Understanding the core limitations of second-order correlation-based functionals through: functional, orbital, and eigenvalue-driven analysis

Density Functional Theory has long struggled to obtain the exact exchange-correlational (XC) functional. Numerous approximations have been designed with the hope of achieving chemical accuracy. However, designing a functional involves numerous methodologies, which has a greater possibility for error accumulation if the functionals are poorly formulated. This study aims to investigate the performance and limitations of second-order correlation functionals within the framework of density functional theory. Specifically, we focus on three major classes of density functional approximations that incorporate second-order energy expressions: \textit{ab initio} (primarily G"{o}rling-Levy) functionals, adiabatic connection models, and double-hybrid functionals. The principal objectives of this research are to evaluate the accuracy of second-order correlation functionals, to understand how the choice of reference orbitals and eigenvalues affects the performance of these functionals, to identify the intrinsic limitations of second-order energy expressions, especially when using arbitrary orbitals or non-canonical configurations, and propose strategies for improving their accuracy. By addressing these questions, we aim to provide deeper insights into the factors governing the accuracy of second-order correlation functionals, thereby guiding future functional development.