⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 110201 Splitting of Degenerate Superatomic Molecular Orbitals Determined by Point Group Symmetry Rui Wang (王瑞)1, Jiarui Li (李佳芮)1, Zhonghua Liu (刘中华)1, Chenxi Wan (万晨曦)1,2, and Zhigang Wang (王志刚)1,2,3* Affiliations 1Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China 2Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China 3Institute of Theoretical Chemistry, College of Chemistry, Jilin University, Changchun 130023, China Received 3 August 2023; accepted manuscript online 8 October 2023; published online 15 November 2023 *Corresponding author. Email: wangzg@jlu.edu.cn Citation Text: Wang R, Li J R, Liu Z H et al. 2023 Chin. Phys. Lett. 40 110201    Abstract We first confirm an idea obtained from first-principles calculations, which is in line with symmetry theory: Although superatomic molecular orbitals (SAMOs) can be classified according to their angular momentum similar to atomic orbitals, SAMOs with the same angular momentum split due to the point group symmetry of superatoms. Based on this idea, we develop a method to quantitatively modulate the splitting spacing of molecular orbitals in a superatom by changing its structural symmetry or by altering geometric parameters with the same symmetry through expansion and compression processes. Moreover, the modulation of the position crossover is achieved between the lowest unoccupied molecular orbital and the highest occupied molecular orbital originating from the splitting of different angular momenta, leading to an effective reduction in system energy. This phenomenon is in line with the implication of the Jahn–Teller effect. This work provides insights into understanding and regulating the electronic structures of superatoms.
cpl-40-11-110201-fig1.png
cpl-40-11-110201-fig2.png
 DOI:10.1088/0256-307X/40/11/110201 © 2023 Chinese Physics Society Article Text Superatoms, as a special class of molecules, provide effective insights into understanding the physical laws of complex clusters.[1-5] This insight is in large part due to the classification of their atom-like molecular orbitals with different angular momenta, which potentially allows simulating or even replacing natural atoms.[1-3,6,7] It is known that atoms with spherical potential maintain their rotationally invariant characteristics, and the Pauli exclusion principle[8] and Hund's rules[9] show that the outermost orbitals are preferentially occupied in a half-full or full state when the number of valence electrons is equal to or twice the number of angular momentum orbitals (2$l$+1). This phenomenon results in a degenerate state for the same angular momentum orbital energy with the same principal quantum number in atoms,[10,11] which is consistent with the results of quantum calculations. As a potential substitute for atoms, although superatoms have high symmetry, they are far from perfect spherical symmetry, similar to atoms, which leads to an important phenomenon that differs from atoms:[1,12] the splitting of molecular orbitals with the same angular momentum.[13-18] Only in early simple modeling for superatoms were these molecular orbitals with the same principal quantum numbers and angular momentum numbers described to be degenerate.[3,6,19] To date, orbital splitting with the same angular momentum has been clearly shown by many studies. For example, in the metal superatom Al$_{13}^{-}$, seven F-type superatomic molecular orbitals ($F$ SAMOs) include quadruple and triple degenerate orbitals,[12] and in the fullerene superatom C$_{60}$, the highest occupied molecular orbital (HOMO), HOMO-1, HOMO-2, HOMO-3, and HOMO-4, are quintuple degenerate H-type SAMOs, and the remaining six $H$ SAMOs are formed by two groups of triple degeneracies in unoccupied molecular orbitals.[20,21] Additionally, although the metal cores in some ligand-protected coinage metal clusters cannot satisfy the magic number rule of electrons, their stability can still be explained by superatomic orbital splitting.[16,22,23] Therefore, understanding the electronic structures of superatoms according to point group symmetry aids in knowing whether the properties of superatoms can conform to general fundamental laws and if the superatoms can be further regulated. External conditions, especially high pressure, can gradually adjust system structures and states, enabling the understanding of numerous physical mechanisms.[24-27] Therefore, as a physical tool for studying the properties of matter, including the evolution of isomerization, high pressure has garnered widespread attention.[25-28] Moreover, theoretical simulation has considerable flexibility, which can facilitate the study of compression processes in pressure and even reverse stretching procedures. This simulation provides an opportunity to understand the characteristics of the evolution of superatomic structure in these two processes by referring to high-pressure experimental techniques. Therefore, in this work, we use first-principles calculations to study the evolution processes of the superatomic structure under axial expansion and compression conditions. By changing symmetry or altering geometric parameters under the same symmetry, modulation data of the SAMOs are obtained. Interestingly, our study achieves not only the quantitative modulation of splitting SAMOs but also exchange modulation between frontier molecular orbitals, which is consistent with the Jahn–Teller effect. These findings can provide an important reference for understanding and modulating SAMOs and related electronic structure properties in superatoms. Carbon-based fullerenes, as a typical superatomic structure, can exist naturally and be synthesized artificially.[2,29,30] The smallest fullerene C$_{20}$ can be observed experimentally and proven to have a simple electronic structure by theoretical calculations.[29,31-33] Moreover, considering the cavity structure, C$_{20}$ can effectively avoid additional complex interactions with embedded objects. Thus, C$_{20}$ is explored in this work, and its structural characteristics are reverified. By using the third-generation dispersion-corrected B3LYP functional [see the computational details in the Supporting Information (SI)], a stable neutral C$_{20}$ structure with $D_{3d}$ symmetry is obtained [Fig. 1(a) and Table S1 in the SI] and this result is consistent with previous reports.[31-34] Further molecular orbital analysis indicates that the superatomic electron configuration of the neutral structure is $1S^{2}1P^{6}1D^{10}1F^{2}$. As can be seen in Fig. 1(b), the $S$ SAMO is always nondegenerate; therefore, its exploration is omitted in this study. The $P$, $D$, and $F$ SAMOs are split. Specifically, the $P$ SAMOs consist of a nondegenerate ($A_{\rm 2u}$) orbital and double-degenerate ($E_{\rm u}$) orbitals, and the $D$ SAMOs consist of a nondegenerate ($A_{\rm 1g}$) orbital and two groups of double-degenerate ($E_{\rm g}$) orbitals. For frontier orbitals, the occupied $F$ SAMO is a nondegenerate ($A_{\rm 1u}$) orbital, and the unoccupied $F$ SAMOs consist of an $A_{\rm 2u}$ orbital and $E_{\rm u}$ orbitals. These calculated results are in accordance with group symmetry theory.[16,35]
cpl-40-11-110201-fig1.png
Fig. 1. Diagrams of the energy levels of neutral C$_{20}$ and its dianionic C$_{20}^{2-}$ clusters. (a) Geometric structures of C$_{20}$ and C$_{20}^{2-}$. The C$_{20}$ structure with $D_{3d}$ symmetry has four different C–C bond lengths (1.409, 1.448, 1.454, and 1.520 Å), which are marked with corresponding colors. The length of the C–C bond in the dianionic C$_{20}^{2-}$ structure ($I_{h}$ symmetry) is 1.466 Å, as shown in parentheses. [(b), (c)] Diagrams of the energy levels of C$_{20}$ ($D_{3d}$) and C$_{20}^{2-}$ ($I_{h}$), respectively. The types of angular momenta of SAMOs are labeled and the parentheses are their irreducible representations. Their corresponding SAMO shapes are shown on the left. The SAMO shapes of C$_{20}^{2-}$ take spin-up ($\alpha$) electrons as an example.
As a molecule, if the superatom has $I_{h}$ symmetry, its $F$ SAMOs can be split into quadruple degenerate ($G_{\rm u}$) orbitals and triple-degenerate ($T_{\rm 2u}$)[36] orbitals according to group symmetry theory.[35] Obviously, for a neutral C$_{20}$ structure with $D_{3d}$ symmetry, the lowest energy $F$ SAMOs are not quadruple degenerate. From Fig. 1(b), this phenomenon occurs because the two outermost electrons on $F$ SAMO are not sufficient to stabilize neutral C$_{20}$ with $I_{h}$ symmetry. This finding leads us to a reasonable assumption that the $F$ SAMOs, housed with four electrons, can satisfy their energy degeneracy after injecting the neutral C$_{20}$ with two electrons, ensuring rationality in the electronic structure. For this reason, we investigated dianionic C$_{20}^{2-}$ with $I_{h}$ symmetry, and verified its structural stability by optimization and frequency calculations (see Table S1 in the SI for details). As shown in Fig. 1(c), the superatomic electron configuration of C$_{20}^{2-}$ is $1S^{2}1P^{6}1D^{10}1F^{4}$. Interestingly, unlike neutral C$_{20}$ ($D_{3d}$), the $P$ and $D$ SAMOs of dianionic C$_{20}^{2-}$ ($I_{h}$) maintain triple and quintuple degeneracy, respectively. The remaining four valence electrons are housed on $F$ SAMOs in half-full states to maintain the quadruple degeneracy of the orbitals. In addition, the result is consistent with that at the equivalent basis set with diffusion functions (for details see Fig. S1 in the SI). These results suggest that there is a strict matching relationship between different symmetries and electron configuration rules. To systematically explore the influences of symmetry reduction on superatoms, we achieved the symmetry changes in C$_{20}^{2-}$ from $I_{h}$ to its subgroup $D_{5d}$ by axial expansion and compression along the planes consisting of two five-membered rings [marked by the black circles in Fig. 2(a)] perpendicular to the symmetry axis of $C_{5}/S_{10}$. The two opposite planes can be chosen arbitrarily, thus helping experimental locating operations, and these processes may be achieved by means of an atomic force microscope that generates force on bilayer graphene sandwiched with fullerenes.[37] This axial compression method can obtain changes in the geometrical parameters of the same order of magnitude as those calculated for the axial compression including the environment, which can be used to discuss the evolution process of the structure.[38,39] Considering that these changes are geometric structure evolution processes of sphere-like structures, they can be described using the deflection accepted by theoretical simulations of pressure properties.[28] Herein, the deflection is defined as the ratio of the change in displacement ($\varDelta$) of axial expansion or compression to the diameter ($2R$) of the sphere ($\varDelta/2R$). The deflection value varies from 0.0 to 0.097 (or $-$0.097) when the relative displacement is expanded (or compressed) in intervals of 0.1 Å. Furthermore, restrictive optimizations of the dianionic C$_{20}^{2-}$ structures corresponding to each deflection value were performed, and their structural characteristics were analyzed. All these corresponding results are shown in Fig. 2.
cpl-40-11-110201-fig2.png
Fig. 2. Characteristics of the evolution of geometric and electronic structures during axial expansion and compression. (a) Characteristics of the geometric structures and SAMOs during axial expansion and compression. The initial structure ($\varDelta /2R=0.0$) has $I_{h}$ symmetry. The intermediate structural diagram shows that the planes perpendicular to the $C_{5}$/$S_{10}$ symmetric axis (circled by black dotted lines) are expanded or compressed along the red and green arrows, respectively. After expansion or compression, the structures have $D_{5d}$ symmetry, except for the $C_{s}$ symmetry with a deflection of $-$0.097. The gray lines display hybridized orbitals. (b) Diagram of the expanded version (energy scale ranging from 2.5–7.5 eV) of the light green background in (a). The structure of $D_{5d}^{\ast}$ ($-$0.097) is constructed by the average bond lengths of the $C_{s}$ symmetric structure; its electronic structure information is obtained by the single point calculation at the same level. (c) Variations in structural geometric parameters during axial expansion and compression. For $D_{5d}$ symmetry, there are three C–C bond lengths, i.e., $b_{1}$, $b_{2}$, and $b_{3}$, which are represented by red spheres, gray diamonds, and yellow stars, respectively. The average bond lengths of C–C bonds in the structure $C_{s}$ ($-$0.097) are counted.
From Fig. 2(a), the initial structure ($I_{h}$) shows that the irreducible representations of the occupied $S$, $P$, $D$, and $F$ SAMOs correspond to $A_{\rm g}$, $T_{\rm 1u}$, $H_{\rm g}$, and $G_{\rm u}$, respectively. Upon axial expansion and compression, the $P$ SAMOs split into $E_{\rm 1u}$ and $A_{\rm 2u}$, the $D$ SAMOs split into $A_{\rm 1g}$, $E_{\rm 1g}$, and $E_{\rm 2g}$, and the $F$ SAMOs split into $E_{\rm 1u}$ and $E_{\rm 2u}$. Notably, the irreducible representations attributed to these orbitals are derived from structural property analysis by first-principles calculations and are confirmed to be consistent with the $D_{5d}$ symmetry in group symmetry theory.[16,35] This finding implies that the reduction in the C$_{20}^{2-}$ symmetry from $I_{h}$ to $D_{5d}$ leads to the splitting of molecular orbitals with the same angular momentum. Figure 2(a) clearly shows that the energy differences of these split SAMOs originating from the same angular momentum gradually increase as expansion and compression proceed. For example, during expansion, the energies of the $D(E_{\rm 2g})$ SAMOs gradually increase and exceed that of $E_{\rm 1g}$, while the energy of the $D(A_{\rm 1g})$ SAMOs gradually decreases and reaches the lowest energy position among the $D$ SAMOs. In contrast, during compression, the energies of $D$ ($A_{\rm 1g}$) increase and those of $D(E_{\rm 2g})$ SAMOs decrease, reaching the highest and lowest energy positions among $D$ SAMOs, respectively. These phenomena illustrate that SAMOs split in a pattern that conforms to the point group symmetry theory and the energy difference in these split orbitals gradually increases with expansion and compression. The above discussion about electron structure takes spin-up electrons as examples and a similar result can also be seen in orbitals occupied by spin-down electrons (for details see Fig. S2 in the SI). Clearly, this phenomenon is attributed to quantitative changes in the geometric parameters of these two processes while maintaining symmetry (having qualitatively consistent molecular orbital splitting patterns). These findings suggest that the energy positions of SAMOs can be modulated by axial expansion or compression on the geometric structure, which subsequently changes the electron energy level. Moreover, with the SAMO splitting behavior as described above, the lowest unoccupied molecular orbital (LUMO), which fails to reflect SAMO features (hybridized orbitals), changes from triple degeneracy in the $I_{h}$ symmetric structure to nondegeneracy in the $D_{5d}$ symmetric structure upon compression. The energy gradually decreases with further compression. In turn, the energies of the highest occupied molecular orbital (HOMO) and the adjacent HOMO-1 gradually increase. This result illustrates that the HOMO–LUMO gap gradually decreases with compression, and the HOMO and LUMO may overlap or even exchange under continuous compression conditions. Intriguingly, when the deflection is $-$0.097, the structure cannot maintain $D_{5d}$ symmetry, but it can maintain $C_{s}$ symmetry. To understand the reason for this phenomenon, we analyzed frontier orbitals for structures with deflections of $-$0.073 ($D_{5d}$), $-$0.097 ($D_{5d}^{\ast}$), and $-$0.097 ($C_{s}$), as shown in Fig. 2(b). The symbol $D_{5d}^{\ast}$ represents a special structure with $D_{5d}$ symmetry, which is constructed by the average bond length of the $C_{s}$ symmetric structure (see Table S2 in the SI for details). From the energy level diagrams and molecular orbital shapes, LUMO is an $F$ SAMO and HOMO is a hybridized orbital in the $D_{5d}^{\ast}$ structure, corresponding to HOMO and LUMO of the $D_{5d}$ ($-$0.073) structure, respectively. This finding clearly indicates that further compression ($\varDelta/2R < -0.073$) can result in orbital exchange. From the above analysis, the exchange behavior between the HOMO and LUMO should further split the double-degenerate $F(E_{\rm 2u})$ SAMOs in the $D_{5d}$ symmetric structure into nondegenerate orbitals. In this case, although the subgroups of $D_{5d}$ symmetry include $D_{5}$, $C_{5v}$, etc., the nondegeneracy of the $F$ SAMOs can only be realized when the symmetry is reduced to a subgroup of $C_{s}$. Indeed, the actual calculation is in accordance with the above analysis; that is, when the deflection is $-$0.097, the structural symmetry is reduced to $C_{s}$. We find that the reduction in structural symmetry not only results in the nondegeneracy of molecular orbitals but also destroys the SAMO feature of the HOMO (marked by the red background). In addition, the total energies of the $C_{s}$ ($-$0.097) and $D_{5d}^{\ast}$ ($-$0.097) structures are compared. As shown in Fig. 2(c), the energy of the former is lower than that of the latter by approximately 0.2 eV (see Fig. S3 in the SI for details), which is consistent with the Jahn–Teller effect, making the clusters tend toward low-energy states with low symmetry.[40-42] During axial expansion and compression, the SAMO splitting pattern, with its trend in the change in energy and the adjustment of frontier orbitals in accordance with the Jahn–Teller effect, can also be observed at the equivalent basis set with diffusion functions (for details see Fig. S4 in the SI). To explore the influences of the geometric parameters on the changes in the splitting energies of SAMOs, the changes in three characteristic C–C bond lengths during expansion and compression were counted. In the lateral two C–C bond lengths, $b_{2}$ is gradually longer and $b_{3}$ is gradually shorter as expansion proceeds. As compression proceeds, $b_{2}$ and $b_{3}$ progressively shorten and lengthen, respectively (see Fig. S3 and Table S2 in the SI for details). This result shows that adjusting the magnitude of the $b_{2}$-to-$b_{3}$ ratio under the same symmetry can be used as an important parameter to modulate the splitting spacing of molecular orbitals. In summary, we present the splitting of SAMOs with the same angular momenta during axial expansion and compression on superatoms, which is governed by point group symmetry. Moreover, further expansion and compression can change the geometrical parameters under the same symmetry to quantitatively modulate the spacing of split SAMOs. Accordingly, the crossover behaviors of the HOMO and LUMO originating from different angular momenta reduce the structural symmetry and reduce the structural energy. This phenomenon is related to the Jahn–Teller effect of reducing cluster symmetry so that the structure tends toward relatively low-energy states. In this study, the concept of SAMO splitting is extended to the entire superatomic domain, further facilitating the regulation of the corresponding electronic configuration. The present work draws renewed attention to the progress made in previous molecular physics studies and provides new perspectives for promoting the continuous development of superatomic research. Acknowledgments. The authors wish to acknowledge Wanrong Huang for the discussion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11974136 and 11674123). Z. Wang also acknowledges the assistance of the High-Performance Computing Center of Jilin University and the National Supercomputing Center in Shanghai. References Formation of Al13 I- : Evidence for the Superhalogen Character of Al13Atomlike, Hollow-Core–Bound Molecular Orbitals of C60Special and General SuperatomsGold-Caged Metal Clusters with Large HOMO−LUMO Gap and High Electron AffinitySuper Atomic Clusters: Design Rules and Potential for Building Blocks of MaterialsElectronic Structure of the Superatom: A Quasiatomic System Based on a Semiconductor HeterostructureAtomic clusters: Building blocks for a class of solidsÜber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der SpektrenSuperatoms: Electronic and Geometric Effects on ReactivitySymmetry and Electronic Structure of Noble-Metal Nanoparticles and the Role of RelativityAu 40 : A large tetrahedral magic clusterPoint Group Symmetry Analysis of the Electronic Structure of Bare and Protected Metal NanocrystalsSuperatomic Orbital Splitting in Coinage Metal NanoclustersMultipoles and SymmetryElectronic Shell Structure and Abundances of Sodium ClustersElectronic structures and spectral properties of endohedral fullerenesQuantum Mechanics, Group Theory, and C60Can an Electron-Shell Closing Model Explain the Structure and Stability of Ligand-Stabilized Metal Clusters?Toward Controlling the Electronic Structures of Chemically Modified Superatoms of Gold and SilverSuperatomic states under high pressureMaterials discovery at high pressuresNanomechanics of Carbon Tubes: Instabilities beyond Linear ResponseBuckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element methodNon-linear elastic response of fullerene balls under uniform and axial deformationsGas-phase production and photoelectron spectroscopy of the smallest fullerene, C20The Smallest Stable Fullerene, M@C28 (M = Ti, Zr, U): Stabilization and Growth from Carbon VaporJahn-Teller distortions in solid C20 and other fullerene structuresVibrational and electronic properties of neutral and negatively charged C 20 clustersBound electronic states of the smallest fullerene C20− anionWhat Are the Ground State Structures of C20 and C24 ? An Explicitly Correlated Ab Initio Approachπ-Systems in three dimensions C 60 Molecular BearingsHigh-pressure-induced electronic and structural transition of superatomsUltra-low friction of graphene/C60/graphene coatings for realistic rough surfacesStability of polyatomic molecules in degenerate electronic states - I—Orbital degeneracyConcerning Jahn-Teller EffectsJahn–Teller distortions in transition metal compounds, and their importance in functional molecular and inorganic materials
[1] Bergeron D E, Castleman Jr A W, Morisato T, and Khanna S N 2004 Science 304 84
[2] Feng M, Zhao J, and Petek H 2008 Science 320 359
[3] Luo Z X and Castleman A W 2014 Acc. Chem. Res. 47 2931
[4] Gao Y, Bulusu S, and Zeng X C 2005 J. Am. Chem. Soc. 127 15680
[5] Jena P and Sun Q 2018 Chem. Rev. 118 5755
[6] Inoshita T, Ohnishi S, and Oshiyama A 1986 Phys. Rev. Lett. 57 2560
[7] Khanna S N and Jena P 1995 Phys. Rev. B 51 13705
[8] Pauli W 1925 Z. Phys. 31 765
[9]Hund F 1981 Linienspektren und Periodisches System der Elemente (Springer: Berlin)
[10]Cowan R D 1981 The Theory of Atomic Structure and Spectra (Oakland, CA: University of California Press)
[11]Sobelman I I 1992 Atomic Spectra and Radiative Transitions (New York: Springer-Verlag)
[12] Reber A C and Khanna S N 2017 Acc. Chem. Res. 50 255
[13] Häkkinen H, Moseler M, Kostko O, Morgner N, Hoffmann M A, and von Issendorff B 2004 Phys. Rev. Lett. 93 093401
[14] Jiang D E and Walter M 2011 Phys. Rev. B 84 193402
[15] Kaappa S, Malola S, and Hakkinen H 2018 J. Phys. Chem. A 122 8576
[16] Kang S Y, Nan Z A, and Wang Q M 2022 J. Phys. Chem. Lett. 13 291
[17]Cotton F A 1991 Chemical Applications of Group Theory (Chichester: John Wiley & Sons)
[18] Gelessus A, Thiel W, and Weber W 1995 J. Chem. Educ. 72 505
[19] Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y, and Cohen M L 1984 Phys. Rev. Lett. 52 2141
[20] Guha S and Nakamoto K 2005 Coord. Chem. Rev. 249 1111
[21] Rioux F 1994 J. Chem. Educ. 71 464
[22] Jung J, Kim H, and Han Y K 2011 J. Am. Chem. Soc. 133 6090
[23] Omoda T, Takano S, and Tsukuda T 2021 Small 17 e2001439
[24] Wang R, Yang X, Huang W, Liu Z, Zhu Y, Liu H, and Wang Z 2023 iScience 26 106281
[25] Zhang L, Wang Y, Lv J, and Ma Y 2017 Nat. Rev. Mater. 2 17005
[26] Yakobson B I, Brabec C J, and Bernholc J 1996 Phys. Rev. Lett. 76 2511
[27] Cao G X and Chen X 2006 Phys. Rev. B 73 155435
[28] Xiao T and Liao K 2003 Nanotechnology 14 1197
[29] Prinzbach H, Weiler A, Landenberger P, Wahl F, Worth J, TScott L T, Gelmont M, Olevano D, and Issendorff B V 2000 Nature 407 60
[30] Dunk P W, Kaiser N K, Mulet-Gas M, Rodriguez-Fortea A, Poblet J M, Shinohara H, Hendrickson C L, Marshall A G, and Kroto H W 2012 J. Am. Chem. Soc. 134 9380
[31] Adams G B, Sankey O F, Page J B, and O'Keeffe M 1993 Chem. Phys. 176 61
[32] Galli G, Gygi F, and Golaz J C 1998 Phys. Rev. B 57 1860
[33] Yang Y F, Klaiman S, Gromov E V, and Cederbaum L S 2018 Phys. Chem. Chem. Phys. 20 17434
[34] Manna D and Martin J M 2016 J. Phys. Chem. A 120 153
[35]Carter R L 1997 Molecular Symmetry and Group Theory (New York: John Wiley & Sons)
[36] Fowler P W and Woolrich J 1986 Chem. Phys. Lett. 127 78
[37] Miura K, Kamiya S, and Sasaki N 2003 Phys. Rev. Lett. 90 055509
[38] Wang R, Liu Z, Yu F, Li J, and Wang Z 2023 J. Chem. Phys. 158 244703
[39] Li H X and Branicio P S 2019 Carbon 152 727
[40] Jahn H A and Teller E 1937 Proc. R. Soc. A 161 220
[41] Pearson R G 1975 Proc. Natl. Acad. Sci. USA 72 2104
[42] Halcrow M A 2013 Chem. Soc. Rev. 42 1784