The bond-breaking and bond-making puzzle: many-body perturbation versus density-functional theory

Diatomic molecules at dissociation provide a prototypical situation in which the ground-state cannot be described by a single Slater determinant. For the paradigmatic case of H$_2$-dissociation we compare state-of-the-art many-body perturbation theory in the $GW$ approximation and density-functional theory (DFT) in the exact-exchange plus random-phase approximation for the correlation energy (RPA). Results from the recently developed renormalized second-order perturbation theory (rPT2) are also reported. For an unbiased comparison and to prevent spurious starting point effects both RPA and $GW$ are iterated to {\it full} self-consistency (i.e. sc-RPA and sc-$GW$). Both include topologically identical diagrams for the exchange and correlation energy but are evaluated with a non-interacting Kohn-Sham and an interacting $GW$ Green function, respectively. This has profound consequences for the kinetic and the correlation energy. $GW$ and rPT2 are both accurate at equilibrium, but fail at dissociation, in contrast to sc-RPA. This failure demonstrates the need of including higher order correlation diagrams in sc-$GW$. Our results indicate that RPA-based DFT is a strong contender for a universally applicable electronic-structure theory. F. Caruso {\it et al.} arxiv.org/abs/1210.8300.