Exploring Aromaticity Effects on Electronic Transport in Cyclo[n]carbon Single-Molecule Junctions

Cyclo[n]carbon (Cn) is one member of the all-carbon allotrope family with potential applications in next-generation electronic devices. By employing first-principles quantum transport calculations, we have investigated the electronic transport properties of single-molecule junctions of Cn, with n = 14, 16, 18, and 20, connected to two bulk gold electrodes, uncovering notable distinctions arising from the varying aromaticities. For the doubly aromatic C14 and C18 molecules, slightly deformed complexes at the singlet state arise after bonding with one Au atom at each side; in contrast, the reduced energy gaps between the highest occupied and the lowest unoccupied molecular orbitals due to the orbital reordering observed in the doubly anti-aromatic C16 and C20 molecules lead to heavily deformed asymmetric complexes at the triplet state. Consequently, spin-unpolarized transmission functions are obtained for the Au-C14/18-Au junctions, while spin-polarized transmission appears in the Au-C16/20-Au junctions. Furthermore, the asymmetric in-plane π-type hybrid molecular orbitals of the Au-C16/20-Au junctions contribute to two broad but low transmission peaks far away from the Fermi level (Ef), while the out-of-plane π-type hybrid molecular orbitals dominate two sharp transmission peaks that are adjacent to Ef, thus resulting in much lower transmission coefficients at Ef compared to those of the Au-C14/18-Au junctions. Our findings are helpful for the design and application of future cyclo[n]carbon-based molecular electronic devices.

Understanding how electrons propagate through C n is essential for their practical applications in electronic devices.Molecular junctions consisting of a single or a few molecules sandwiched between two electrodes provide a platform for studying the electronic transport properties at the molecular scale.Zhang et al. reported current-voltage (I-V) characteristics of C 18 molecular junctions with one-dimensional (1D) carbon chain electrodes, two-dimensional (2D) graphene electrodes, and three-dimensional (3D) bulk silver electrodes; however, they neglected the atomic changes of C 18 possibly brought by their contact with the electrodes [29].Other studies have focused on the I-V characteristics like negative differential resistance [30], or spintronic transport characteristics [27,28,31] like the spin filtering effect and the magnetoresistance of C 18 sandwiched between graphene nanoribbons.The electronic transport properties of other C n remain unexplored, particularly in junctions formed with conventional metal electrodes like gold.Moreover, previous studies have demonstrated the variations in the electronic structures of isolated C n based on their aromaticities [9,[16][17][18]; therefore, how the aromaticity affects the electronic propagation is very intriguing.
Herein, we present a systematic study of the electronic transport properties of C n (n = 14, 16, 18, 20) molecular junctions with gold electrodes using the non-equilibrium Green's function formalism combined with density functional theory (that is, the NEGF + DFT approach) [32][33][34][35][36][37][38][39].Our calculations reveal distinct junction geometries and spin states resulting from different C n aromaticities, which lead to significantly different electronic transport performances between the doubly aromatic C 14/18 and the doubly anti-aromatic C 16/20 after connecting to the gold electrodes.Furthermore, we extend our calculations to larger Au-C n -Au junctions involving more carbon atoms.

Results and Discussion
The optimized atomic structures of cyclo[n]carbons (n = 14, 16, 18, 20) at the ωB97XD/6-311+G(d,p) level are displayed in Figure S1.Our calculations indicate that C 14 is an intermediate with a tiny bond length alternation (BLA) of 0.047 Å and a large bond angle alternation (BAA) of 25.67 • , possessing a C 7h symmetry; in contrast, C 16 , C 18 , and C 20 , which are, respectively, of C 8h , D 9h , and C 10h symmetries, exhibit polyynic structures consisting of alternating triple and single bonds with a much larger BLA (0.15 Å, 0.12 Å, and 0.14Å, respectively) and nearly zero or no BAA (0.70 • , 0, and 0.14 • , respectively).The less symmetric atomic structures of C n with n = 16, 18, and 20 may result from the symmetry breaking event, a consequence of the second-order Jahn-Teller effect [13].Our calculations agree well with previous experimental observations [8][9][10][11] and theoretical calculations applying HF (Hartree-Fock) [40], CCSD (coupled-cluster singles and doubles) [21,41], ab initio CASSCF (complete active space self-consistent field) [20], and DFT [12,14,16,17,20,21,26].Despite sharing a similar ring structure with benzene, these C n molecules possess two types of orthogonal π MOs.One is the out-of-plane π (π out ) MOs (see the upper panel of Figure 1a), and the other is the in-plane π (π in ) MOs (see the bottom panel of Figure 1a).These two types of π orbitals are separately occupied by electrons [17].Consequently, depending on the occupancy of these two types of π MOs, C n can exhibit double aromaticity, double anti-aromaticity, or conflicting aromaticity [42].
The frontier molecular orbitals (FMOs) and their corresponding energies of C n (n = 14, 16, 18, 20) are presented in Figure S1.Our calculations suggest C 14/18 is doubly aromatic, containing 14/18 (4k + 2) π in and π out electrons.In contrast, C 16/20 is doubly anti-aromatic, since it has 16/20 (4k) π in and π out electrons.Furthermore, upon closer examination of their FMOs and energies, we observe that for the doubly aromatic C 14/18 , π orbitals appear as degenerate or nearly degenerate pairs of the same orientation and semblable structures distinguished by rotation.For instance, the HOMO and HOMO−1 orbitals of C 18 are degenerate π in orbitals.However, for the doubly anti-aromatic C 16/20 , obeying this rule will result in the occupancy of the two types of π orbitals by 4k + 2/4k − 2 electrons.Therefore, in order to preserve the double anti-aromaticity, orbital reordering occurs where one of the π out orbitals pairs with one of the π in orbitals as HOMO−1 and HOMO, while the other π out orbital related to the rotation pairs with the left π in orbital as LUMO and LUMO + 1 (see the orbital reordering process in C 20 , as shown in Figure 1b).This orbital reordering is likely to reduce the HOMO-LUMO gaps of the doubly anti-aromatic molecules.Indeed, the HOMO-LUMO gaps of the doubly anti-aromatic C 16/20 are, respectively, calculated to be 5.87 eV and 5.67 eV, which are much smaller compared to those of the doubly aromatic C 14/18 , which are 7.14 eV and 6.76 eV.The frontier molecular orbitals (FMOs) and their corresponding energies of Cn (n = 14, 16, 18, 20) are presented in Figure S1.Our calculations suggest C14/18 is doubly aromatic, containing 14/18 (4k + 2) πin and πout electrons.In contrast, C16/20 is doubly anti-aromatic, since it has 16/20 (4k) πin and πout electrons.Furthermore, upon closer examination of their FMOs and energies, we observe that for the doubly aromatic C14/18, π orbitals appear as degenerate or nearly degenerate pairs of the same orientation and semblable structures distinguished by rotation.For instance, the HOMO and HOMO−1 orbitals of C18 are degenerate πin orbitals.However, for the doubly anti-aromatic C16/20, obeying this rule will result in the occupancy of the two types of π orbitals by 4k + 2/4k − 2 electrons.Therefore, in order to preserve the double anti-aromaticity, orbital reordering occurs where one of the πout orbitals pairs with one of the πin orbitals as HOMO−1 and HOMO, while the other πout orbital related to the rotation pairs with the left πin orbital as LUMO and LUMO + 1 (see the orbital reordering process in C20, as shown in Figure 1b).This orbital reordering is likely to reduce the HOMO-LUMO gaps of the doubly anti-aromatic molecules.Indeed, the HOMO-LUMO gaps of the doubly anti-aromatic C16/20 are, respectively, calculated to be 5.87 eV and 5.67 eV, which are much smaller compared to those of the doubly aromatic C14/18, which are 7.14 eV and 6.76 eV.
Having illustrated the atomic and electronic structures of the isolated Cn molecules, we now move on to explore the electronic transport characteristics of the Au-Cn-Au molecular junctions.We construct the junctions by attaching the Cn molecules to the gold electrodes through one gold adatom on both sides without any other anchor groups, because it has been experimentally demonstrated that carbon atoms with radially oriented π orbitals can directly bond to the gold electrodes [43,44].Before probing the junction transport properties, we first explore the interactions between the Cn molecules and Au atoms.Hence, we construct isolated Au-Cn-Au complexes to examine the atomic and electronic changes in the Cn molecules after bonding to the Au atoms.The structures at both the singlet and triplet states can be obtained.By comparing the total energy of the optimized structures and considering the possibility of the Cn molecules bonding with the gold electrodes, we identify the most probable Au-Cn-Au (n = 14, 16, 18, 20) configurations appearing in the junctions (see Figure 2).Large Au-C bonding energies are obtained, of 1.60 eV, 1.45 eV, 1.39 eV, and 1.29 eV for n = 14, 16, 18, and 20, respectively.This suggests the formation of strong covalent Au-C bonds [45], and also results in severe deformations of Cn.A similar deformation has also been reported for the C18 molecular junction with graphene nanoribbon electrodes [31,46].Meanwhile, clear distinctions between the doubly aromatic and doubly anti-aromatic Cn molecules are observed after bonding with Having illustrated the atomic and electronic structures of the isolated C n molecules, we now move on to explore the electronic transport characteristics of the Au-C n -Au molecular junctions.We construct the junctions by attaching the C n molecules to the gold electrodes through one gold adatom on both sides without any other anchor groups, because it has been experimentally demonstrated that carbon atoms with radially oriented π orbitals can directly bond to the gold electrodes [43,44].Before probing the junction transport properties, we first explore the interactions between the C n molecules and Au atoms.Hence, we construct isolated Au-C n -Au complexes to examine the atomic and electronic changes in the C n molecules after bonding to the Au atoms.The structures at both the singlet and triplet states can be obtained.By comparing the total energy of the optimized structures and considering the possibility of the C n molecules bonding with the gold electrodes, we identify the most probable Au-C n -Au (n = 14, 16, 18, 20) configurations appearing in the junctions (see Figure 2).Large Au-C bonding energies are obtained, of 1.60 eV, 1.45 eV, 1.39 eV, and 1.29 eV for n = 14, 16, 18, and 20, respectively.This suggests the formation of strong covalent Au-C bonds [45], and also results in severe deformations of C n .A similar deformation has also been reported for the C 18 molecular junction with graphene nanoribbon electrodes [31,46].Meanwhile, clear distinctions between the doubly aromatic and doubly anti-aromatic C n molecules are observed after bonding with the Au atoms, including the more asymmetrical atomic structure and the spin-polarized triplet state for the Au-C 16/20 -Au complexes.We attribute this difference to the lower HOMO-LUMO gaps of the doubly anti-aromatic C 16/20 molecules, which are easier to overcome via the exchange energy acting on electrons with the same spin, favoring a structure at a triplet state after bonding with the Au atoms.
Then, we construct the junction structures with the obtained Au-C n -Au complexes, which are shown in the upper panels of Figure 3 and Figure S2.And the corresponding equilibrium transmission functions around the Fermi Level (E f ) are also plotted in the bottom panels of Figure 3 and Figure S2, with E f being set to zero.The transmission coefficients at E f (T(E f )) are, respectively, calculated to be 0.53, 0.18 (spin-up)/0.01(spindown), 0.50, and 0.16 (spin-up)/0.01(spin-down)for the C 14 , C 16 , C 18 , and C 20 junctions.Remarkable spin-polarized features with a much smaller T(E f ) are observed for the Au-C 16/20 -Au junctions compared to the Au-C 14/18 -Au junctions.This observation is somewhat perplexing because the doubly anti-aromatic molecules are expected to exhibit a higher T(E f ) due to their smaller HOMO-LUMO gaps.Attempting to elucidate this peculiar phenomenon, conducting eigenchannels were calculated to identify the dominating MOs (depicted in Figure 3 and Figure S2) [47].Then, we construct the junction structures with the obtained Au-Cn-Au complexes, which are shown in the upper panels of Figures 3 and S2.And the corresponding equilibrium transmission functions around the Fermi Level (Ef) are also plotted in the bottom panels of Figures 3 and S2, with Ef being set to zero.The transmission coefficients at Ef (T(Ef)) are, respectively, calculated to be 0.53, 0.18 (spin-up)/0.01(spin-down), 0.50, and 0.16 (spin-up)/0.01(spin-down)for the C14, C16, C18, and C20 junctions.Remarkable spin-polarized features with a much smaller T(Ef) are observed for the Au-C16/20-Au junctions compared to the Au-C14/18-Au junctions.This observation is somewhat perplexing because the doubly anti-aromatic molecules are expected to exhibit a higher T(Ef) due to their smaller HOMO-LUMO gaps.Attempting to elucidate this peculiar phenomenon, conducting eigenchannels were calculated to identify the dominating MOs (depicted in Figures 3 and S2) [47].Then, we construct the junction structures with the obtained Au-Cn-Au complexes, which are shown in the upper panels of Figures 3 and S2.And the corresponding equilibrium transmission functions around the Fermi Level (Ef) are also plotted in the bottom panels of Figures 3 and S2, with Ef being set to zero.The transmission coefficients at Ef (T(Ef)) are, respectively, calculated to be 0.53, 0.18 (spin-up)/0.01(spin-down), 0.50, and 0.16 (spin-up)/0.01(spin-down)for the C14, C16, C18, and C20 junctions.Remarkable spin-polarized features with a much smaller T(Ef) are observed for the Au-C16/20-Au junctions compared to the Au-C14/18-Au junctions.This observation is somewhat perplexing because the doubly anti-aromatic molecules are expected to exhibit a higher T(Ef) due to their smaller HOMO-LUMO gaps.Attempting to elucidate this peculiar phenomenon, conducting eigenchannels were calculated to identify the dominating MOs (depicted in Figures 3 and S2) [47].For the doubly aromatic C 14/18 , spin-unpolarized and highly similar transmission functions are obtained for their single-molecule junctions with Au electrodes, which display two apparent resonance peaks below E f (see Figures 3a and S2a).In the case of the Au-C 14 -Au junction, the transmission peaks located at −0.69 eV and −0.65 eV can be decomposed into multiple eigenchannels, resulting in their heights exceeding unity.Due to the strong coupling between C n and the two Au adatoms, MO switching happens, and some MOs heavily hybridize with each other or with some orbitals of the Au adatoms, generating new orbitals contributing to the junction transmission.The shapes of the eigenchannels suggest that after connecting to the gold electrodes, HOMO−1 and HOMO exchange their energetic order.The HOMO and LUMO of the isolated deformed C 14 molecule (C 14_D ) hybridize and contribute to one eigenchannel of the peak centered at −0.69 eV (see FMOs and their hybrid shown in Figure 4a), along with the other one, which is dominated by the HOMO.Moreover, besides the one eigenchannel contributed by the HOMO, the HOMO−1 of C 14_D combining with the 6s and 5d z2 , 5d yz orbitals of the Au adatoms forms two new orbitals contributing to the peak at −0.65 eV.And the sharpness of this peak arises from the nodes of the HOMO−1 at the C atoms bonded to the Au adatoms, leading to reduced coupling.Despite the large transmission coefficients of these two peaks, they decay rapidly towards E f and thus contribute negligibly to the transmission around E f .In contrast, the LUMO of C 14_D , which exhibits a much stronger coupling with the 6s orbitals of the Au adatoms due to its π in character, results in a broadened and steady transmission between −0.5 eV and 1 eV and thus dominates T(E f ).The overall shape of the transmission spectrum of the Au-C 18 -Au molecular junction is similar to that of the Au-C 14 -Au junction (see the dominating MOs of the isolated deformed C 18 molecule (C 18_D ) contributing to the transmission peaks shown in Figures S2a and S3a), except that the two transmission peaks change their places because of the different coupling effects between C 18_D and C 14_D with the Au electrodes; additionally, they also move towards E f due to the smaller HOMO-LUMO gap of C 18_D .
exchange their energetic order.The HOMO and LUMO of the isolated deformed C14 mol ecule (C14_D) hybridize and contribute to one eigenchannel of the peak centered at −0.69 eV (see FMOs and their hybrid shown in Figure 4a), along with the other one, which i dominated by the HOMO.Moreover, besides the one eigenchannel contributed by the HOMO, the HOMO−1 of C14_D combining with the 6s and 5dz2, 5dyz orbitals of the Au ada toms forms two new orbitals contributing to the peak at −0.65 eV.And the sharpness o this peak arises from the nodes of the HOMO−1 at the C atoms bonded to the Au adatoms leading to reduced coupling.Despite the large transmission coefficients of these two peaks, they decay rapidly towards Ef and thus contribute negligibly to the transmission around Ef.In contrast, the LUMO of C14_D, which exhibits a much stronger coupling with the 6s orbitals of the Au adatoms due to its πin character, results in a broadened and steady transmission between −0.5 eV and 1 eV and thus dominates T(Ef).The overall shape of the transmission spectrum of the Au-C18-Au molecular junction is similar to that of the Au C14-Au junction (see the dominating MOs of the isolated deformed C18 molecule (C18_D contributing to the transmission peaks shown in Figures S2a and S3a), except that the two transmission peaks change their places because of the different coupling effects between C18_D and C14_D with the Au electrodes; additionally, they also move towards Ef due to the smaller HOMO-LUMO gap of C18_D.For the doubly anti-aromatic C16/20, the difference between the spin-up and spin down population of the Cn molecules in the junctions is calculated to be 1.53, leading to remarkable spin-polarized transmission functions (see Figures 3b and S2b).This spin-un polarized to spin-polarized transition of the C16/20 molecules indicates their much stronge electronic couplings with the Au adatoms so that the FMOs of the isolated Au-C16/20-Au complexes are employed to analyze the dominating MOs of the transmission peaks (see For the doubly anti-aromatic C 16/20 , the difference between the spin-up and spindown population of the C n molecules in the junctions is calculated to be 1.53, leading to remarkable spin-polarized transmission functions (see Figures 3b and S2b).This spinunpolarized to spin-polarized transition of the C 16/20 molecules indicates their much stronger electronic couplings with the Au adatoms so that the FMOs of the isolated Au-C 16/20 -Au complexes are employed to analyze the dominating MOs of the transmission peaks (see Figure 4b and Figure S3b).The transmission function of the Au-C 16 -Au junction reveals two prominent peaks for both the spin-up and spin-down channels located at −1 eV and −0.11 eV (spin-up)/0.29 eV (spin-down).For the spin-up channel, the peak away from E f is dominated by one hybrid MO composed of the HOMO−3 and HOMO−2 orbitals of the isolated Au-C 16 -Au complex.While its π in character exhibits a significant coupling to one gold electrode, leading to the broadening of the peak, its asymmetric nature also results in a much lower peak height.Inspecting the HOMO−3 and HOMO−2 orbitals, we can find that these two MOs exhibit similar orbital shape patterns and adjacent energies.This observation suggests that a potential rule for MO hybridization may exist for these doubly anti-aromatic C n molecules in the junctions.The spin-up HOMO−1 and HOMO orbitals are another pair that satisfies this criterion; indeed, they hybridize and dominate the transmission peak centered at −0.11 eV.However, as this π out MO has less asymmetry, this hybrid MO contributes to the relatively higher peak decaying rapidly towards E f .Likewise, for the spin-down channel, the asymmetric hybrid π in MO formed by HOMO−1 and HOMO dominates the broadened yet low peak centered at −1 eV, while the less asymmetric hybrid π out MO composed of the LUMO and LUMO + 1 dominates the relatively high but sharp peak located at 0.29 eV.Therefore, although both the spin-up and spin-down resonance peaks are much closer to E f , their asymmetric π out nature makes them decay quickly towards E f , thus the total transmission at E f is still less than that of the Au-C 14/18 -Au junctions.The overall shape of the transmission function of the Au-C 20 -Au molecular junction is nearly identical to that of the Au-C 16 -Au junction, as shown in Figures S2b and S3b.
Finally, we further explore the electronic transport properties of larger Au-C n -Au junctions containing more carbon atoms (n = 22, 24, 26, 28).For the doubly aromatic C 22/26 , they form slightly deformed complexes at the singlet state after bonding with two Au atoms like C 14/18 , and their molecular junctions share similar spin-unpolarized transmission functions with the Au-C 14/18 -Au junctions, possessing a large T(E f ) (see Figure S4a,c).Meanwhile, the reduced HOMO-LUMO gaps of the doubly anti-aromatic C 24/28 molecules lead to heavily deformed and asymmetric complexes at the triplet state after bonding to two Au atoms, so that their molecular junctions have spin-polarized transmission functions and a small T(E f ), resembling the transport characteristics observed in the Au-C 16/20 -Au junctions (see Figure S4b,d).

Computational Methods
Geometry optimization and electronic structure calculations of the isolated C n and Au-C n -Au complexes were carried out by employing the Gaussian16 DFT package with the range-separated hybrid functional ωB97XD [48], which has been proven to provide the correct qualitative polyynic structure of cyclo[n]carbons (n > 14) [11,16,20,21].The 6-311+G(d,p) basis set is employed for carbon atoms; the SDD basis set is employed for gold atoms [49][50][51].Frequency analysis is performed to verify the nature of the stationary points.For comparison, the generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE) formulation [52] is also used for calculating the molecular orbitals (MOs) of the Au-C n -Au complexes and the deformed C n molecules in the gas phase.
Quantum transport calculations were performed using TRANSISESTA code, which is a practical implementation of the NEGF + DFT approach, employing SIESTA as the DFT engine [53][54][55].The atom cores are described using improved Troullier-Martins norm-conserving pseudopotentials (PPs), and the wave functions of the valence electrons are expanded over a finite-range numerical orbital basis set [56].The default doublezeta plus polarization (DZP) basis set is used for Au atoms and a user-defined DZP is constructed for C atoms.The PBE GGA is used as the approximate exchange-correlation functional.Although GGA, like PBE, often underestimates the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) since it is unable to describe the derivative discontinuity of the DFT potential [57], we have compared the MOs of the Au-C n -Au complexes and the deformed C n molecules at both the ωb97XD and PBE levels (see Figure S5); their similar MO shapes and energy orders suggest that our calculations are able to qualitatively describe the transport properties of the Au-C n -Au molecular junctions.The unit cell of the extended molecule is composed of the central C n molecule, the gold adatoms, and ten Au (111) atomic layers with a 4 × 5 in-plane supercell.The C n molecule is placed in the y-z plane, and the transport is defined along the z-axis.We always consider periodic boundary conditions in the plane transverse to the transport.An equivalent energy cutoff of 200.0 Ry is taken for the real-space mesh, and the Brillouin zone is sampled using a 4 × 4 × 1 k-point mesh.The spin-polarized transmission function T σ (E) for the spin-up and spin-down electrons (σ =↑/↓) is evaluated as follows: where Ω 2DBZ is the area of the two-dimensional Brillouin zone (2DBZ) that is orthogonal to the transport direction.The k-dependent transmission coefficient T σ → k ; E is defined as follows: where G r M,σ is the retarded Green's function matrix of the extended molecule, and Γ L/R, σ represents the broadening function matrix describing the interaction between the left/right electrode and the extended molecule.In a spin-unpolarized junction, the same transmission functions are obtained for both the spin-up and spin-down channels.

Conclusions
We have systemically investigated the electronic transport properties of cyclo[n]carbon (n = 14,16,18,20) single-molecule junctions connected to bulk gold electrodes by employing the NEGF+DFT approach.Our calculations show that the isolated C 14 and C 18 molecules exhibit double aromaticity, while the C 16 and C 20 molecules in the gas phase display double anti-aromaticity with a smaller HOMO-LUMO gap associated with orbital reordering.The bonding to Au atoms induces the deformation of the C n , especially for the doubly antiaromatic C 16/20 molecules for which the corresponding more asymmetric complexes are in the triplet state due to their reduced HOMO-LUMO gaps.Distinguished spin states and junction geometries originating from different aromaticities lead to significantly different transport properties between these doubly aromatic and anti-aromatic C n molecules when they form Au-C n -Au single-molecule junctions.In detail, the Au-C 14/18 -Au junctions demonstrate a spin-unpolarized transport with a relatively large transmission around E f .In contrast, spin-polarized transmission occurs in the Au-C 16/20 -Au molecular junctions, and the asymmetric hybrid π in MOs contribute to two broad but low peaks well below E f , while the two peaks adjacent to E f are dominated by hybrid π out MOs decaying fast towards E f , which result in a much smaller T(E f ).We further extend our studies to larger junctions with more carbon atoms, and obtain very similar transport characteristics for cyclo[n]carbons with n = 22, 24, 26, and 28.These findings are beneficial to design and construct molecular electronic devices with cyclo[n]carbon molecules.

Figure 1 .
Figure 1.(a) Visualizations of the πout (upper) and πin MOs (bottom) of the C20 molecule.(b) The orbital reordering process in the anti-aromatic C20 molecule.

Figure 1 .
Figure 1.(a) Visualizations of the π out (upper) and π in MOs (bottom) of the C 20 molecule.(b) The orbital reordering process in the anti-aromatic C 20 molecule.
Molecules 2024, 29, x FOR PEER REVIEW 4 of 10 the Au atoms, including the more asymmetrical atomic structure and the spin-polarized triplet state for the Au-C16/20-Au complexes.We attribute this difference to the lower HOMO-LUMO gaps of the doubly anti-aromatic C16/20 molecules, which are easier to overcome via the exchange energy acting on electrons with the same spin, favoring a structure at a triplet state after bonding with the Au atoms.

Figure 2 .
Figure 2. The atomic structures for the most probable Au-Cn-Au (n = 14, 16, 18, 20) complexes and their spin states appearing in the corresponding molecular junctions.The arrows point to the C atoms that are bonded with the Au atoms after coupling.

Figure 3 .
Figure 3.The atomic junction structures (upper panel) and the corresponding equilibrium transmission spectra (bottom panel) of the Au-C14-Au (a) and Au-C16-Au (b) molecular junctions, together with the conducting eigenchannels for selected transmission peaks.

Figure 2 .
Figure 2. The atomic structures for the most probable Au-C n -Au (n = 14, 16, 18, 20) complexes and their spin states appearing in the corresponding molecular junctions.The arrows point to the C atoms that are bonded with the Au atoms after coupling.

Figure 2 .
Figure 2. The atomic structures for the most probable Au-Cn-Au (n = 14, 16, 18, 20) complexes and their spin states appearing in the corresponding molecular junctions.The arrows point to the C atoms that are bonded with the Au atoms after coupling.

Figure 3 .
Figure 3.The atomic junction structures (upper panel) and the corresponding equilibrium transmission spectra (bottom panel) of the Au-C14-Au (a) and Au-C16-Au (b) molecular junctions, together with the conducting eigenchannels for selected transmission peaks.

Figure 3 .
Figure 3.The atomic junction structures (upper panel) and the corresponding equilibrium transmission spectra (bottom panel) of the Au-C 14 -Au (a) and Au-C 16 -Au (b) molecular junctions, together with the conducting eigenchannels for selected transmission peaks.

Figure 4 .
Figure 4. FMOs and the hybrid MOs of the isolated C14_D molecule (a) and the Au-C16-Au complex (b).

Figure 4 .
Figure 4. FMOs and the hybrid MOs of the isolated C 14_D molecule (a) and the Au-C 16 -Au complex (b).

:
Optimized atomic structures and FMOs of the isolated C n molecules with n = 14, 16, 18, and 20 calculated at the ωB97XD/6-311+G(d, p) level; Figure S2: The atomic structures (upper panel) and the corresponding equilibrium transmission spectra (bottom panel) of the Au-C 18 -Au (a) and Au-C 20 -Au (b) molecular junctions, together with the conducting eigenchannels for selected transmission peaks; Figure S3: FMOs and the hybrid MOs of the isolated C 18_D molecule (a) and the Au-C 20 -Au complex (b); Figure S4: The equilibrium transmission spectra of the Au-C 22 -Au (a), Au-C 24 -Au (b), Au-C 26 -Au (c) and Au-C 28 -Au (d) molecular junctions are displayed in the left panels, and their corresponding atomic structures are shown in the right panels; Figure S5: FMOs of the isolated C 14_D/18_D molecules (a) and the Au-C 16/20 -Au complexes (b) calculated at both the ωB97XD and PBE levels; Cartesian coordinates of the Au-C n -Au molecular junctions (unit: Å).Author Contributions: Conceptualization, P.Y. and S.H.; theoretical calculations and data analysis, P.Y., H.P., and Y.W. (Yudi Wang); writing-original draft, P.Y.; visualization, P.Y., H.P., J.L., and Y.D.; writing-review and polishing, S.H. and Y.W. (Yongfeng Wang).All authors have read and agreed to the published version of the manuscript.Funding: This project was funded by the National Natural Science Foundation of China (grant no.21933002).Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.Data Availability Statement: Data are available from the corresponding author on request.