Exact Mo̷ller-Plesset Adiabatic Connection Correlation Energy Densities

The Mo̷ller-Plesset adiabatic connection (MPAC) provides a powerful tool for developing density functional theory (DFT)-like approximations that map Hartree-Fock densities to the wave function-based correlation energy, thereby leveraging both wave function and DFT concepts for electronic structure approximations. A key object in this context is the correlation energy density, which represents the local (pointwise) contribution to the total correlation energy. While well-studied in DFT, it remains largely unexplored in the wave function framework. Here, we introduce a rigorous formulation of the wave function-based correlation energy density within MPAC, implement it via full configuration interaction calculations, and analyze its behavior and physically meaningful contributions for representative small (di)atomic systems. We define this quantity by employing a general gauge strategy, from which the conventional DFT correlation energy density gauge also arises. We then discuss the resulting commonalities and differences between correlation energy densities in the DFT and wave function frameworks and derive the small-interaction (MP2) limit of the latter in terms of Hartree-Fock orbitals. Finally, we show how these newly introduced energy densities can serve as new approximation targets in both machine-learning-assisted and traditional electronic structure methods for mapping HF-density-based features to correlation energy within the wave function framework.