionals based on RPA are described in references [307, 308, 309, 310]. For energy and gradients, the resolution-of-the-identity (RI) approximation is used to approximate the two-electron repulsion integrals in the correlation treatment and is combined with an imaginary frequency integration. The RI approximation is also employed by default for the computation of the Coulomb integrals for the HF energy. For the energy, it is optional to use RI for the Fock exchange integrals (’RI-K’), while RI-K for the gradients is not available yet. Open shell systems and the frozen core approximation may be used in RPA energy and gradient calculations. Two-component RPA energy calculations are only possible for Kramers-restricted closed-shell systems. ECPs are presently not compatible with RIRPA gradients. Neither RPA energy nor gradients support symmetry at the moment. The gradients may be used together with the scripts jobex (for structure optimizations) and NumForce (for numerical harmonic vibrational frequencies). A frozen core is recommended for geometry optimization and vibrational frequency calculations to reduce computational costs with minimal impact on the final results.