Multireference Dynamic Correlation Energy from the Combined Particle–Hole and Particle–Particle Adiabatic Connection with Random Phase Approximation

We present a formally exact adiabatic connection (AC) framework for the correlation energy of multireference wave functions, derived using the fermionic operator algebra. This framework can be formulated in both the particle–hole (ph) and particle–particle (pp) representations by exploiting the corresponding decompositions of the two-electron reduced density matrix. While the phAC formalism is well established, here we derive the ppAC formula and provide explicit working expressions based on the extended random-phase approximation. The second-order multireference pp correlation energy introduced by Tucholska et al. [ J. Phys. Chem. Lett. 2024, 15, 12001] emerges naturally as the lowest-order approximation in this context. Exploiting the equivalence of ph and pp correlation amplitudes, we propose a combined ffAC method that incorporates both contributions and avoids double counting of correlation. We test all the considered AC methods, i.e., ph- and ppAC, their linearized variants ph- and ppAC0, and the ph-pp combined methods ffAC and ffAC0, on a variety of challenging cases, including multiple bond dissociations, atomic excitations, singlet–triplet gaps of organic biradicals, and singlet and triplet excitation energies of organic chromophores. Across all systems, the linearized ffAC0 method consistently provides the most accurate results. Its accuracy matches or exceeds that of NEVPT2, yet it is significantly more efficient, requiring only one- and two-electron reduced density matrices.