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are no longer valid because the projection operator breaks the unitary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.