Leveraging configuration interaction singles for qualitative descriptions of ground and excited states: state-averaging, linear-response, and spin-projection

TL;DR

This work tackles the persistent limitations of configuration interaction singles (CIS) in excited-state descriptions, notably the lack of orbital relaxation and ground-state bias. It develops a unified variational framework that incorporates state-specific and state-averaged orbital optimization, linear-response corrections via double-CIS (DCIS) schemes, and spin-projection with rigorous gradients and Hessians, optimized robustly by trust-region augmented Hessian (TRAH) methods. Benchmark results show that spin projection alone (ECIS) often worsens weakly correlated excitation energies, but combining spin projection with state averaging (SAECIS) or DCIS corrections (EDCIS) yields substantial improvements, especially for Rydberg states; in strongly correlated regimes, SACIS and SAECIS provide complementary benefits and can capture near-degeneracy features that CIS misses. Overall, the study clarifies how orbital relaxation, state averaging, and symmetry restoration contribute to low-cost yet qualitatively reliable excited-state descriptions and highlights TRAH’s crucial role for robust convergence, with potential for integrating dynamical correlation in future work.

Abstract

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, partially 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 (state-specific and state-averaged), linear-response orbital relaxation via double-CIS schemes, and spin-symmetry breaking and restoration. In spin-projected state-averaged formulations, standard multistate parametrizations are no longer valid because the projection operator breaks the unitary invariance of orbital rotations and induces nonorthogonal couplings among states.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 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-CI 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.

Figures (4)

• Figure 1: Comparison between convergence behaviors of DIIS and TRAH. (a) Formaldehyde with SACIS. (b) Formaldehyde with SAECIS. (c) Hydrogen fluoride with SACIS. (d) Hydrogen fluoride with SAECIS.
• Figure 2: Same as Figure \ref{['fig:conv']} but as a function of $N_{\rm Fock}$.
• Figure 3: Excitation energy error from EOM-CCSDT in eV. $\bigcirc$ and $\blacksquare$ denote valence and Rydberg excitations.
• Figure 4: Potential curves of hydrogen fluoride computed by different methods. (a) Ground state $S_0$. (b) Energy error from FCI for $S_0$. (c) First excited state $S_1$. (d) Error from the FCI excitation energy.