  The construction of density-functional approximations is explored by modeling
the adiabatic connection em locally, using energy densities defined in terms of
the electrostatic potential of the exchange-correlation hole. These local
models are more amenable to the construction of size-consistent approximations
than their global counterparts. In this work we use accurate input local
ingredients to assess the accuracy of range of local interpolation models
against accurate exchange-correlation energy densities. The importance of the
strictly-correlated electrons (SCE) functional describing the strong coupling
limit is emphasized, enabling the corresponding interpolated functionals to
treat strong correlation effects. In addition to exploring the performance of
such models numerically for the helium and beryllium isoelectronic series and
the dissociation of the hydrogen molecule, an approximate analytic model is
presented for the initial slope of the local adiabatic connection. Comparisons
are made with approaches based on global models and prospects for future
approximations based on the local adiabatic connection are discussed.
