Intriguing Electronic Structures of C8 and C12 Carbon Rings

Technical Summary: Intriguing Electronic Structures of C8 and C12 Carbon Rings

Problem Statement
Monocyclic carbon rings ($C_{4n}$) represent a challenging class of carbon allotropes where the interplay between geometric structure and electronic properties often defies standard theoretical predictions. While the synthesis and characterization of $C_{18}$ in 2019 confirmed a polyynic (alternating bond lengths) ground state, contradicting some earlier Density Functional Theory (DFT) predictions of cumulenic (equal bond lengths) structures, the electronic nature of smaller rings like $C_8$ and $C_{12}$ remains complex. Specifically, the behavior of excited states, the validity of Baird's rule for open-shell systems, and the potential for "disjoint diradical" character in carbon allotropes have not been systematically explored with high-level ab initio methods. Furthermore, the violation of Hund's rule—a phenomenon previously observed only in graphene—has not been definitively reported in other carbon allotropes or double-aromatic molecules.

Methodology
The authors employed state-of-the-art ab initio electronic structure methods to investigate the ground and low-lying excited states of $C_8$ and $C_{12}$ rings.

• Geometric Optimization: Closed-shell ground states were optimized using Coupled Cluster Singles and Doubles (CCSD) with the cc-pVTZ basis set. For open-shell diradical states, particularly the cumulenic structures where restricted Hartree-Fock references are unstable, the Equation-of-Motion Spin-Flip Coupled Cluster (EOM-SF-CCSD) method was utilized.
• Excitation Energies: Vertical excitation energies were calculated using the Equation-of-Motion Coupled Cluster for electron excitations (EE-EOM-CCSD).
• Pathway Analysis: To understand the interconnection between polyynic and cumulenic geometries and the mechanism of Hund's rule violation, Minimum Energy Paths (MEPs) were computed for $C_8$ using Multi-Reference Configuration Interaction with Single and Double excitations (MRCISD) plus the generalized Davidson correction (+Q) with the cc-pVDZ basis set.
• Vibronic Coupling: Potential energy surface cuts were analyzed along specific vibrational modes ($A_{2g}$) to elucidate symmetry breaking and state mixing.

Key Contributions and Results

1. Ground State vs. Excited State Geometries:

   • The global ground states for both $C_8$ and $C_{12}$ are confirmed as closed-shell, polyynic structures (unequal C-C bond lengths), consistent with anti-aromaticity predicted by Hückel's rule.
   • In contrast, the energetically lowest-lying triplet states ($^3A_{2g}$) exhibit stable, cumulenic geometries (equal C-C bond lengths). This supports Baird's rule, which predicts aromaticity for open-shell systems with $4n$ $\pi$-electrons.
   • The geometric differences between the polyynic ground state and the cumulenic triplet state are surprisingly small (e.g., bond angle changes of ~2.7° for $C_8$), yet they induce dramatic changes in electronic character.

2. Violation of Hund's Rule:

   • A central finding is the violation of Hund's multiplicity rule in the cumulenic structures of both $C_8$ and $C_{12}$. Contrary to the expectation that the triplet state should be lower in energy than the singlet, the lowest open-shell singlet state ($^1A_{2g}$) is found to be lower in energy than its triplet counterpart ($^3A_{2g}$).
   • This is the first reported instance of Hund's rule violation in carbon allotropes other than graphene and the first in double-aromatic molecules.
   • The mechanism is attributed to strong vibronic coupling. The two singly occupied natural orbitals (SONOs) in the cumulenic structure are disjoint (localized on alternating carbon atoms) and quasi-degenerate. The open-shell singlet state interacts strongly with a nearby closed-shell singlet state ($^1A_{1g}$), causing a symmetry breaking via an $A_{2g}$ vibrational mode that lowers the singlet energy below the triplet.

3. Disjoint Diradical Character:

   • The study identifies the low-lying open-shell states of $C_8$ and $C_{12}$ as stable disjoint diradicals. Visualization of the SONOs reveals that in the cumulenic triplet state, the unpaired electrons are distributed over disjoint sets of atoms (odd vs. even carbon sites) with nodes on the connecting atoms.
   • This disjoint nature results in negligible kinetic exchange, facilitating the near-degeneracy of singlet and triplet states and enabling the violation of Hund's rule.

4. Relaxation Pathways and Excitation Cycles:

   • The authors mapped the Minimum Energy Paths connecting polyynic and cumulenic structures.
   • The polyynic triplet state relaxes directly to the stable cumulenic triplet minimum.
   • The polyynic first excited singlet (open-shell) relaxes to a closed-shell cumulenic state.
   • Conversely, the lowest energy cumulenic singlet (open-shell) is a transition state that relaxes back to the polyynic closed-shell ground state.
   • The paper proposes a theoretical cycle to access the cumulenic structure: electron impact excitation from the polyynic ground state to the polyynic triplet, followed by relaxation to the cumulenic triplet. Due to weak spin-orbit coupling in carbon, this cumulenic triplet is expected to be long-lived before potentially undergoing intersystem crossing back to the singlet and returning to the polyynic ground state.

Significance
The paper claims that these findings significantly enrich the understanding of carbon allotropes by demonstrating that small geometric changes (polyynic to cumulenic) can lead to profound electronic restructuring, including the emergence of stable disjoint diradicals and the violation of fundamental electronic rules like Hund's rule. The work highlights the necessity of high-level multi-reference and coupled cluster methods to accurately describe these systems, as popular methods like DFT have historically provided conflicting predictions for carbon rings. The identification of stable, aromatic, disjoint diradicals in carbon rings suggests potential, though not explicitly detailed, relevance for designing materials with unique magnetic or optoelectronic properties, specifically noting the potential for long-lived triplet states in carbon rings.