Aufbau Suppressed Coupled Cluster Theory for Doubly Excited States

We generalize the Aufbau suppressed coupled cluster formalism into the realm of doubly excited states by deriving, implementing, and testing a wave function initialization strategy that allows the zeroth order wave function to match the largest configurations of a doubly excited reference wave function while maintaining the method’s overall asymptotic cost parity with ground state singles and doubles theory. Starting from state-averaged complete active space self consistent field references, this approach produces highly accurate excitation energies for states dominated by a single doubly excited determinant, as well as states in glyoxal and similar molecules where two different doubly excited determinants have large weights. Typical excitation energy errors in both types of states are on the order of 0.15 eV, with the largest observed error being 0.3 eV. These errors stand in stark contrast to equation of motion methods, where typical errors are 4 to 6 eV at the singles and doubles level and 0.4 to 0.8 eV at the full triples level. It remains an open question how best to generalize the Aufbau suppression approach into an even wider variety of multi-configurational double excitations, but these early results offer strong motivation for further investigation.

💡 Research Summary

The paper presents a significant extension of the Aufbau Suppressed Coupled Cluster (ASCC) methodology, originally devised for singly excited states, to the challenging domain of doubly excited electronic states. Doubly excited states are notoriously difficult for conventional excited‑state quantum‑chemical methods: time‑dependent density‑functional theory fails to describe them within the adiabatic approximation, linear‑response coupled‑cluster (LR‑CC) and equation‑of‑motion CC (EOM‑CC) at the singles‑and‑doubles (SD) level produce errors of 4–6 eV, and even full triples (EOM‑CCSDT) often leave errors of 0.4–0.8 eV. State‑specific approaches such as Δ‑CC work well for single‑configuration doubles but struggle with multi‑configurational cases, while multireference methods (CASSCF, CASPT2) depend heavily on the size and choice of the active space and scale poorly with system size.

ASCC tackles these problems by deliberately suppressing the contribution of the ground‑state Aufbau determinant and constructing a wave‑function that contains only the dominant configurations of the target excited state at zeroth order. This is achieved through a double‑exponential ansatz: an exponential of a de‑excitation operator S† is applied to the usual coupled‑cluster exponential e^T. For a single‑configuration (1‑CSF) doubly excited state, the authors derive analytically that choosing T^(0)=c₁S + c₂S² with η=√2, c₁=√2 and c₂=0 reproduces exactly the desired doubly excited determinant after the action of the two exponentials. Thus the zeroth‑order wave function is identical to the reference double excitation, while the remaining correlation is captured by solving for the first‑order amplitudes in T.

The authors also address a simple two‑configuration (2‑CSF) case, exemplified by the glyoxal molecule where two different doubly excited determinants have comparable weights. Here they introduce two distinct de‑excitation operators S₁₁† and S₁₂† with independent scaling parameters γ and ω. By expanding T^(0) as a linear combination of S₁₁, S₁₁², S₁₂, S₁₂², and the mixed product S₁₁S₁₂, they obtain a set of six algebraic equations that uniquely determine γ, ω and the coefficients c₁…c₅, ensuring that the zeroth‑order wave function matches the multi‑configurational reference.

To keep the computational cost comparable to conventional CCSD, the authors partition the full cluster operator T into three parts: T_N (non‑primary excitations), T_M (mixed primary–non‑primary excitations), and T_P (pure primary excitations). The primary part T_P contains the zeroth‑order terms derived above, while T_N and T_M are truncated to at most first order. By separating the Hamiltonian into a zeroth‑order Fock‑only piece H^(0) and a residual H^(1), they formulate similarity‑transformed amplitude equations that involve only O(N⁵) tensor contractions, preserving the favorable scaling of ground‑state CCSD.

The method is tested on a set of small molecules using state‑averaged CASSCF wave functions with minimal active spaces as the starting point. For states dominated by a single doubly excited determinant (e.g., formaldehyde, acetone, benzene), the ASCC(SD) approach yields excitation energies with a mean absolute error of ~0.15 eV and a worst‑case error of 0.30 eV. For the more demanding multi‑configurational double excitations in glyoxal and related systems, the same level of accuracy is retained, demonstrating that the 2‑CSF extension works as intended. These errors are dramatically smaller than those of EOM‑CCSD (4–6 eV) and even EOM‑CCSDT (0.4–0.8 eV), and comparable to, or better than, Δ‑CC for single‑configuration cases while handling multi‑configurational cases that Δ‑CC cannot.

The paper acknowledges several open issues. The current formulation handles only up to two dominant configurations; a general scheme for three or more configurations remains to be developed. Sensitivity to the choice of the active space, although reduced compared with CASPT2, still influences accuracy. Future work is suggested on automated active‑space selection, incorporation of higher‑order excitations (triples, quadruples) within the ASCC framework, and efficient parallel implementations to enable applications to larger, chemically relevant systems.

In summary, the authors deliver a novel state‑specific coupled‑cluster methodology that retains the low O(N⁵) scaling of CCSD while achieving excitation‑energy accuracies for doubly excited states that rival or surpass existing high‑level methods. This represents a promising avenue for reliable modeling of photochemical processes, singlet fission, delayed fluorescence, and other phenomena where doubly excited configurations play a central role.