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The many-body theory of interacting electrons poses an intrinsicallydifficult problem that requires simplifying assumptions. For the determinationof electronic screening properties of the Coulomb interaction, the Random PhaseApproximation (RPA) provides such a simplification. Here, we explicitly showthat this approximation is justified for band structures with sizeable bandgaps. This is when the electronic states responsible for the screening areenergetically far away from the Fermi level, which is equivalent to a shortelectronic propagation length of these states. The RPA contains exactly thosediagrams in which the classical Coulomb interaction covers all distances,whereas neglected vertex corrections involve quantum tunneling through thebarrier formed by the band gap. Our analysis of electron-electron interactionsprovides a real-space analogy to Migdal's theorem on the smallness of vertexcorrections in electron-phonon problems. An important application is theincreasing use of constrained Random Phase Approximation (cRPA) calculations ofeffective interactions. We find that their usage of Kohn-Sham energies alreadyaccounts for the leading local (excitonic) vertex correction in insulators.

Random Phase Approximation for gapped systems: role of vertex corrections and applicability of the constrained random phase approximation