. 2025 May 29;21(11):5533–5544. doi: 10.1021/acs.jctc.5c00428

1. Contributions in the Expansion of the Beyond-RPA Correlation Energy up to Terms Quadratic in the RPA Double Excitation Amplitudes T .

ph contributions	pp/hh contributions
E^RPA_c = 2∑_aibj (ai\|bj) T _ij E _c = −∑_aibj (ai\|bj) T _ji	E^2e_c = 2∑_aibjkl (ij\|kl) T _ik T _lj
E^2b_c = −4∑_aibjck (ai\|bj) T _ik T _jk E _c = 2∑_aibjck (ai\|bj) T _ki T_jk	E^2f_c = −∑_aibjkl (ij\|kl)T _ik T _jl
E^2g_c = −4∑_aibjck (ij\|ab) T _jk T _ki E _c = −4∑_aibjck (ij\|ab) T _kj T _ik	E^2k_c = 2∑_aibjcd (ab\|cd) T _ij T _ji
E²ⁱ_c = 2∑_aibjck (ij\|ab) T _jk T _ik	E^2l_c = −∑_aibjcd (ab\|cd) T _ij T _ij

The terms are divided into the particle-hole (ph) and particle-particle/hole–hole (pp/hh) subsets according to the type of the Coulomb integral contracted with the 2-electron reduced density matrix (2-RDM).

ph contributions	pp/hh contributions
E^RPA_c = 2∑_aibj (ai\|bj) T _ij E _c = −∑_aibj (ai\|bj) T _ji	E^2e_c = 2∑_aibjkl (ij\|kl) T _ik T _lj
E^2b_c = −4∑_aibjck (ai\|bj) T _ik T _jk E _c = 2∑_aibjck (ai\|bj) T _ki T_jk	E^2f_c = −∑_aibjkl (ij\|kl)T _ik T _jl
E^2g_c = −4∑_aibjck (ij\|ab) T _jk T _ki E _c = −4∑_aibjck (ij\|ab) T _kj T _ik	E^2k_c = 2∑_aibjcd (ab\|cd) T _ij T _ji
E²ⁱ_c = 2∑_aibjck (ij\|ab) T _jk T _ik	E^2l_c = −∑_aibjcd (ab\|cd) T _ij T _ij