Analytical Nuclear Gradients for State-Averaged Configuration Interaction Singles Variants: Application to Conical Intersections

We derive analytical nuclear gradients for state-averaged orbital-optimized configuration interaction singles (SACIS) and its spin-projected extension (SAECIS), enabling efficient geometry optimization and minimum-energy conical intersection (MECX) searches within a low-cost CIS-based framework. The formulation employs a Lagrangian approach and explicitly removes null-space contributions in the coupled perturbed equations to ensure numerically stable gradients. For twisted-pyramidalized ethylene, both SACIS and SAECIS qualitatively reproduce the correct conical intersection topology, in sharp contrast to conventional CIS and ECIS. Benchmark calculations on twelve MECXs demonstrate that both methods reproduce geometries with mean RMSDs below 0.1~{\AA} relative to high-level reference methods. SACIS captures the essential degeneracy through variational orbital relaxation, which alleviates ground-state Hartree--Fock (HF) orbital bias and effectively incorporates static correlation through localization effects; notably, spin projection is found to be non-essential for the qualitative description of these intersections. Overall, SACIS and SAECIS provide qualitatively reliable CX descriptions at mean-field computational cost in a black-box manner. Given their comparable accuracy and the additional overhead associated with spin projection, SACIS offers a more favorable cost-performance balance for general applications, whereas SAECIS may become advantageous when higher excited states with significant double-excitation character are involved.