Abstract
Owing to the presence of strong static correlation effects, accurate prediction of the electronic properties (e.g., the singlettriplet energy gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and realspace representation of active orbitals) of cyclacenes with n fused benzene rings (n = 4–100) has posed a great challenge to traditional electronic structure methods. To meet the challenge, we study these properties using our newly developed thermallyassistedoccupation density functional theory (TAODFT), a very efficient method for the study of large systems with strong static correlation effects. Besides, to examine the role of cyclic topology, the electronic properties of cyclacenes are also compared with those of acenes. Similar to acenes, the ground states of cyclacenes are singlets for all the cases studied. In contrast to acenes, the electronic properties of cyclacenes, however, exhibit oscillatory behavior (for n ≤ 30) in the approach to the corresponding properties of acenes with increasing number of benzene rings. On the basis of the calculated orbitals and their occupation numbers, the larger cyclacenes are shown to exhibit increasing polyradical character in their ground states, with the active orbitals being mainly localized at the peripheral carbon atoms.
Introduction
Carbon nanotubes (CNTs) are promising nanomaterials, which have been extensively studied by many researchers^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Due to different combinations of structural variation, CNTs can exhibit a wide range of electronic and optical properties, which can be of great use in the design of novel techniques^{13}. CNTs are also polyfunctional macromolecules, where specific reactions can occur at various sites with different efficiencies^{10}. There are three major types of CNTs: armchair CNTs, chiral CNTs, and zigzag CNTs, which are distinguished by the geometrical vector (n, m), with n and m being integers. CNTs can behave as either metals or semiconductors depending on their chiral angles, diameters, and lengths. Therefore, a further investigation of how these factors affect the properties of CNTs is essential for the comprehensive understanding of these materials^{13,15}.
In particular, it is useful to study the basic repeating units of CNTs, which still need further fundamental research exploration^{12}. The targeting units of the present study, a series of ncyclacenes, consisting of n fused benzene rings forming a closed loop (see Fig. 1), are the shortest (n, 0) zigzag CNTs with hydrogen passivation, which have attracted considerable interest in the research community due to their fascinating electronic properties^{12,15,16,17,18,19,20,21,22,23,24,25,26,27}. As ncyclacenes belong to the category of catacondensed aromatics (i.e., molecules that have no carbon atoms belonging to more than two rings), each carbon atom is on the periphery of the conjugated system^{17}. Before ncyclacenes are intensively connected to zigzag CNTs, they have been studied mainly due to the research curiosity in highly conjugated cyclic systems. The studies of ncyclacenes can also be important for atomiclevel structural control in the synthesis of CNTs. In addition, bottomup approaches to the synthesis of CNTs not only provide a fundamental understanding of the relationship between the design of CNTs and their electronic properties, but also greatly lower the synthetic temperatures^{13}. While zigzag CNTs may be synthesized from cycloarylenes by devising the cutout positions of CNTs^{14}, it remains important to systematically investigate the properties of ncyclacenes, which can be useful for exploring the possible utility of their cylindrical cavities in hostguest chemistry^{19}.
The structure of ncyclacene has two types of components: an arenoid belt (composed of fused benzene rings) and two peripheral circuits (the top and bottom peripheral circuits)^{23}. The peripheral circuits are of two types: 4k and 4k + 2 (where k is an integer), depending on the number of benzene rings in ncyclacene. In previous studies, it has been shown that ncyclacene with evennumber benzene rings (4k type) is more stable than that with oddnumber benzene rings (4k + 2 type)^{17,18,20,23}. Therefore, the nature of peripheral circuits (i.e., the cryptoannulenic effect) is expected to be responsible for the properties of ncyclacene. Besides, the structure of ncyclacene can also be regarded as two fused trannulenes (i.e., circular, alltrans cyclic polyene ribbons)^{19,21}. From the bond length analysis of ncyclacene, there is bond length alternation in the benzene ring, and the aromaticity is reduced due to the structural strain, which can hence be responsible for the properties of ncyclacene.
Even though there has been a keen interest in ncyclacenes, the studies of their electronic properties are scarce. While ncyclacene may be synthesized via an intramolecular cyclization of nacene (a chainlike molecule with n linearly fused benzene rings, e.g., see Fig. 1 of ref. 23), the synthetic procedure has been very challenging, and has not succeeded in producing pure ncyclacene^{23,24,26}, possibly due to its highly strained structure and highly reactive nature^{12,26}. As the stabilities of annulated polycyclic saturated hydrocarbons decrease rapidly with the number of fused benzene rings^{16}, the synthesis of larger ncyclacenes should be even more difficult.
To date, the reported properties of ncyclacenes are based on theoretical calculations. Nevertheless, accurate prediction of the electronic properties of larger ncyclacenes has been very challenging for traditional electronic structure methods, due to the presence of strong static correlation effects^{25}. KohnSham density functional theory (KSDFT)^{28} with conventional (i.e., semilocal^{29,30,31,32}, hybrid^{33,34,35,36,37,38,39,40}, and doublehybrid^{41,42,43,44}) exchangecorrelation (XC) density functionals can yield unreliable results for systems with strong static correlation effects^{45}. Highlevel ab initio multireference methods^{25,46,47,48,49,50,51,52,53} are typically required to accurately predict the properties of larger ncyclacenes. However, as the number of electrons in ncyclacene quickly increases with increasing n, there have been very few studies on the properties of larger ncyclacenes using multireference methods, due to their prohibitively high cost.
To circumvent the formidable computational expense of highlevel ab initio multireference methods, we have recently developed thermallyassistedoccupation density functional theory (TAODFT)^{54,55}, a very efficient electronic structure method for studying the properties of large groundstate systems (e.g., containing up to a few thousand electrons) with strong static correlation effects^{56,57,58}. In contrast to KSDFT, TAODFT is a density functional theory with fractional orbital occupations, wherein strong static correlation is explicitly described by the entropy contribution (see Eq. (26) of ref. 54), a function of the fictitious temperature and orbital occupation numbers. Note that the entropy contribution is completely missing in KSDFT. Recently, we have studied the electronic properties of zigzag graphene nanoribbons (ZGNRs) using TAODFT^{56}. The ground states of ZGNRs are found to be singlets for all the widths and lengths studied. The longer ZGNRs should possess increasing polyradical character in their ground states, with the active orbitals being mainly localized at the zigzag edges. Our results are in good agreement with the available experimental and highly accurate ab initio data. Besides, on the basis of our TAODFT calculations, the active orbital occupation numbers for the ground states of ZGNRs should exhibit a curve crossing behavior in the approach to unity (singly occupied) with increasing ribbon length. Very recently, the curve crossing behavior has been confirmed by highly accurate ab initio multireference methods^{53}!
TAODFT has similar computational cost as KSDFT for singlepoint energy and analytical nuclear gradient calculations, and reduces to KSDFT in the absence of strong static correlation effects. Besides, existing XC density functionals in KSDFT may also be adopted in TAODFT. Relative to highlevel ab initio multireference methods, TAODFT is computationally efficient, and hence very powerful for the study of large polyradical systems. In addition, the orbital occupation numbers from TAODFT, which are intended to simulate the natural orbital occupation numbers (NOONs) [i.e., the eigenvalues of oneelectron reduced density matrix]^{59}, can be very useful for assessing the possible polyradical character of systems. Recent studies have demonstrated that the orbital occupation numbers from TAODFT are qualitatively similar to the NOONs from highlevel ab initio multireference methods, giving promise for applying TAODFT to large polyradical systems^{53,54,56,57}.
Due to its computational efficiency and reasonable accuracy for large systems with strong static correlation effects, in this work, TAODFT is adopted to study the electronic properties of ncyclacenes (n = 4–100). As ncyclacenes have not been successfully synthesized, no experimental data are currently available for comparison. Therefore, our results are compared with the available highlevel ab initio data as well as those obtained from various XC density functionals in KSDFT. In addition, as ncyclacene can be considered as an interconnection of nacene, the electronic properties of ncyclacene are also compared with those of nacene to assess the role of cyclic topology.
Computational Details
All calculations are performed with a development version of QChem 4.0^{60}, using the 6–31 G(d) basis set with the fine grid EML (75, 302), consisting of 75 EulerMaclaurin radial grid points and 302 Lebedev angular grid points. Results are calculated using KSLDA (i.e., KSDFT with the LDA XC density functional^{29,30}) and TAOLDA (i.e., TAODFT with the LDA XC density functional and the LDA θdependent density functional (see Eq. (41) of ref. 54) with the fictitious temperature θ = 7 mhartree (as defined in ref. 54). Note that KSLDA is simply TAOLDA with θ = 0, and hence it is important to assess the performance of KSLDA here to assess the significance of TAOLDA.
The ground state of ncyclacene/nacene (n = 4–100) is obtained by performing spinunrestricted KSLDA and TAOLDA calculations for the lowest singlet and triplet energies of ncyclacene/nacene on the respective geometries that were fully optimized at the same level of theory. The singlettriplet energy (ST) gap of ncyclacene/nacene is calculated as (E_{T} − E_{S}), the energy difference between the lowest triplet (T) and singlet (S) states of ncyclacene/nacene.
Results and Discussion
SingletTriplet Energy Gap
Figure 2 shows the ST gap of ncyclacene as a function of the number of benzene rings, calculated using spinunrestricted KSLDA and TAOLDA. The results are compared with the available data^{25}, calculated using the completeactivespace secondorder perturbation theory (CASPT2)^{46} (a highlevel ab initio multireference method) as well as the M06L functional^{32} (a popular semilocal XC density functional) and the B3LYP functional^{33,34} (a popular hybrid XC density functional) in KSDFT (see Table S1 in Supplementary Information).
As can be seen, the anticipated evenodd oscillations in the ST gaps may be attributed to the cryptoannulenic effects of ncyclacenes^{17,18,20,23}. However, the amplitudes of the evenodd oscillations are considerably larger for KSDFT with the XC functionals, which are closely related to the degree of spin contamination (as discussed in ref. 25). In general, the larger fraction of HartreeFock (HF) exchange adopted in the XC functional in KSDFT, the higher the degree of spin contamination for systems with strong static correlation effects. For example, the ST gap obtained with KSB3LYP is unexpectedly large at n = 10, due to the high degree of spin contamination^{25}.
On the other hand, as commented in ref. 25, the ST gaps obtained with CASPT2 are rather sensitive to the choice of active space. Since the complete πvalence space was not selected as the active space (due to the prohibitively high cost), the CASPT2 results here should be taken with caution. Recent studies have shown that a sufficiently large active space should be adopted in highlevel ab initio multireference calculations^{47,49,53} for accurate prediction of the electronic properties of systems with strong static correlation effects, which can, however, be prohibitively expensive for large systems. Note that the ST gap obtained with CASPT2 unexpectedly increases at n = 12, possibly due to the insufficiently large active space adopted in the calculations^{25}.
To assess the role of cyclic topology, Figs 3 and 4 show the ST gap of ncyclacene/nacene as a function of the number of benzene rings, calculated with spinunrestricted TAOLDA (see Table S2 in Supplementary Information). Similar to nacenes, the ground states of ncyclacenes remain singlets for all the cases investigated. In contrast to nacene, the ST gap of ncyclacene, however, displays oscillatory behavior for small n, and the oscillation vanishes gradually with increasing n. For small n, ncyclacene with evennumber benzene rings exhibits a larger ST gap (i.e., greater stability) than that with oddnumber benzene rings. For sufficiently large n (n > 30), the ST gap of ncyclacene converges monotonically from below to the ST gap of nacene (which monotonically decreases with increasing n). At the level of TAOLDA, the ST gaps of the largest ncyclacene and nacene studied (i.e., n = 100) are essentially the same (0.49 kcal/mol). On the basis of the ST gaps obtained with TAOLDA, the cryptoannulenic effect and structural strain of ncyclacene are more important for the smaller n, and less important for the larger n.
Due to the symmetry constraint, the spinrestricted and spinunrestricted energies for the lowest singlet state of ncyclacene/nacene, calculated using the exact theory, should be identical^{50,54,55,56}. Recent studies have shown that KSDFT with conventional XC density functionals cannot satisfy this condition for the larger ncyclacene/nacene, due to the aforementioned spin contamination^{25,47,49,50,54,55,56}. To assess the possible symmetrybreaking effects, spinrestricted TAOLDA calculations are also performed for the lowest singlet energies on the respective optimized geometries. Within the numerical accuracy of our calculations, the spinrestricted and spinunrestricted TAOLDA energies for the lowest singlet state of ncyclacene/nacene are essentially the same (i.e., essentially no unphysical symmetrybreaking effects occur in our spinunrestricted TAOLDA calculations).
Vertical Ionization Potential, Vertical Electron Affinity, and Fundamental Gap
At the lowest singlet state (i.e., the groundstate) geometry of ncyclacene/nacene (containing N electrons), TAOLDA is adopted to calculate the vertical ionization potential IP_{v} = E_{N−1} − E_{N}, vertical electron affinity EA_{v} = E_{N} − E_{N+1}, and fundamental gap E_{g} = IP_{v} − EA_{v} = E_{N+1} + E_{N−1} − 2E_{N} using multiple energydifference methods, with E_{N} being the total energy of the Nelectron system.
With increasing number of benzene rings in ncyclacene, IP_{v} oscillatorily decreases (see Fig. 5), EA_{v} oscillatorily increases (see Fig. 6), and hence E_{g} oscillatorily decreases (see Fig. 7). However, these oscillations are damped and eventually disappear with increasing n (see Tables S3 to S5 in Supplementary Information). For sufficiently large n (n > 30), the IP_{v} and E_{g} values of ncyclacene converge monotonically from above to those of nacene (which monotonically decrease with increasing n), while the EA_{v} value of ncyclacene converges monotonically from below to that of nacene (which monotonically increases with increasing n). Note also that the E_{g} value of ncyclacene (n = 13–54) is within the most interesting range (1 to 3 eV), giving promise for applications of ncyclacenes in nanophotonics.
Symmetrized von Neumann Entropy
To investigate the possible polyradical character of ncyclacene/nacene, we calculate the symmetrized von Neumann entropy (e.g., see Eq. (9) of ref. 50).
for the lowest singlet state of ncyclacene/nacene as a function of the number of benzene rings, using TAOLDA. Here, f_{i} the occupation number of the i^{th} orbital obtained with TAOLDA, which ranges from 0 to 1, is approximately the same as the occupation number of the i^{th} natural orbital^{53,54,55,56,57,58}. For a system without strong static correlation ({ f_{i}} are close to either 0 or 1), S_{vN} provides insignificant contributions, while for a system with strong static correlation ({ f_{i}} are fractional for active orbitals and are close to either 0 or 1 for others), S_{vN} increases with the number of active orbitals.
As shown in Fig. 8, the S_{vN} value of ncyclacene oscillatorily increases with increasing number of benzene rings (see Table S6 in Supplementary Information). Nonetheless, the oscillation is damped and eventually disappears with the increase of n. For sufficiently large n (n > 30), the S_{vN} value of ncyclacene converges monotonically from above to that of nacene (which monotonically increases with increasing n). Therefore, similar to nacenes^{47,49,50,51,53,54,55,56,57,58}, the larger ncyclacenes should possess increasing polyradical character.
Active Orbital Occupation Numbers
To illustrate the causes of the increase of S_{vN} with n, we plot the active orbital occupation numbers for the lowest singlet state of ncyclacene as a function of the number of benzene rings, calculated using TAOLDA. Here, the highest occupied molecular orbital (HOMO) is the (N/2)^{th} orbital, and the lowest unoccupied molecular orbital (LUMO) is the (N/2 + 1)^{th} orbital, where N is the number of electrons in ncyclacene. For brevity, HOMO, HOMO − 1, …, and HOMO − 15, are denoted as H, H − 1, …, and H − 15, respectively, while LUMO, LUMO + 1, …, and LUMO + 15, are denoted as L, L + 1, …, and L + 15, respectively.
As presented in Fig. 9, the number of fractionally occupied orbitals increases with increasing cyclacene size, clearly indicating that the polyradical character of ncyclacene indeed increases with the cyclacene size. Similar to the previously discussed properties, the active orbital occupation numbers of ncyclacene also exhibit oscillatory behavior, showing wavepacket oscillations.
RealSpace Representation of Active Orbitals
For the lowest singlet states of some representative ncyclacenes (n = 4–7), we explore the realspace representation of active orbitals (e.g., HOMOs and LUMOs), obtained with TAOLDA. Similar to previous findings for nacenes^{47,49,50,56}, the HOMOs and LUMOs of ncyclacenes are mainly localized at the peripheral carbon atoms (see Fig. 10).
Conclusions
In conclusion, we have studied the electronic properties of ncyclacenes (n = 4–100), including the ST gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and realspace representation of active orbitals, using our newly developed TAODFT, a very efficient electronic structure method for the study of large systems with strong static correlation effects. To assess the effects of cyclic nature, the electronic properties of ncyclacenes have also been compared with those of nacenes. Similar to nacenes, the ground states of ncyclacenes are singlets for all the cases investigated. In contrast to nacenes, the electronic properties of ncyclacenes, however, display oscillatory behavior for small n (n ≤ 30) in the approach to the corresponding properties of nacenes with increasing number of benzene rings, which to the best of our knowledge have never been addressed in the literature. The oscillatory behavior may be related to the cryptoannulenic effect and structural strain of ncyclacene, which have been shown to be important for small n, and unimportant for sufficiently large n. On the basis of several measures (e.g., the smaller ST gap, the smaller E_{g}, and the larger S_{vN}), for small n, ncyclacene with oddnumber benzene rings should possess stronger radical character than that with evennumber benzene rings. In addition, based on the calculated orbitals and their occupation numbers, the larger ncyclacenes are expected to possess increasing polyradical character in their ground states, where the active orbitals are mainly localized at the peripheral carbon atoms.
Since TAODFT is computationally efficient, it appears to be a promising method for studying the electronic properties of large systems with strong static correlation effects. Nevertheless, as with all approximate electronic structure methods, a few limitations remain. Relative to the exact full configuration interaction (FCI) method^{61}, TAOLDA (with θ = 7 mhartree) is not variationally correct (i.e., overcorrelation can occur), and hence, the orbital occupation numbers from TAOLDA may not be the same as the NOONs from the FCI method. To assess the accuracy of our TAOLDA results, as the computational cost of the FCI method is prohibitive, the electronic properties of ncyclacenes from relatively affordable ab initio multireference methods are called for.
Additional Information
How to cite this article: Wu, C.S. et al. Electronic Properties of Cyclacenes from TAODFT. Sci. Rep. 6, 37249; doi: 10.1038/srep37249 (2016).
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Acknowledgements
This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST1042628M002011MY3), National Taiwan University (Grant No. NTUCDP105R7818), the Center for Quantum Science and Engineering at NTU (Subproject Nos: NTUERP105R891401 and NTUERP105R891403), and the National Center for Theoretical Sciences of Taiwan. We are grateful to the Computer and Information Networking Center at NTU for the partial support of highperformance computing facilities.
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 ChunShian Wu
 & PeiYin Lee
These authors contributed equally to this work.
Affiliations
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
 ChunShian Wu
 , PeiYin Lee
 & JengDa Chai
Department of Chemistry, National Taiwan University, Taipei 10617, Taiwan
 ChunShian Wu
Center for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan
 JengDa Chai
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Contributions
C.S.W. and P.Y.L. contributed equally to this work. J.D.C. conceived and designed the project. C.S.W. and J.D.C. performed the calculations. P.Y.L. and J.D.C. wrote the paper. All authors performed the data analysis.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to JengDa Chai.
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Further reading

1.
Scientific Reports (2017)
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