Abstract
It has been extremely difficult for traditional theoretical methods to adequately predict the properties of systems possessing radical character (i.e., multireference systems), especially for multireference systems at the nanoscale. To circumvent this, we employ thermallyassistedoccupation density functional theory (TAODFT) to predict the electronic properties of Möbius cyclacenes, with the number of fused benzene rings (n) ranging from 8 to 100. In addition, to investigate the significance of Möbius topology, we also compare these properties with the respective properties of cyclacenes and acenes, containing the same number of fused benzene rings. From our TAODFT results, Möbius cyclacenes, cyclacenes, and acenes have singlet ground states for all the cases examined. However, unlike acenes, the electronic properties of Möbius cyclacenes and cyclacenes display clear oscillation patterns when n is small (e.g., n ≤ 10 for Möbius cyclacenes and n ≤ 23 for cyclacenes), and converge to the respective properties of acenes when n greatly exceeds 30. The polyradical character of the ground states of Möbius cyclacenes should increase with the molecular size, intimately correlated with the localization of active orbitals at the edges of molecules.
Introduction
Recently, graphene nanoribbons (GNRs) have emerged as promising quasionedimensional (Q1D) materials for nextgeneration electronic nanodevices^{1,2,3,4,5,6,7}. Because of the effect of edges and quantum confinement, GNRs can exhibit band gaps for transistor operation with exceptional switching speed and high carrier mobility. However, the properties of GNRs are intimately correlated with the geometrical arrangements of GNRs; they are rendered into semimetal without band gaps when widening into their twodimensional parent material, graphene. Particularly, GNRs with zigzag edges (ZGNRs) are expected to host edgelocalized states, which may serve as key elements for graphenebased electronics and spintronics. Therefore, a comprehensive structural analysis of GNRs is of fundamental importance for building highperformance GNRbased nanodevices.
While the local geometries of nanomaterials have profound consequences on their properties, such as the surface states affecting the bulk crystals, and the edge states affecting graphene, the global topology of Möbius graphene nanoribbons (MGNRs), with nonorientable global invariants, can significantly influence their electronic properties. Note that a MGNR can be constructed by joining the two ends of a ZGNR with a single halftwist. Accordingly, a MGNR has only one edge and one surface. Due to their intriguing topologies, MGNRs have attracted extensive attention from the research community. Theoretical predictions of their electronic^{8,9,10}, magnetic^{11,12}, transport^{13}, and thermal^{14,15} properties have been made in recent years. Furthermore, MGNRs were predicted to behave as topological insulators^{16,17}. Whereas the conducting edge states of ZGNRs are very fragile, for systems with nontrivial topology, such as MGNRs, the Hamiltonians for the edge states are invariant to small perturbations, bringing tremendous possibilities to realize topological insulators^{18}.
In particular, the properties of the fundamental repeating units of MGNRs may require further investigation. In the present work, our investigation focuses on a series of Möbius ncyclacenes (see Fig. 1). Note that Möbius ncyclacene can be constructed by joining the two ends of nacene (i.e., a Q1D molecule containing n fused benzene rings^{19,20}) with a single halftwist. Möbius ncyclacenes, which belong to the category of aromatic hydrocarbons, are delocalized πconjugated systems. Aromaticity is a key concept in chemistry in that aromatic molecules display enhanced chemical stability and induced aromatic ring currents^{21}. The peculiar chemical properties of aromatics are amplified by Möbius topologies, where the topics of Möbius aromaticity have recently attracted considerable interest^{22,23}. Aromaticity is characterized by the number of delocalized πelectrons. For the untwisted case, i.e., ncyclacene (as shown in Fig. 1 of ref.^{24}), there are two annulene peripheral circuits joined by transannular bonds to form n fused benzene rings. The annulene periphery, following Hückel’s rule, is only stable (aromatic), when the number of πelectrons is 4k + 2, with k being an integer (corresponding to an odd number of benzene rings). Möbius ncyclacene, however, has only one edge and thus only one annulene periphery of twofold the size as the comparing the untwisted case. Therefore, the single annulene periphery forming Möbius ncyclacene always includes 4k πelectrons, irrespective of the number of benzene rings, which violates Hückel’s rule. However, Zimmerman^{25} and Heilbronner^{26} argued that for the Möbius topology, these molecules rather need 4k πelectrons to achieve aromaticity and stability. The single halftwist of the structure was concluded to be correlated with the interesting properties of Möbius ncyclacenes.
Note that the Möbius strip was first discovered by August Ferdinand Möbius (a German mathematician) in 1858. While the first prediction on the possibility of forming Möbius aromaticity was made in 1964^{26}, the first stable Möbius aromatic hydrocarbon was synthesized only very recently^{27}. Although the prospects of successful synthesis of Möbius cyclacenes may not be ideal^{28}, progress has been made in synthesizing different kinds of microscopic Möbius strips: the Möbius structure formed by NbSe_{2} crystals was obtained under unconventional growth conditions^{29}; the design and synthesis of the first triply twisted Möbius annulene was recently proposed^{30}. Besides, twisted GNRs were obtained inside carbon nanotubes (CNTs), offering the possibility of building MGNRs^{31}. Recently, the successful onsurface generation of triangulene was reported^{32}. Note that triangulene, consisting of six fused benzene rings, belongs to the category of topologically nontrivial structures, such as Möbius ncyclacenes. Studying Möbius ncyclacenes may be crucial for the atomically controlled bottomup fabrication of MGNRs as well. Therefore, a comprehensive study of the properties of Möbius ncyclacenes is of essential importance. These properties are expected to be constructive for probing the potential applications of their quantum effects induced by the nontrivial topological configurations.
As of now, the studies of Möbius ncyclacenes have mainly been performed theoretically; yet it remains rather difficult to make reliable prediction on the electronic properties of the larger Möbius ncyclacenes, possibly due to their πconjugation and polyradical character. Note that KohnSham density functional theory (KSDFT)^{33} employing traditional (semilocal^{10,11} and hybrid^{34,35,36}) exchangecorrelation (XC) functionals may not reliably predict the properties of multireference (MR) systems (i.e., systems possessing radical character)^{37,38,39}. Note that πconjugated polyradical systems usually require ab initio MR electronic structure methods^{19,20,40,41,42}. However, calculations based on ab initio MR electronic structure methods are computationally infeasible for MR systems at the nanoscale (especially for geometry relaxation). Accordingly, it remains extremely difficult for traditional theoretical methods to adequately predict the properties of the larger Möbius ncyclacenes.
To achieve a favorable balance between cost and performance for studying MR systems at the nanoscale, thermallyassistedoccupation density functional theory (TAODFT)^{43} has recently been proposed. From the physical statements provided in Section III.E of ref.^{43} and the numerical results given in Section IV of ref.^{43}, the static correlation energy of a system can be adequately described by the entropy contribution (which can be expressed by the fictitious temperature (θ) and orbital occupation numbers in TAODFT), even for TAODFT employing a local XC density functional. Just like the static correlation energy of a system, the entropy contribution in TAODFT, which is always nonpositive, is negligible for a singlereference (SR) system (i.e., a system possessing nonradical character), and can significantly lower the total energy of a MR system. Accordingly, TAODFT reduces to KSDFT for SR systems, and outperforms KSDFT for MR systems. Existing semilocal and hybrid XC functionals in KSDFT may be employed in TAODFT as well^{44,45}. Very recently, a selfconsistent scheme for determining the fictitious temperature in TAODFT has also been proposed for a diverse range of applications^{46}.
Since TAODFT is similar to KSDFT in computational efficiency, TAODFT has been widely applied to study the electronic properties of various MR systems at the nanoscale^{24,47,48,49,50,51,52}. Therefore, in the present study, we adopt TAODFT to investigate the electronic properties of Möbius ncyclacenes (n = 8–100). Besides, to assess the significance of Möbius topology, we also compare the electronic properties of Möbius ncyclacenes with the respective properties of ncyclacenes^{24,53,54} and nacenes^{19,20,43,44,47,55}.
Computational Details
We perform all calculations with QChem 4.3^{56}, adopting the 6–31 G(d) basis set and the numerical grid containing 75 radial points in the EulerMaclaurin quadrature and 302 angular points in the Lebedev grid. Results are obtained from TAOLDA^{43} (TAODFT adopting the local density approximation (LDA) XC functional^{57,58} and the LDA θdependent functional) with θ = 7 mhartree.
Here, we briefly explain the reason that θ = 7 mhartree is chosen. In our previous study^{43}, TAOLDA (with some fictitious temperature θ) has been shown to perform reasonably well for MR systems, providing that the corresponding orbital occupation numbers are close to the natural orbital occupation numbers (NOONs). In such a situation, the strong static correlation effects can be adequately described by the entropy contribution of TAOLDA. However, this implies that a θ related to the NOONs should be adopted. For simplicity and computational efficiency, TAOLDA with a systemindependent θ is favorable. Accordingly, in our previous study^{43}, the optimal θ value has been defined as the largest θ value for which TAOLDA performs similarly to KSLDA (i.e., KSDFT with the LDA XC functional, which is TAOLDA with θ = 0) for SR systems, yielding an optimal θ = 7 mhartree based on the numerical investigations. TAOLDA (θ = 7 mhartree), though not optimal for all systems, has been shown to consistently improve upon KSLDA for MR systems, while performing similarly to KSLDA for SR systems. Besides, in a recent study^{48}, the orbital occupation numbers obtained from TAOLDA (θ = 7 mhartree) have been shown to be qualitatively similar to the NOONs obtained from the activespace variational twoelectron reduceddensitymatrix (RDMCASSCF) method^{41} (i.e., an accurate MR electronic structure method), yielding a similar trend for the radical character of the 24 alternant PAHs (polycyclic aromatic hydrocarbons) studied. Due to its computational efficiency and reasonable accuracy, we adopt TAOLDA (θ = 7 mhartree) in the present study.
Note also that TAODFT has been extended to the generalizedgradient approximation (GGA) XC functionals^{44}. However, in TAODFT, the GGAs improve upon the LDA mainly for the properties governed by shortrange XC effects, not for the properties governed by strong static correlation effects. As shown in our previous study^{44}, in TAODFT, the GGAs have similar performance as the LDA for the electronic properties of nacenes (i.e., systems with strong static correlation effects). Since ncyclacene can be regarded as an interconnection of nacene, and Möbius ncyclacene can be regarded as ncyclacene with a single halftwist, we expect that the electronic properties of Möbius ncyclacene/ncyclacene/nacene obtained with the LDA and GGAs in TAODFT should remain similar, especially for very large n.
Results and Discussion
SingletTriplet Energy Gap
To determine the ground state of Möbius ncyclacene/ncyclacene/nacene (n = 8–100), we perform calculations based on spinunrestricted TAOLDA to obtain the lowest singlet and lowest triplet states of Möbius ncyclacene/ncyclacene/nacene, with the corresponding geometries being completely relaxed. Subsequently, we calculate the singlettriplet energy gap of Möbius ncyclacene/ncyclacene/nacene as
with E_{T} and E_{S} being the lowest triplet and lowest singlet energies, respectively, of Möbius ncyclacene/ncyclacene/nacene.
From our results (see Figs 2 and 3), Möbius ncyclacenes, ncyclacenes, and nacenes have singlet ground states for all the cases studied (i.e., n = 8–100). As n increases, the E_{ST} value of nacene decreases monotonically. By contrast, the smaller Möbius ncyclacenes (e.g., n ≤ 10) and the smaller ncyclacenes (e.g., n ≤ 23) exhibit cryptoannulenic effects^{8}, displaying oscillatory patterns in the respective E_{ST} values. Based on the oscillation amplitudes of the E_{ST} values, ncyclacenes should exhibit more prominent cryptoannulenic effects than Möbius ncyclacenes. When n greatly exceeds 30, the E_{ST} values of Möbius ncyclacene and ncyclacene monotonically converge from below to the E_{ST} value of nacene (see Supplementary Information (SI), Table S1 for details).
As mentioned previously, for πconjugated systems, such as Möbius ncyclacenes, the predictions from KSDFT employing traditional XC density functional can be problematic^{37,38,39}. For example, Möbius ncyclacenes (n = 8–20) were previously predicted to have triplet ground states, based on KSB3LYP (the B3LYP hybrid functional in KSDFT)^{36}. The contradictory results obtained from KSB3LYP can be artifacts, intimately correlated with spin contamination (i.e., the artificial mixing of different electronic spinstates)^{10,11,24,36,43,44,47,52,53,59,60}. Note that spin contamination is not a systematic error, and hence, the energy difference between spinstates can be adversely influenced.
Because of the symmetry constraint^{24,43,44,47,55}, for the exact theory, the lowest singlet energy of Möbius ncyclacene/ncyclacene/nacene obtained from spinrestricted calculations should be the same as the corresponding energy obtained from spinunrestricted calculations. However, this condition may not be satisfied by KSDFT with traditional XC functionals (because of the spin contamination mentioned above), as shown in recent studies on Möbius ncyclacenes^{10,11,36}, ncyclacenes^{24,53}, and nacenes^{19,20,43,44,47,55}. To assess whether this condition can be satisfied by TAOLDA, we additionally perform calculations based on spinrestricted TAOLDA to obtain the lowest singlet states of Möbius ncyclacenes, ncyclacenes, and nacenes, with the corresponding geometries being fully relaxed. The lowest singlet energy of Möbius ncyclacene/ncyclacene/nacene obtained from spinrestricted TAOLDA is found to be numerically identical to the corresponding energy obtained from spinunrestricted TAOLDA, implying that our calculations based on spinunrestricted TAOLDA do not lead to unphysical symmetrybreaking solutions.
Vertical Ionization Potential, Vertical Electron Affinity, and Fundamental Gap
Here, we investigate the possibility of Möbius ncyclacene/ncyclacene/nacene for photovoltaic applications. At the completely relaxed geometry of the ground state (i.e., the lowest singlet state) of Möbius ncyclacene/ncyclacene/nacene, we perform calculations based on spinunrestricted TAOLDA to obtain the vertical ionization potential
vertical electron affinity
and fundamental gap
where E_{total}(neutral), E_{total}(cation), and E_{total}(anion) are the total energies of the neutral, cationic, and anionic states.
As shown in Fig. 4, with the increase of n, the IP_{v} value of nacene decreases monotonically, the EA_{v} value of nacene increases monotonically, and hence, the E_{g} value of nacene decreases monotonically. By contrast, as n increases, the IP_{v} values of Möbius ncyclacene and ncyclacene decrease with oscillatory patterns, and the EA_{v} values of Möbius ncyclacene and ncyclacene increase with oscillatory patterns. Note that the oscillation amplitudes of the IP_{v} and EA_{v} values of ncyclacenes are larger than those of Möbius ncyclacenes, implying that ncyclacenes should possess more prominent cryptoannulenic effects than Möbius ncyclacenes. Besides, with increasing n, these oscillatory patterns reduce gradually, and disappear eventually. When n greatly exceeds 30, the IP_{v} values of Möbius ncyclacene and ncyclacene monotonically converge from above to the IP_{v} value of nacene, the EA_{v} values of Möbius ncyclacene and ncyclacene monotonically converge from below to the EA_{v} value of nacene, and the E_{g} values of Möbius ncyclacene and ncyclacene monotonically converge from above to the E_{g} value of nacene (see SI, Tables S2 to S4 for details). Particularly, the E_{g} value of Möbius ncyclacene (n = 13–55) is between 1 and 3 eV (i.e., the desirable regime for photovoltaic applications).
Symmetrized von Neumann Entropy
Here, we examine the potential polyradical character of Möbius ncyclacene/ncyclacene/nacene by calculating the symmetrized von Neumann entropy^{24,44,47,55}
for the ground state of Möbius ncyclacene/ncyclacene/nacene using spinunrestricted TAOLDA. In Eq. (5), f_{i,σ} (i.e., the occupation number of the i^{th} σspin orbital (σ = α or β) calculated using spinunrestricted TAOLDA), which takes a value between zero and one, is close to the occupation number of the i^{th} σspin natural orbital^{43,44,45,48}. For a SR system ({f_{i,σ}} are approximately equal to either zero or one), S_{vN} is very small. Nevertheless, for a MR system ({f_{i,σ}} are very different from either zero or one for active spinorbitals (i.e., fractionally occupied spinorbitals), and are approximately equal to either zero or one for others), S_{vN} increases with the number of fractionally occupied spinorbitals.
As presented in Fig. 5, the S_{vN} value of nacene increases monotonically with the molecular size. By contrast, with the increase of n, the S_{vN} values of Möbius ncyclacene and ncyclacene increase with oscillatory patterns. However, these oscillatory patterns reduce gradually, and disappear eventually, as n increases. When n greatly exceeds 30, the S_{vN} values of Möbius ncyclacene and ncyclacene monotonically converge from above to the S_{vN} value of nacene (see SI, Table S5 for details). Accordingly, just like previous findings for ncyclacenes^{24,54} and nacenes^{19,20,43,44,45,47,55}, the polyradical character of the ground states of Möbius ncyclacenes should increase with the molecular size.
Occupation Numbers of Active Orbitals
The occupation numbers of active orbitals in TAODFT provide valuable information for directly assessing the polyradical character of Möbius ncyclacene^{43,44,45,48}. To further demonstrate the reasons of the increase of S_{vN} with the size of molecule, the occupation numbers of active orbitals for the ground state of Möbius ncyclacene, obtained with spinrestricted TAOLDA, are plotted in Fig. 6. For Möbius ncyclacene (with N electrons), the highest occupied molecular orbital (i.e., the (N/2)^{th} orbital) and lowest unoccupied molecular orbital (i.e., the (N/2 + 1)^{th} orbital) are referred to as the HOMO and LUMO, respectively^{24,43,47}. As shown, with the increase of n, more and more orbitals have an occupation number close to one (i.e., more and more spinorbitals have an occupation number close to 0.5), supporting that the polyradical character of the ground states of Möbius ncyclacenes should increase with the molecular size.
Visualization of Active Orbitals
The active orbitals, such as HOMOs and LUMOs, for the ground states of a few illustrative Möbius ncyclacenes (n = 10–15), calculated using spinrestricted TAOLDA, are plotted in Figs 7 and 8. As shown, the active orbitals are mostly concentrated at the edges of Möbius ncyclacenes. Note that the visualization of active orbitals for the ground states of a few illustrative nacenes (as presented in Fig. 14 of ref.^{47}) and ncyclacenes (as presented in Fig. 10 of ref.^{24}), obtained with spinrestricted TAOLDA, can be found in previous studies for comparison. Just like previous findings for ncyclacenes^{24,54} and nacenes^{19,20,47,55}, the increasing polyradical character of the larger Möbius ncyclacenes should be intimately related to the localization of active orbitals at the edges of molecules, clearly increasing with the increase of n.
GroundState Geometry
There is a close relationship between curvature and topology in nanocarbon systems^{61}. To investigate the effect of the groundstate geometry of Möbius ncyclacene on the electronic properties, we examine the Gaussian curvature of Möbius ncyclacene. Note that since nacene possesses a planar geometry, and ncyclacene possesses a cylindrical geometry, the Gaussian curvature of nacene/ncyclacene is zero everywhere.
In the present study, we adopt the initial starting geometry of Möbius ncyclacene given by the one with a twist evenly distributed along the whole chain^{28}, where the length and width of the chain are taken from the length and width, respectively, of nacene. After full relaxation, the groundstate geometry of Möbius ncyclacene (with n = 100 as an illustrative example), colored by Gaussian curvature, obtained with spinunrestricted TAOLDA, is plotted. Here, to determine the discrete approximation to the Gaussian curvature, we employ a local least squares surface fit of cubic polynomials for knearest neighbors (with k = 50). As shown in Fig. 9, the groundstate (i.e., energetically preferred) geometry of Möbius ncyclacene is composed mostly of an essentially untwisted open chain (see the red region with zero Gaussian curvature) plus a highly twisted stripe (see the blue region with negative Gaussian curvature), i.e., the twist is not evenly distributed along the whole chain.
Conclusions
In summary, the electronic properties (e.g., E_{ST}, IP_{v}, EA_{v}, E_{g}, S_{vN}, and the occupation numbers and visualization of active orbitals) of Möbius ncyclacenes (n = 8–100) have been studied using the recently proposed TAODFT, because of its computational efficiency and reasonable accuracy for the study of MR systems at the nanoscale. Since the ground states of the larger Möbius ncyclacenes have been shown to possess increasing polyradical character, calculations based on traditional XC functionals in KSDFT may not adequately predict the properties of Möbius ncyclacenes, and calculations based on ab initio MR electronic structure methods are computationally intractable for MR systems at the nanoscale (e.g., the larger Möbius ncyclacenes). Consequently, the use of TAODFT in the present study is well justified. Based on our TAODFT results, the larger Möbius ncyclacenes, which have the smaller E_{ST} values, smaller E_{g} values, larger S_{vN} values, and more pronounced polyradical character, should exhibit stronger static correlation effects than the smaller Möbius ncyclacenes.
To examine the significance of Möbius topology, we have also compared the electronic properties of Möbius ncyclacenes with the respective properties of ncyclacenes and nacenes. Möbius ncyclacenes, ncyclacenes, and nacenes have singlet ground states for all the cases studied (i.e., n = 8–100). However, unlike nacenes, the electronic properties of Möbius ncyclacenes and ncyclacenes reveal oscillatory patterns when n is small (e.g., n ≤ 10 for Möbius ncyclacenes and n ≤ 23 for ncyclacenes), and converge to the respective properties of nacenes when n greatly exceeds 30. These oscillatory patterns should be intimately correlated with the cryptoannulenic effects of Möbius ncyclacenes and ncyclacenes, which have been found to be significant only for the smaller n. Just like previous findings for ncyclacenes and nacenes, the increasing polyradical character of the larger Möbius ncyclacenes should be intimately related to the localization of active orbitals at the edges of molecules, clearly increasing with the increase of n.
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Acknowledgements
This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST1072628M002005MY3), National Taiwan University (Grant Nos NTUCC107L892906; NTUCCP106R891706; NTUCDP105R7818), and the National Center for Theoretical Sciences of Taiwan.
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The project was conceived and designed by J.H.C. and J.D.C. The calculations were performed by J.H.C. The data analysis was performed by J.H.C. and J.D.C. The manuscript was written by J.H.C. and J.D.C.
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Supplementary Material to: Electronic Properties of Möbius Cyclacenes Studied by ThermallyAssistedOccupation Density Functional Theory
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Chung, JH., Chai, JD. Electronic Properties of Möbius Cyclacenes Studied by ThermallyAssistedOccupation Density Functional Theory. Sci Rep 9, 2907 (2019). https://doi.org/10.1038/s41598019395244
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DOI: https://doi.org/10.1038/s41598019395244
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