We present a graphical analysis of the adiabatic connections underlyingdouble-hybrid density-functional methods that employ second-order perturbationtheory. Approximate adiabatic connection formulae relevant to the constructionof these functionals are derived and compared directly with those calculatedusing accurate ab initio methods. The discontinuous nature of the approximateadiabatic integrands is emphasized, the discontinuities occurring atinteraction strengths which mark the transitions between regions that are: (i)described predominantly by second- order perturbation theory (ii) described bya mixture of density-functional and second-order perturbation theorycontributions and (iii) described purely by density-functional theory.Numerical examples are presented for a selection of small molecular systems andvan der Waals dimers. The impacts of commonly used approximations in each ofthe three sections of the adiabatic connection are discussed along withpossible routes for the development of improved double-hybrid methodologies.
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