Second-Order Perturbative Treatment of Spin–Orbit Coupling and Ground-State Electron Correlation

Relativistic effects, particularly spin-orbit coupling (SOC), are essential for accurately describing heavy-element systems, but their efficient treatment within correlated wave function frameworks remains a significant challenge. In this work, we develop four second-order approaches that simultaneously treat electron correlation and SOC within a coupled-cluster-based perturbative framework for closed-shell systems. SOC is introduced as a zeroth-order operator, and correlation effects are included through second-order perturbative expansions starting from a scalar-relativistic Hartree-Fock (SR-HF) reference. The methods consist of one SR-HF-based standard second-order perturbation theory with spin-orbit treated at zeroth order (SOPT2) and three variants derived from spin-orbit coupled cluster singles (SOCCS), all of which reduce to second-order Mo̷ller-Plesset perturbation theory (MP2) in the absence of SOC. Benchmark results for closed-shell sixth- and seventh-row atoms, ions, and halides show that the SOCCS-based schemes significantly improve upon SOPT2, with the CI-like variant achieving the highest accuracy. Accurate 2Π SOC splittings are also obtained using the equation-of-motion coupled-cluster theory at the singles and doubles level (EOM-CCSD) with perturbatively generated cluster operators. Overall, these approaches offer an efficient and reliable framework for incorporating SOC into correlated electronic structure calculations for systems with strong relativistic effects.