Dynamic Electron Correlation Energy for Multireference Wave Function Methods from One- and Two-Electron Reduced Density Matrices

Abstract Efficiently recovering dynamic correlation in strongly correlated systems without incurring prohibitive computational costs remains a central challenge in quantum chemistry. In this perspective, we review and benchmark methods capable of recovering dynamic correlation for multireference wave functions exclusively from low-order reduced density matrices and densities. These approaches require at most the two-electron reduced density matrix of the reference wave function and fall into two categories: density functional theory-based methods and purely ab initio multireference adiabatic connection methods. The former include MC-srDFT, which recovers dynamic correlation through a short-range exchange–correlation functional depending on the charge and spin densities, as well as MC-PDFT and MC-srPDFT, which employ translated functionals that additionally depend on the on-top pair density. Within the post-CASSCF framework, we perform a direct, head-to-head benchmark of these approaches under identical computational settings (including active spaces and basis sets) against challenging multireference problems, including singlet–triplet gaps in organic biradicals, excitation energies, and spin-state splittings in iron complexes. Among the DFT-based methods, MC-srPDFT emerges as the most accurate, underscoring the benefit of incorporating the on-top pair density. However, all considered DFT-based methods fail to provide reliable spin-state energetics for transition-metal complexes. Conversely, linearized AC0 rivals or outperforms more computationally expensive second-order perturbation theory approaches across all benchmark sets. We discuss these findings in the context of alternative formulations and existing literature, highlighting critical limitations and identifying promising directions for the future development of scalable multireference methods.