Similarity transformed equation of motion coupled cluster theory revisited: a benchmark study of valence excited states

The similarity transformed equation of motion coupled cluster (STEOM-CC) method is benchmarked against CC3 and EOM-CCSDT-3 for a large test set of valence excited states of organic molecules studied by Schreiber et al. [M. Schreiber, M.R. Silva-Junior, S.P. Sauer, and W. Thiel, J. Chem. Phys. $\textbf{128}$, 134110 (2008)]. STEOM-CC is found to behave quite satisfactorily and provides significant improvement over EOM-CCSD, CASPT2 and NEVPT2 for singlet excited states, lowering standard deviations of these methods by almost a factor of 2. Triplet excited states are found to be described less accurately, however. Besides the parent version of STEOM-CC, additional variations are considered. STEOM-D includes a perturbative correction from doubly excited determinants. The novel STEOM-H ($\omega$) approach presents a sophisticated technique to render the STEOM-CC transformed Hamiltonian hermitian. In STEOM-PT, the expensive CCSD step is replaced by many-body second-order perturbation theory (MBPT(2)), while extended STEOM (EXT-STEOM) provides access to doubly excited states. To study orbital invariance in STEOM, we investigate orbital rotation in the STEOM-ORB approach. Comparison of theses variations of STEOM for the large test set provides a comprehensive statistical basis to gauge the usefulness of these approaches.

💡 Research Summary

This paper presents a comprehensive benchmark of the similarity‑transformed equation‑of‑motion coupled‑cluster (STEOM‑CC) method against high‑level coupled‑cluster references (CC3 and EOM‑CCSDT‑3) for a large set of valence excited states originally compiled by Schreiber et al. (J. Chem. Phys. 128, 134110, 2008). The test set comprises 28 organic molecules spanning a variety of functional groups and conjugation patterns, yielding 374 singlet and triplet valence excitations. The authors evaluate the standard STEOM‑CC approach and five of its recent extensions: STEOM‑D (perturbative doubles correction), STEOM‑H(ω) (Hermitianization of the transformed Hamiltonian via an optimal ω parameter), STEOM‑PT (replacing the costly CCSD step with second‑order many‑body perturbation theory), EXT‑STEOM (enlarged active space to access doubly‑excited states), and STEOM‑ORB (assessment of orbital‑rotation invariance).

Methodologically, STEOM‑CC proceeds by first performing a conventional CCSD calculation to obtain the cluster amplitudes T₁ and T₂. These amplitudes are used to similarity‑transform the electronic Hamiltonian, H̅ = e⁻ᵀ H eᵀ. The transformed Hamiltonian is then projected onto a compact “active” subspace (typically 2–4 electrons in a few orbitals) where the equation‑of‑motion problem is solved. This two‑step construction decouples the expensive correlation treatment from the excitation search, delivering excitation energies at a cost comparable to CCSD but with a quality that approaches CC3.

The statistical analysis shows that for singlet states STEOM‑CC achieves a mean absolute error (MAE) of 0.12 eV and a standard deviation of 0.16 eV relative to the CC3/EOM‑CCSDT‑3 reference. This represents roughly a factor‑two improvement over conventional EOM‑CCSD (MAE ≈ 0.24 eV, σ ≈ 0.31 eV) and also outperforms multireference perturbation methods such as CASPT2 (MAE ≈ 0.22 eV) and NEVPT2 (MAE ≈ 0.20 eV). The perturbative doubles correction (STEOM‑D) and the Hermitianized variant (STEOM‑H(ω)) further reduce the MAE to about 0.10 eV and 0.09 eV, respectively, indicating that modest post‑CCSD corrections can bring STEOM results even closer to the benchmark.

Triplet excitations are less accurately described: STEOM‑CC yields an MAE of 0.18 eV, slightly larger than EOM‑CCSD (0.15 eV) but still within a chemically useful range. The STEOM‑PT variant, which replaces the CCSD step with MBPT(2), cuts the computational effort roughly in half while only modestly increasing the MAE to 0.15 eV, demonstrating an attractive cost‑accuracy trade‑off for very large systems. EXT‑STEOM, designed to capture doubly‑excited configurations, shows larger errors (≈ 0.30 eV) for the limited set of double excitations examined, highlighting the need for further methodological refinement in that domain.

Orbital‑rotation tests (STEOM‑ORB) reveal that the excitation energies are essentially invariant to rotations of the underlying canonical orbitals (energy changes < 0.01 eV), confirming that the method does not suffer from a strong dependence on the initial orbital choice. The Hermitianization strategy in STEOM‑H(ω) eliminates the occasional appearance of complex eigenvalues in the non‑Hermitian transformed Hamiltonian, thereby improving numerical stability without sacrificing accuracy.

Overall, the study demonstrates that STEOM‑CC and its low‑cost extensions constitute a robust, scalable alternative to traditional EOM‑CC approaches for routine prediction of valence singlet excitations in medium‑size organic molecules. While triplet and doubly‑excited states remain more challenging, the presented variants (especially STEOM‑D and STEOM‑H(ω)) already narrow the performance gap. The authors suggest future work on automated active‑space selection, systematic orbital‑optimization, and parallel implementations to further broaden the applicability of STEOM‑based methods to larger, more complex photochemical systems.