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We reexamine the recently introduced basis-set correction theory based ondensity-functional theory consisting in correcting the basis-set incompletenesserror of wave-function methods using a density functional. We use aone-dimensional model Hamiltonian with delta-potential interactions which hasthe advantage of making easier to perform a more systematic analysis than forthree-dimensional Coulombic systems while keeping the essence of the slow basisconvergence problem of wave-function methods. We provide some mathematicaldetails about the theory and propose a new variant of basis-set correctionwhich has the advantage of being suited to the development of an adaptedlocal-density approximation. We show indeed how to develop a local-densityapproximation for the basis-set correction functional which is automaticallyadapted to the basis set employed, without resorting to range-separateddensity-functional theory as in previous works, but using instead a finiteuniform electron gas whose electron-electron interaction is projected on thebasis set. The work puts the basis-set correction theory on firmer grounds andprovides an interesting strategy for the improvement of this approach.

Basis-set correction based on density-functional theory: Rigorous framework for a one-dimensional model