Stochastic Resolution of Identity to Linear Response Second Order Coupled Cluster with Spin-Orbit Couplings

We develop a stochastic Resolution of the Identity (sRI) approach to the second order coupled-cluster linear response (LR-CC2) framework for the efficient calculation of spin-orbit coupling (SOC) constants in molecular systems. Conventional RI-CC2 methodology, limited by its substantial computational cost and storage requirements associated with the treatment of two-electron spin-orbit integrals, is typically restricted to small systems. By introducing stochastic orbital techniques, the sRI-CC2 method significantly reduces the computational scaling of deterministic RI-CC2 from $\mathcal{O}(N^5)$ to $\mathcal{O}(N^3),$ with $N$ being the number of electrons. Evaluations on a series of representative organic molecules demonstrate that sRI-CC2 yields SOC constants in strong agreement with RI-CC2 results, with deviations generally falling within acceptable tolerances. The convergence behavior with respect to the number of stochastic orbitals is systematically analyzed, confirming the numerical robustness of the methodology. These results establish sRI-CC2 as a computationally efficient and reliable approach for investigating spin-orbit coupling effects in medium-to-large molecular systems, thereby extending the applicability of high-level electronic structure methods in photophysical and photochemical simulations.