5- and 6-membered rings: A natural orbital functional study

TL;DR

This work benchmarks the recent electron-pairing-based natural orbital functionals GNOF and its modification GNOFm on a 12-molecule set of 5- and 6-membered rings, focusing on dynamic correlation. The authors compute complete-basis-set (CBS) limit correlation energies and compare against CCSD(T), using an exponential extrapolation $E(X) = E_{\mathrm{CBS}} + a_1 \cdot \exp(-a_2 X)$ across cc-pVDZ–cc-pV5Z (with special treatment for Thiophene). The results show that GNOFm systematically improves over GNOF, achieving CBS-limit energies within roughly $50\,mE_h$ of CCSD(T) for most systems, demonstrating that NOFs can capture dynamic correlation without active-space selection. Collectively, the study supports NOFs as a robust and scalable alternative to traditional multireference and perturbative methods for molecules where dynamic correlation dominates, and it paves the way for further refinement and broader benchmarking of the GNOF family.

Abstract

The Global Natural Orbital Functional (GNOF) provides a straightforward approach to capture most electron correlation effects without needing perturbative corrections or limited active spaces selection. In this work, we evaluate both the original GNOF and its modified variant GNOFm on a set of twelve 5- and 6-membered molecular rings, systems characterized primarily by dynamic correlation. This reference set is vital as it comprises essential substructures of more complex molecules. We report complete-basis-set limit correlation energies for GNOF, GNOFm, and the benchmark CCSD(T) method. Across the Dunning basis sets, both functionals deliver a balanced and accurate description of the molecular set, with GNOFm showing small but systematic improvements while preserving the overall robustness of the original formulation. These results confirm the reliability of the GNOF family and its ability to capture dynamic correlation effects.

Figures (7)

• Figure 1: 5- and 6-membered molecular rings studied along this work, as well as the corresponding numbering employed later on.
• Figure 2: Correlation energies ($E-E_\mathrm{HF}$) in mE$_{h}$ for the selected set of molecules, obtained by using GNOF (dotted lines), GNOFm (dashed lines), and CCSD(T) (solid lines) with the cc-pVXZ basis sets, $X=2, 3, 4, 5$ being the cardinal number of the basis set. Molecules ordered according to the numbering given in Fig. \ref{['fig:systems']}.
• Figure 3: Correlation energies ($E-E_\mathrm{HF}$) in mE$_{h}$ for the selected set of molecules, obtained by using GNOF (dotted lines) and GNOFm (dashed lines) with the cc-pVXZ basis sets ($X=2, 3, 4, 5$), together with the resulting complete basis set (CBS) limit estimates. Details corresponding to the latter are given throughout the text, as well as in Fig. \ref{['fig:Ecorr_cbslimit']} and Table \ref{['table1']}.
• Figure 4: Complete basis set (CBS) extrapolated correlation energies ($E-E_\mathrm{HF}$) in mE$_{h}$ for the 12 molecular systems, computed using GNOF, GNOFm, and CCSD(T). An exponential extrapolation scheme, $E(X) = E_{\mathrm{CBS}} + a_1 \cdot \exp(-a_2 X)$, was employed with $X = 2, 3, 4, 5$ as the cardinal number of the basis set. For Thiophene, the extrapolation was performed using $X = 2, 3, 4$.
• Figure S1: Correlation energies ($E-E_{HF}$) in mE$_{h}$ for the selected set of molecules, obtained by using CCSD(T)/cc-pVXZ with $X=2, 3, 4, 5$, together with the resulting CBS limit energies. Details corresponding to the latter are given throughout the text. CBS limit estimate corresponding to Thiophene (no. 3) was obtained by using $X = 2, 3, 4$ results.