The Singlet-Triplet Gap of Cyclobutadiene: The CIPSI-Driven CC($P$;$Q$) Study

TL;DR

This work assesses the CIPSI-driven CC($P$;$Q$) methodology for accurately describing cyclobutadiene’s automerization pathway, focusing on the challenging balance between nondynamical correlation in the singlet ground state and dynamical correlation in the triplet. By identifying leading triply excited determinants with CIPSI and incorporating them into the $P$ space, while evaluating remaining triples via the noniterative $\delta(P;Q)$ correction, the authors achieve near-CCSDT energetics at dramatically reduced cost, outperforming noniterative triples corrections near the barrier. Across cc-pVDZ and cc-pVTZ basis sets, the approach yields singlet–triplet gaps within 0.1–0.3 kcal/mol of CCSDT, with especially strong performance near the automerization transition state where T3 effects are large. The results demonstrate the practicality and scalability of CIPSI-driven CC($P$;$Q$) for accurately capturing multireference effects in biradicals and point to potential extensions to higher CC levels and excited states, offering significant speedups over full CCSDT.

Abstract

An accurate determination of singlet-triplet gaps in biradicals, including cyclobutadiene in the automerization barrier region where one has to balance the substantial nondynamical many-electron correlation effects characterizing the singlet ground state with the predominantly dynamical correlations of the lowest-energy triplet, remains a challenge for many quantum chemistry methods. High-level coupled-cluster (CC) approaches, such as the CC method with a full treatment of singly, doubly, and triply excited clusters (CCSDT), are often capable of providing reliable results, but the routine application of such methods is hindered by their high computational costs. We have recently proposed a practical alternative to converging the CCSDT energetics at small fractions of the computational effort, even when electron correlations become stronger and connected triply excited clusters are larger and nonperturbative, by merging the CC($P$;$Q$) moment expansions with the selected configuration interaction methodology abbreviated as CIPSI. We demonstrate that one can accurately approximate the highly accurate CCSDT potential surfaces characterizing the lowest singlet and triplet states of cyclobutadiene along the automerization coordinate and the gap between them using tiny fractions of triply excited cluster amplitudes identified with the help of relatively inexpensive CIPSI Hamiltonian diagonalizations.

Figures (6)

• Figure 1: The PECs (in kcal/mol) characterizing the lowest-energy singlet and triplet states of cyclobutadiene along the $D_{\rm 2h}$-symmetric automerization pathway, defined using the interpolation formula given by Eq. (\ref{['eq:ell']}) and parameterized by dimensionless variable $\lambda$, resulting from the full CCSDT (red solid circles and lines), active-space DEA-EOMCC(4p-2h)$\{N_\text{u}\}$ (green solid diamonds and lines), and perturbatively corrected and extrapolated CIPSI (blue solid squares and lines) calculations using the cc-pVDZ basis set described in the main text. For each of the three methods, the energy of the singlet ground state at the reactant (R, $\lambda = 0$) geometry is set to 0. The numbers in the middle, colored in the same way as the corresponding PECs, are the unsigned values of the singlet--triplet gaps determined at the $\lambda= 1$ TS structure.
• Figure 2: Graphical illustration of the convergence of the CC($P$) (red lines and circles) and CC($P$;$Q$) (black lines and squares) energies characterizing the lowest singlet state of cyclobutadiene, as described by the cc-pVDZ basis set, toward their CCSDT parents as functions of the actual numbers of determinants $N_\mathrm{det(out)}$ that define the sizes of the terminal wave functions $|\Psi^{(\text{CIPSI})}\rangle$ generated in the underlying CIPSI runs at (a) $\lambda = 0$, (b) $\lambda = 0.2$, (c) $\lambda = 0.4$, (d) $\lambda = 0.6$, (e) $\lambda = 0.8$, and (f) $\lambda = 1$. The insets show the percentages of the $S_z=0$$A_{g}(D_{2\text{h}})$-symmetric triply excited determinants captured by CIPSI as functions of $N_\text{det(out)}$.
• Figure 3: Graphical illustration of the convergence of the CC($P$) (red lines and circles) and CC($P$;$Q$) (black lines and squares) energies characterizing the lowest triplet state of cyclobutadiene, as described by the cc-pVDZ basis set, toward their CCSDT parents as functions of the actual numbers of determinants $N_\mathrm{det(out)}$ that define the sizes of the terminal wave functions $|\Psi^{(\text{CIPSI})}\rangle$ generated in the underlying CIPSI runs at (a) $\lambda = 0$, (b) $\lambda = 0.2$, (c) $\lambda = 0.4$, (d) $\lambda = 0.6$, (e) $\lambda = 0.8$, and (f) $\lambda = 1$. The insets show the percentages of the $S_z=1$$B_{1g}(D_{2\text{h}})$-symmetric triply excited determinants captured by CIPSI as functions of $N_\text{det(out)}$.
• Figure 4: Graphical illustration of the convergence of the CC($P$) (red lines and circles) and CC($P$;$Q$) (black lines and squares) singlet--triplet gaps $\Delta E_\text{S--T}= E_\text{S} - E_\text{T}$ of cyclobutadiene, as described by the cc-pVDZ basis set, toward their CCSDT parents as functions of the CIPSI input parameter $N_{\text{det(in)}}$ (common to the calculations for the lowest singlet and triplet states) at (a) $\lambda = 0$, (b) $\lambda = 0.2$, (c) $\lambda = 0.4$, (d) $\lambda = 0.6$, (e) $\lambda = 0.8$, and (f) $\lambda = 1$. The insets show the percentages of the triply excited determinants of the $S_z=0$$A_{g}(D_{2\text{h}})$ (blue lines and circles) and $S_z=1$$B_{1g}(D_{2\text{h}})$ (green lines and circles) symmetries captured by the underlying CIPSI runs as functions of $N_{\text{det(in)}}$.
• Figure 5: Convergence of the CC($P$) and CC($P$;$Q$) energies $E$, reported as ($E + 154.0$) hartree, of the lowest singlet [panels (a) and (b)] and triplet [panels (c) and (d)] states of cyclobutadiene, as described by the cc-pVDZ basis set, and the $\Delta E_\text{S--T}$ gaps between them [panels (e) and (f)] toward their CCSDT counterparts with the CIPSI wave function termination parameter $N_\text{det(in)}$ at selected values of the dimensionless variable $\lambda$ defining the automerization coordinate via the interpolation formula given by Eq. (\ref{['eq:ell']}). The CC($P$) results are reported in panels (a), (c), and (e). Panels (b), (d), and (f) show the corresponding CC($P$;$Q$) data.