Leveraging Configuration Interaction Singles for Qualitative Descriptions of Ground and Excited States: State-Averaging, Linear-Response, and Spin-Projection.

While configuration interaction singles (CIS) provides a computationally efficient description of excited states, it systematically overestimates excitation energies and performs poorly for strongly correlated systems, partly due to the lack of orbital relaxation and the strong ground-state bias of Hartree-Fock orbitals. To address these limitations, we present a unified variational framework that extends CIS by incorporating orbital optimization in state-specific and state-averaged forms (SSCIS and SACIS), linear-response orbital relaxation via a double-CIS scheme (DCIS), and spin-symmetry breaking and restoration (ECIS). In spin-projected state-averaged formulations, standard multistate parametrizations are no longer valid because the projection operator breaks the unitary invariance of orbital rotations and induces nonorthogonal couplings among states. By formulating a rigorous state-averaged objective in the projected subspace, we derive analytic electronic gradients and Hessians and enable robust optimization using a trust-region augmented Hessian algorithm. Benchmark calculations show that spin projection alone significantly exacerbates the CIS overestimation in weakly correlated systems, whereas combining spin projection with state averaging or double-CIS corrections substantially reduces errors, particularly for Rydberg excitations. We further demonstrate that state averaging and spin projection provide complementary and essential benefits in strongly correlated regimes, as illustrated by the bond dissociation of hydrogen fluoride and nitrogen.