Analyzing correlation effects through a two-legged approximation of the upside-down adiabatic connection

In Kohn-Sham density functional theory, the adiabatic connection expresses the exchange-correlation energy (E_xc) of a system through scaling the strength of the interactions between electrons. However, this adiabatic connection is a difficult object to calculate, and thus a simplified version of the curve was made, known as the two-legged approximation. This technique found its use in hybrid functionals for E_xc as a way to determine a density-dependent weighting parameter for Hartree-Fock exchange, improving simpler E_xc approximations. This work seeks to develop an analogue of this two-legged approximation within the context of the strictly correlated limit, an alternative reference point for density functional theory (as opposed to the usual non-interacting Kohn-Sham system). In this limit, the interactions between electrons are scaled in strength to an energy minimum, only limited by needing to recreate the density of the true system through adjustment of the external potential. By analyzing the two-legged approximation to the adiabatic connection for this strictly correlated limit, we identify trends for different Kohn-Sham energy components as model system parameters vary. Similarly to the Kohn-Sham two-legged approximation, these trends can help assign a number to characterize how strongly correlated a system is, as well as a scheme to characterize the sources of this correlation.