The coupling-constant-averaged exchange-correlation hole of spherical atoms calculated from the effective potential derived from the coordinate-scaling relation

Density functional theory's accuracy in assessing electronic structures hinges on the exchange-correlation (XC) approximation. Fundamentally, the XC hole, describing electron-electron interaction effects around a reference electron, offers a rigorous definition for the XC approximation. This gives a unique lens to gauge the quality of various XC approximations. To compute the XC hole, an external potential (v_λ), reliant on a coupling constant (λ), is essential to ensure consistent density across the adiabatic connection. Achieving this entails optimizing the general Lieb functional against v_λ over diverse λ values. When paired with the high-caliber coupled-cluster method at λ=1 as a benchmark, this technique is termed the Lieb+CCSD(T) method, which is computationally intensive. In our research, we utilized the coordinate scaling relation of the XC potential for approximating v_λ. Subsequently, XC holes for helium and specific second-period atoms were determined using different XC potentials and juxtaposed against the precise Lieb+CCSD(T) results. Our study paves the way for using the coordinate-scaling-derived potential in systems where the Lieb+CCSD(T) approach is computationally prohibitive.