Abstract
It has been challenging to adequately investigate the properties of nanosystems with radical nature using conventional electronic structure methods. We address this challenge by calculating the electronic properties of linear carbon chains (lCC[n]) and cyclic carbon chains (cCC[n]) with n = 10–100 carbon atoms, using thermallyassistedoccupation density functional theory (TAODFT). For all the cases investigated, lCC[n]/cCC[n] are groundstate singlets, and cCC[n] are energetically more stable than lCC[n]. The electronic properties of lCC[n]/cCC[n] reveal certain oscillation patterns for smaller n, followed by monotonic changes for larger n. For the smaller carbon chains, oddnumbered lCC[n] are more stable than the adjacent evennumbered ones; cCC[\(4m+2\)]/cCC[4m] are more/less stable than the adjacent oddnumbered ones, where m are positive integers. As n increases, lCC[n]/cCC[n] possess increasing polyradical nature in their ground states, where the active orbitals are delocalized over the entire length of lCC[n] or the whole circumference of cCC[n].
Introduction
Carbon is the most versatile element in forming various structures. In bulk phase, graphite and diamond, which are wellknown materials, have been used for centuries. In nanoforms, fullerenes and graphene have been studied in detail for decades. In general, nanostructures can be classified into three categories: zerodimensional (0D), onedimensional (1D), and twodimensional (2D) nanomaterials. Carbon forms all these nanostructures with unique shapes and properties. Over the past few decades, carbon nanomaterials have been widely studied, and applied in diverse industries^{1,2,3}.
A number of carbon nanostructures have been synthesized and applied in different fields. The 0D carbon nanomaterials include clusters, quantum dots, nanoflakes, and buckyballs^{3}. Among them, the C\(_{60}\) fullerene molecule (containing 12 pentagons and 20 hexagons), where the carbon atoms are \(sp^{2}\)–\(sp^{3}\)hybridized, has been a popular carbon nanomaterial^{1}. The discovery of C\(_{60}\) has led to the flourishment of carbon nanomaterials in various ways.
Graphite is a bulk layered material, where the \(sp^{2}\)hybridized carbon atoms in each layer are arranged in a hexagonal lattice. The 2D carbon nanomaterial, graphene, can be obtained by mechanically exfoliating a single layer of carbon atoms from graphite^{2}. Thus, graphene, which is a perfect arrangement of hexagons made up of \(sp^{2}\)hybridized carbon atoms in a 2D planar surface, can be the thinnest (i.e., singleatomthick) material synthesized ever. Graphene is a zerogap semiconductor or semimetal with massless Dirac fermions with linear dispersion at low energy. Because of the Diraccone feature, graphene has huge potential in electronics applications^{2}. The discovery of graphene has also led to the discovery of other 2D materials. Besides, if a graphene sheet can be rolled up to form a seamless cylinder, one obtains a carbon nanotube (CNT), which belongs to the class of 1D nanostructures. Note that CNTs were first observed by Iijima in 1991^{4}, well before the separation and characterization of graphene. On the basis of the direction of rolling (chirality), CNTs can be classified into three groups: chiral, zigzag, and armchair CNTs, which can have rather different electronic properties.
Apart from these well studied carbon nanoallotropes, linear carbon chains (i.e., also belonging to the class of 1D nanostructures), where the carbon atoms are sphybridized, have recently gained much attention because of their interesting physical and chemical properties^{5,6,7,8,9,10,11,12,13,14,15,16,17,18}. A linear carbon chain consisting of n carbon atoms (for brevity, denoted as lCC[n] (see Fig. 1a)) is an ideal 1D carbon nanomaterial. Kroto et al. originally designed an experiment for explaining the formation mechanism of carbon chains in outer space, which led to the serendipitous discovery of C\(_{60}\)^{1}. In attempting to synthesize lCC[n] (i.e., the holy grail of truly 1D carbon allotropes), significant progresses have been made recently^{6,11,14}. Owing to their high reactivity, pristine linear carbon chains have not yet been reported. However, the linear carbon chains (containing up to 100 carbon atoms) supported inside multiwalled CNTs were realized in 2003 by Zhao et al.^{6}. In 2015, Andrade et al.^{11} observed the carbon chains inside CNTs under high pressures. In 2016, Shi et al.^{14} demonstrated the largescale syntheses of linear carbon chains inside doublewalled CNTs by using the confined space inside the tube as a nanoreactor to grow ultralong carbon chains (containing up to 6400 carbon atoms) in large quantities^{14,15}. Linear carbon chains can be potential candidates for nanodevices, molecular electronics, and the building blocks of novel hybrid nanomaterials (e.g., sp–\(sp^{2}\) and sp–\(sp^{3}\) hybridized materials) by integrating with other nanostructures^{8,9,17}. On the theoretical side, a number of relevant calculations are available^{9}. The calculations showed that the linear carbon chains inside singlewalled CNTs can modify the electronic properties of pristine singlewalled CNTs significantly. It has been found that the chirality of the enclosing nanotubes can affect the properties of linear carbon chains^{18}. Besides, recent theoretical studies showed that linear carbon chains have excellent mechanical and electrical properties^{9,10,14}.
On the other hand, cyclic carbon chains (for brevity, denoted as cCC[n] (see Fig. 1b)), which are the monocyclic isomers of linear carbon chains, have also attracted considerable attention in recent years^{19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38}. Note that cCC[n] (where the carbon atoms are also sphybridized) are hypothesized to be the building blocks of fullerenes in the initial stages of growth^{21}. Interestingly, cCC[\(4m+2\)] (where m are positive integers) have been found to possess high stability^{24,25}. Among them, cCC[18] has recently been synthesized^{26}, and some of the electronic properties of cCC[18] have been reported. There are also a few theoretical studies on the electronic properties and applications of cCC[18] and other cyclic carbon chains^{22,27,28,29,30,31,32,33,34,35,36,37,38}. For example, cCC[18] has been found to possess electronacceptor properties, and can be dubbed as the smallest allcarbon electron acceptor^{28}.
In general, it remains extremely difficult to synthesize both lCC[n] and cCC[n]. Accordingly, theoretical studies are complementary, and may provide immense information on the properties of lCC[n] and cCC[n]. In a recent theoretical study, it has been reported that very small lCC[n] (n = 5–10) with even n values possess diradical nature in their ground states^{16}. Therefore, it can be anticipated that the larger lCC[n] and cCC[n] could also possess radical character due to the low dimensionality of lCC[n] and cCC[n], respectively^{39}. Despite its success in describing some groundstate properties, KohnSham density functional theory (KSDFT)^{40} can yield unreliable results for systems with radical nature^{41}, when the conventional XC density functionals are employed. Typically, multireference (MR) electronic structure methods, such as the completeactivespace selfconsistentfield (CASSCF), completeactivespace secondorder perturbation theory (CASPT2), and related methods^{42,43,44,45,46,47,48}, are required to reliably predict the energy and related properties of systems with radical nature. Nonetheless, accurate MR electronic structure methods can be prohibitively expensive for large systems, and hence may not be practical for studying the properties of the larger lCC[n] and cCC[n].
Recently, thermallyassistedoccupation density functional theory (TAODFT)^{49}, which is a density functional theory with fractional orbital occupation numbers, has been formulated to tackle such challenging problems (i.e., nanosystems with radical nature), wherein an entropy contribution term (i.e., a function of the fictitious temperature \(\theta \) and orbital occupation numbers) can approximately describe strong static correlation even when the simplest local density approximation (LDA) XC density functional is employed. In TAODFT, one can also adopt more sophisticated XC density functionals, such as semilocal^{50}, global hybrid^{51}, and rangeseparated hybrid^{51,52} XC density functionals. Besides, aiming to improve the accuracy of TAODFT for a wide range of applications, a selfconsistent scheme determining the fictitious temperature \(\theta \) in TAODFT has been recently proposed^{53}. Since TAODFT is a computationally efficient electronic structure method, a number of strongly correlated electron systems at the nanoscale have been studied using TAODFT in recent years^{16,54,55,56,57,58,59,60,61,62}. Besides, TAODFT has been recently shown to be useful in describing the vibrational spectra of molecules with radical nature^{63}. In addition, TAODFT and related methods have recently been employed to investigate the electronic properties of several nanosystems with radical nature^{64,65}, and have also been combined with the linearscaling divideandconquer approach for the study of large systems with strong static correlation effects^{66}.
Therefore, in the present study, we carry out TAODFT calculations to investigate the electronic properties of lCC[n]/cCC[n] with n = 10–100. Specifically, we report the electronic properties, such as the singlettriplet energy gap, singletquintet energy gap, vertical ionization potential, vertical electron affinity, fundamental gap, symmetrized von Neumann entropy, active orbital occupation numbers, realspace representation of active orbitals, and relative stability of lCC[n]/cCC[n]. For considerably large values of n, we show that both lCC[n] and cCC[n] are polyradicals in their ground states, playing an important role in determining their electronic properties.
Computational details
All geometry optimizations and other calculations are performed with QChem 4.4^{67}, using the 631G(d) basis set (i.e., a valence doublezeta polarized basis set) with a numerical quadrature that consists of 75 points in the EulerMaclaurin radial grid and 302 points in the Lebedev angular grid. We carry out calculations using TAOLDA^{49}, which is TAODFT employing the LDA XC and \(\theta \)dependent density functionals, with the recommended fictitious temperature \(\theta \) = 7 mhartree^{49}.
In several recent studies^{46,48,49,54,55}, the orbital occupation numbers obtained from TAOLDA (with \(\theta \) = 7 mhartree) have been found to be qualitatively similar to the natural orbital occupation numbers obtained from the variational twoelectron reduceddensitymatrixdriven CASSCF (v2RDMCASSCF) method (i.e., an accurate MR electronic structure method), leading to a similar trend for the radical nature of several systems.
Results and discussion
Singlettriplet energy gap and singletquintet energy gap
Aiming to obtain the energetically preferred spin state (i.e., the ground state) of lCC[n]/cCC[n], we obtain the energies of lCC[n]/cCC[n] for the lowest singlet, triplet, and quintet states by optimizing the corresponding structures with spinunrestricted TAOLDA, and thereafter compute the singlettriplet energy gap of lCC[n]/cCC[n] using
and compute the singletquintet energy gap of lCC[n]/cCC[n] using
where \(E_{\text {Q}}\), \(E_{\text {T}}\), and \(E_{\text {S}}\) are the lowest quintet, triplet, and singlet energies, respectively, of lCC[n]/cCC[n] (also see Section I, Figure S1, and Figure S2 in Supplementary Information (SI)). According to their definitions, the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) reported in this work are the adiabatic singlettriplet energy gap and adiabatic singletquintet energy gap, respectively, of lCC[n]/cCC[n]. The \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values of lCC[n]/cCC[n] as functions of the number of carbon atoms are presented in Figs. 2 and 3, respectively (also see Tables S1 and S2 in SI).
Since the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values remain positive, the lCC[n]/cCC[n] studied are all groundstate singlets. The \(E_{\text {ST}}\) values of the smaller lCC[n] exhibit an oddeven oscillation pattern in which the \(E_{\text {ST}}\) values of oddnumbered lCC[n] are larger than those of the adjacent evennumbered ones. However, such oscillations decrease with increasing n, and eventually disappear for considerably large n. Accordingly, the \(E_{\text {ST}}\) values of lCC[n] are oscillatory only for smaller values of n (up to n = 33), and become monotonically decreasing for larger n. The \(E_{\text {SQ}}\) values of lCC[n] decrease essentially monotonically with n (i.e., except only for n = 10).
By contrast, the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values of the smaller cCC[n] display distinct oscillation patterns in which the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values of cCC[\(4m+2\)]/cCC[4m] are larger/smaller than the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values, respectively, of the adjacent oddnumbered ones, where m are positive integers. Nevertheless, with the increase of molecular size, these oscillations are progressively reduced, and eventually absent for sufficiently large n. Therefore, the \(E_{\text {ST}}\) and \(E_{\text {SQ}}\) values of cCC[n] are oscillatory only for smaller n (up to n = 94 for \(E_{\text {ST}}\) and up to n = 85 for \(E_{\text {SQ}}\)), and become monotonically decreasing for larger values of n. Note that the reasons for these oscillation patterns have been recently provided in a quantum Monte Carlo study^{34}.
Understanding the \(E_{\text {ST}}\) values is essential for applications that involve harnessing energy through the singletfission phenomenon^{68}. Consequently, the \(E_{\text {ST}}\) values of lCC[n]/cCC[n] reported in the present study can provide insight into the singletfission phenomenon. For other relevant applications, it is worth mentioning that the electronic transport properties of cCC[18]based molecular devices have been recently studied^{38}.
Vertical ionization potential, vertical electron affinity, and fundamental gap
Here, we assess if lCC[n]/cCC[n] are promising for photovoltaic applications. For a molecule in its ground state, the vertical ionization potential (\(\text {IP}_{v}\)) is the energy required to remove an electron from the molecule without affecting the molecular geometry, the vertical electron affinity (\(\text {EA}_{v}\)) is the energy released when an electron is added to the molecule without affecting the molecular geometry, and the fundamental gap (\(E_{g}\)) is the difference between \(\text {IP}_{v}\) and \(\text {EA}_{v}\) (i.e., \(E_{g} = \text {IP}_{v}  \text {EA}_{v}\)). Accordingly, in this work, we calculate the vertical ionization potential of groundstate lCC[n]/cCC[n] using
the vertical electron affinity of groundstate lCC[n]/cCC[n] using
and the fundamental gap of groundstate lCC[n]/cCC[n] using
where \({E}_{N}\) is the total energy of the Nelectron molecule (i.e., lCC[n]/cCC[n]) at the groundstate (i.e., lowest singlet state) geometry, obtained with spinunrestricted TAOLDA. The \(\text {IP}_{v}\) (Fig. 4a), \(\text {EA}_{v}\) (Fig. 4b), and \(E_{g}\) (Fig. 4c) values of groundstate lCC[n]/cCC[n] are plotted as functions of n (also see Tables S3 and S4 in SI).
As n increases, the \(\text {IP}_{v}\) value of lCC[n] decreases monotonically, showing a very slight oddeven pattern only for very small n. The \(\text {EA}_{v}\)/\(E_{g}\) values of the smaller lCC[n] reveal an oddeven oscillation pattern in which the \(\text {EA}_{v}\)/\(E_{g}\) values of oddnumbered lCC[n] are smaller/larger than those of the adjacent evennumbered ones. Nevertheless, with the increase of n, these oscillations are progressively reduced, and ultimately absent for sufficiently large n. Accordingly, the \(\text {EA}_{v}\)/\(E_{g}\) values of lCC[n] are oscillatory only for smaller n (up to n = 21 for \(\text {EA}_{v}\) and up to n = 15 for \(E_{g}\)), and become monotonically increasing/decreasing for larger values of n.
On the other hand, the \(\text {IP}_{v}\)/\(\text {EA}_{v}\)/\(E_{g}\) values of the smaller cCC[n] display a distinct oscillation pattern in which the \(\text {IP}_{v}\)/\(\text {EA}_{v}\)/\(E_{g}\) values of cCC[\(4m+2\)] are larger/smaller/larger than those of the adjacent oddnumbered ones, where m are positive integers. Nonetheless, such oscillations are gradually reduced, and eventually absent with the increase of molecular size. Therefore, the \(\text {IP}_{v}\)/\(\text {EA}_{v}\)/\(E_{g}\) values of cCC[n] are oscillatory only for smaller values of n (up to n = 61 for \(\text {IP}_{v}\), up to n = 58 for \(\text {EA}_{v}\), and up to n = 46 for \(E_{g}\)), and become monotonically decreasing/increasing/decreasing for larger n.
For each n, the \(E_{g}\) value of cCC[n] is larger than that of lCC[n]. Besides, the \(E_{g}\) values of lCC[n] (with n = 24–100) and cCC[n] (with n = 31–100) are in the range of 1 to 3 eV, showing promise for their applications in nanophotonics.
Symmetrized von Neumann entropy
To assess the radical nature of groundstate (i.e., lowest singlet state) lCC[n]/cCC[n], we calculate the symmetrized von Neumann entropy^{16,50,51,54,56,57,58,59,60,61,62,69}
using spinunrestricted TAOLDA. Here, the occupation number \(f_{i,\sigma }\) of the \(i\)th \(\sigma \)spin orbital (i.e., upspin orbital or downspin orbital) obtained with spinunrestricted TAOLDA, which ranges from 0 to 1, is closely related to the occupation number of the \(i\)th \(\sigma \)spin natural orbital^{49,50,51}. For a molecule with nonradical nature, the occupation numbers of all spinorbitals should be close to either 0 or 1, leading to insignificant contributions to the corresponding \(S_{\text {vN}}\) value. However, for a molecule with pronounced radical nature, the occupation numbers of active spinorbitals (i.e., the spinorbitals that have considerable fractional occupations) can be very different from 0 and 1 (e.g., between 0.1 and 0.9); hence, the corresponding \(S_{\text {vN}}\) value is expected to increase when the occupation numbers of active spinorbitals become closer to 0.5 and/or the number of active spinorbitals increases.
As shown in Fig. 4d, the \(S_{\text {vN}}\) values of groundstate lCC[n] and cCC[n] possess rather different oscillation patterns (also see Tables S3 and S4 in SI). The \(S_{\text {vN}}\) values of the smaller lCC[n] exhibit an oddeven oscillation pattern in which the \(S_{\text {vN}}\) values of oddnumbered lCC[n] are smaller than those of the adjacent evennumbered ones. Nonetheless, when the system size increases, such oscillations are gradually damped, and ultimately absent for considerably large n. Therefore, the \(S_{\text {vN}}\) values of lCC[n] are oscillatory only for smaller n (up to n = 29), and become monotonically increasing for larger n. By contrast, the \(S_{\text {vN}}\) values of the smaller cCC[n] reveal a distinct oscillation pattern in which the \(S_{\text {vN}}\) values of cCC[\(4m+2\)]/cCC[4m] are smaller/larger than those of the adjacent oddnumbered ones, where m are positive integers. Nevertheless, with the increase of molecular size, these oscillations are progressively reduced, and eventually absent for sufficiently large n. Accordingly, the \(S_{\text {vN}}\) values of cCC[n] are oscillatory only for smaller n (up to n = 73), and become monotonically increasing for larger n. Since the \(S_{\text {vN}}\) value can be regarded as a quantitative measure of the radical nature of a molecule, the larger lCC[n]/cCC[n] are expected to exhibit increasing polyradical nature in their ground states.
Active orbital occupation numbers
To understand the reason that the symmetrized von Neumann entropy grows with the size of lCC[n]/cCC[n], it is instructive and informative to present the active orbital occupation numbers of groundstate lCC[n]/cCC[n] (consisting of N electrons), obtained with spinrestricted TAOLDA, wherein the highest occupied molecular orbital (HOMO) is defined as the \((N/2)\)th orbital, the lowest unoccupied molecular orbital (LUMO) is defined as the \((N/2+1)\)th orbital, and so forth^{49,51,58,60,61,62}. Here, the active orbitals are regarded as the orbitals with an occupation number ranging from 0.2 to 1.8 (i.e., the active spinorbitals are regarded as the spinorbitals with an occupation number ranging from 0.1 to 0.9).
As shown in Fig. 5, the active orbital occupation numbers of the smaller lCC[n] (e.g., up to n = 20) reveal oddeven oscillation patterns, indicating that oddnumbered lCC[n] possess nonradical nature (i.e., the occupation numbers of all orbitals are close to either 0 or 2), and evennumbered lCC[n] possess pronounced diradical nature (i.e., the active orbitals are HOMO and LUMO). However, with the increase of the size of lCC[n], the active orbital occupation numbers become closer to 1 and/or the number of active orbitals increases, suggesting an increasing polyradical nature of the larger lCC[n].
In contrast to lCC[n], the active orbital occupation numbers of cCC[n] display very different patterns (see Fig. 6). In particular, the smaller cCC[\(4m+2\)] (e.g., up to \(4m+2\) = 46) possess nonradical nature (i.e., the occupation numbers of all orbitals are close to either 0 or 2), and hence are relatively more stable than cCC[4m], cCC[\(4m+1\)], and cCC[\(4m+3\)], where m are positive integers. By contrast, cCC[4m] (where m are positive integers) possess tetraradical nature (i.e., the active orbitals are HOMO−1, HOMO, LUMO, and LUMO+1). Nevertheless, with the increase of the size of cCC[n], the active orbital occupation numbers become closer to 1 and/or the number of active orbitals increases, suggesting an increasing polyradical nature of the larger cCC[n].
On the basis of the active orbital occupation numbers, the smaller oddnumbered lCC[n] and the smaller cCC[\(4m+2\)] (where m are positive integers) possess nonradical nature in their ground states, being consistent with the analyses of the other electronic properties (e.g., the larger \(E_{\text {ST}}\) values, larger \(E_{\text {SQ}}\) values, larger \(E_{g}\) values, and smaller \(S_{\text {vN}}\) values) of these relatively stable linear carbon chains and cyclic carbon chains, respectively. Accordingly, this study confirms the high stability of the smaller cCC[\(4m+2\)]^{24,25}, including the recently synthesized cCC[18]^{26}. On the other hand, cCC[4m] (where m are positive integers) possess tetraradical nature in their ground states, also showing consistency with the analyses of the other electronic properties (e.g., the smaller \(E_{\text {ST}}\) values, smaller \(E_{\text {SQ}}\) values, smaller \(E_{g}\) values, and larger \(S_{\text {vN}}\) values) of these relatively unstable cyclic carbon chains.
Realspace representation of active orbitals
Here, we plot the realspace representation of the active orbitals, such as HOMO−2, HOMO−1, HOMO, LUMO, LUMO+1, and LUMO+2, for the ground states of some representative lCC[n] (see Figs. 7 and 8) and cCC[n] (see Figs. 9 and 10), obtained with spinrestricted TAOLDA (also see Figures S1 to S6 in SI for more illustrative cases). The realspace representation analysis indicates that the active orbitals are delocalized over the entire length of lCC[n] or the whole circumference of cCC[n]. As the electrical conductivities of molecules consisting of many delocalized electrons are likely to be high^{70}, it can be anticipated that lCC[n]/cCC[n] should be highly conductive due to the presence of delocalized electrons.
Relative stability
We carry out spinunrestricted TAOLDA calculations to investigate the relative stability of lCC[n] and cCC[n] (i.e., the two isomers) in their ground states, which is examined by
Here, \(E_{rel}\) is the relative energy of groundstate lCC[n] with respect to groundstate cCC[n], and \(E_{\text {S}}({\textit{l}CC})\) and \(E_{\text {S}}({\textit{c}CC})\) are the lowest singlet state (i.e., ground state) energies of lCC[n] and cCC[n], respectively.
The \(E_{rel}\) value as a function of the number of carbon atoms is plotted in Fig. 11 (also see Table S5 in SI). As the system size increases, the \(E_{rel}\) values are oscillatory only for smaller values of n (up to n = 59), and become monotonically increasing for larger n. For all the cases investigated, cCC[n] are energetically more stable than lCC[n], showing the significance of cyclic topology.
Conclusions
In conclusion, we have employed TAODFT to investigate the electronic properties (e.g., the singlettriplet energy gap, singletquintet energy gap, vertical ionization potential, vertical electron affinity, fundamental gap, symmetrized von Neumann entropy, active orbital occupation numbers, realspace representation of active orbitals, and relative stability) of lCC[n]/cCC[n] with n = 10–100 carbon atoms. For considerably large n, lCC[n]/cCC[n] are polyradicals in their ground states, playing an important role in determining their electronic properties. In view of their polyradical nature, it can be unreliable to study the properties of the larger lCC[n]/cCC[n] using KSDFT with the traditional XC energy functionals, and it can be computationally intractable to study the properties of the larger lCC[n]/cCC[n] using accurate MR electronic structure methods. Consequently, it is well justified to study the electronic properties of lCC[n]/cCC[n] using TAODFT (i.e., a computationally efficient electronic structure method for nanosystems with radical nature) in this work.
For all the cases investigated, lCC[n] and cCC[n] are groundstate singlets, and cCC[n] are energetically more stable than lCC[n]. The electronic properties of lCC[n] and cCC[n] display peculiar oscillation patterns for smaller values of n, followed by monotonic changes for larger values of n. For the smaller carbon chains, oddnumbered lCC[n] are more stable than the adjacent evennumbered ones, and cCC[\(4m+2\)]/cCC[4m] (where m are positive integers) are more/less stable than the adjacent oddnumbered ones. With the increase of n, lCC[n] and cCC[n] possess increasing polyradical nature in their ground states, with the active orbitals being delocalized over the entire length of lCC[n] or the whole circumference of cCC[n].
On the basis of our TAOLDA results, the smaller cCC[\(4m+2\)] (up to \(4m+2\) = 22, where m are positive integers) possess nonradical nature and sizable singlettriplet energy gaps (e.g., larger than 20 kcal/mol). In view of their high stability, it can be anticipated that these relatively stable cyclic carbon chains, such as cCC[10], cCC[14], cCC[18], and cCC[22], are likely to be synthesized in the near future. Note that among them, cCC[18] have been recently synthesized^{26}.
While the method adopted (i.e., TAOLDA with the fictitious temperature \(\theta \) = 7 mhartree^{49}) is computationally efficient for the study of nanosystems with radical nature, a few limitations remain. First, owing to the use of LDA XC and \(\theta \)dependent density functionals in TAODFT, TAOLDA can yield the selfinteraction error^{41,49, 51}, and hence bond length alternation (BLA) can be suppressed^{30}. Second, the present \(\theta \) value is systemindependent, and hence the static correlation associated with electronic systems can only be described approximately^{53}. Note that the first issue can be greatly resolved by using the longrange corrected hybrid XC functionals^{71,72} in TAODFT^{51,52} for an improved description of nonlocal exchange effects, and the second issue can be greatly resolved by using the respective selfconsistent scheme for determining the \(\theta \) value^{53}. Accordingly, we plan to work in this direction in the future.
On the other hand, to fully understand the impact of molecular geometries (e.g., BLA) on the electronic properties (e.g., the singlettriplet energy gap, singletquintet energy gap, fundamental gap, and active orbital occupation numbers) of lCC[n]/cCC[n], it is essential to employ accurate multireference methods, such as the CASSCF (for static correlation) and possibly, CASPT2 (for both static correlation and dynamical correlation) methods, for geometry optimizations and singlepoint energy calculations on different spin states (e.g., the lowest singlet, triplet, and quintet states) of lCC[n]/cCC[n]. As this task can be computationally intractable, the electronic properties of lCC[n]/cCC[n] calculated using reasonably accurate and relatively inexpensive multireference methods are needed.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
 1.
Kroto, H. W., Heath, J. R., O'Brien, S. C., Curl, R. F. & Smalley, R. E. C\(_{60}\): Buckminsterfullerene. Nature 318, 162–163 (1985).
 2.
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
 3.
Georgakilas, V., Perman, J. A., Tucek, J. & Zboril, R. Broad family of carbon nanoallotropes: Classification, chemistry, and applications of fullerenes, carbon dots, nanotubes, graphene, nanodiamonds, and combined superstructures. Chem. Rev. 115, 4744–4822 (2015).
 4.
Iijima, S. Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991).
 5.
Baeyer, A. Ueber polyacetylenverbindungen. Ber. Dtsch. Chem. Ges. 18, 2269–2281 (1885).
 6.
Zhao, X., Ando, Y., Liu, Y., Jinno, M. & Suzuki, T. Carbon nanowire made of a long linear carbon chain inserted inside a multiwalled carbon nanotube. Phys. Rev. Lett. 90, 187401 (2003).
 7.
Belau, L. et al. Ionization thresholds of small carbon clusters: Tunable VUV experiments and theory. J. Am. Chem. Soc. 129, 10229–10243 (2007).
 8.
Jin, C., Lan, H., Peng, L., Suenaga, K. & Iijima, S. Deriving carbon atomic chains from graphene. Phys. Rev. Lett. 102, 205501 (2009).
 9.
Liu, M., Artyukhov, V. I., Lee, H., Xu, F. & Yakobson, B. I. Carbyne from first principles: Chain of C atoms, a nanorod or a nanorope. ACS Nano 7, 10075–10082 (2013).
 10.
Artyukhov, V. I., Liu, M. & Yakobson, B. I. Mechanically induced metalinsulator transition in carbyne. Nano Lett. 14, 4224–4229 (2014).
 11.
Andrade, N. F. et al. Linear carbon chains under highpressure conditions. J. Phys. Chem. C 119, 10669–10676 (2015).
 12.
Banhart, F. Chains of carbon atoms: A vision or a new nanomaterial?. Beilstein J. Nanotechnol. 6, 559–569 (2015).
 13.
Casari, C. S., Tommasini, M., Tykwinski, R. R. & Milani, A. Carbonatom wires: 1D systems with tunable properties. Nanoscale 8, 4414–4435 (2016).
 14.
Shi, L. et al. Confined linear carbon chains as a route to bulk carbyne. Nat. Mater. 15, 634–639 (2016).
 15.
Shi, L. et al. Electronic band gaps of confined linear carbon chains ranging from polyyne to carbyne. Phys. Rev. Mater. 1, 075601 (2017).
 16.
Seenithurai, S. & Chai, J.D. Effect of Li termination on the electronic and hydrogen storage properties of linear carbon chains: A TAODFT study. Sci. Rep. 7, 4966 (2017).
 17.
Casari, C. S. & Milani, A. Carbyne: From the elusive allotrope to stable carbon atom wires. MRS Commun. 8, 207–219 (2018).
 18.
Heeg, S., Shi, L., Poulikakos, L. V., Pichler, T. & Novotny, L. Carbon nanotube chirality determines properties of encapsulated linear carbon chain. Nano Lett. 18, 5426–5431 (2018).
 19.
Hoffmann, R. Extended Hückel theoryV: Cumulenes, polyenes, polyacetylenes and C\(_{n}\). Tetrahedron 22, 521–538 (1966).
 20.
Yang, S. et al. UPS of 2–30atom carbon clusters: Chains and rings. Chem. Phys. Lett. 144, 431–436 (1988).
 21.
von Helden, G., Gotts, N. G. & Bowers, M. T. Experimental evidence for the formation of fullerenes by collisional heating of carbon rings in the gas phase. Nature 363, 60–63 (1993).
 22.
Handschuh, H., Ganteför, G., Kessler, B., Bechthold, P. S. & Eberhardt, W. Stable configurations of carbon clusters: Chains, rings, and fullerenes. Phys. Rev. Lett. 74, 1095–1098 (1995).
 23.
Kaizu, K. et al. Neutral carbon cluster distribution upon laser vaporization. J. Chem. Phys. 106, 9954 (1997).
 24.
Van Orden, A. & Saykally, R. J. Small carbon clusters: Spectroscopy, structure, and energetics. Chem. Rev. 98, 2313–2358 (1998).
 25.
Wakabayashi, T., Momose, T. & Shida, T. Mass spectroscopic studies of laser ablated carbon clusters as studied by photoionization with 10.5 eV photons under high vacuum. J. Chem. Phys. 111, 6260–6263 (1999).
 26.
Kaiser, K. et al. An \(sp\)hybridized molecular carbon allotrope, cyclo[18]carbon. Science 365, 1299–1301 (2019).
 27.
Lu, T., Chen, Q. & Liu, Z. A thorough theoretical exploration of intriguing characteristics of cyclo[18]carbon: Geometry, bonding nature, aromaticity, weak interaction, reactivity, excited states, vibrations, molecular dynamics and various molecular properties. ChemRxiv, Preprint. https://doi.org/10.26434/chemrxiv.11320130.v2 (2019).
 28.
Stasyuk, A. J., Stasyuk, O. A., Solà, M. & Voityuk, A. A. Cyclo[18]carbon: The smallest allcarbon electron acceptor. Chem. Commun. 56, 352–355 (2020).
 29.
Martin, J. M. L., ElYazal, J. & François, J.P. Structure and vibrational spectra of carbon clusters C\(_{n}\) (\(n\) = 2–10, 12, 14, 16, 18) using density functional theory including exact exchange contributions. Chem. Phys. Lett. 242, 570–579 (1995).
 30.
HeatonBurgess, T. & Yang, W. Structural manifestation of the delocalization error of density functional approximations: C\(_{4N+2}\) rings and C\(_{20}\) bowl, cage, and ring isomers. J. Chem. Phys. 132, 234113 (2010).
 31.
Shi, B., Yuan, L., Tang, T., Yuan, Y. & Tang, Y. Study on electronic structure and excitation characteristics of cyclo[18]carbon. Chem. Phys. Lett. 741, 136975 (2020).
 32.
Pereira, Z. S. & da Silva, E. Z. Spontaneous symmetry breaking in cyclo[18]carbon. J. Phys. Chem. A 124, 1152–1157 (2020).
 33.
Li, M. et al. Potential molecular semiconductor devices: CycloC\(_{n}\) (\(n\) = 10 and 14) with higher stabilities and aromaticities than acknowledged cycloC\(_{18}\). Phys. Chem. Chem. Phys. 22, 4823–4831 (2020).
 34.
Hong, I. et al. Competition between Hückel’s rule and JahnTeller distortion in small carbon rings: A quantum Monte Carlo study. J. Phys. Chem. A 124, 3636–3640 (2020).
 35.
Brémond, É, PérezJiménez, ÁJ., Adamo, C. & SanchoGarcía, J. C. sphybridized carbon allotrope molecular structures: An ongoing challenge for densityfunctional approximations. J. Chem. Phys. 151, 211104 (2019).
 36.
Zou, W., Tao, Y. & Kraka, E. Systematic description of molecular deformations with CremerPople puckering and deformation coordinates utilizing analytic derivatives: Applied to cycloheptane, cyclooctane, and cyclo[18]carbon. J. Chem. Phys. 152, 154107 (2020).
 37.
Baryshnikov, G. V., Valiev, R. R., Kuklin, A. V., Sundholm, D. & Ågren, H. Cyclo[18]carbon: Insight into electronic structure, aromaticity, and surface coupling. J. Phys. Chem. Lett. 10, 6701–6705 (2019).
 38.
Zhang, L., Li, H., Feng, Y. P. & Shen, L. Diverse transport behaviors in cyclo[18]carbonbased molecular devices. J. Phys. Chem. Lett. 11, 2611–2617 (2020).
 39.
Brus, L. Size, dimensionality, and strong electron correlation in nanoscience. Acc. Chem. Res. 47, 2951–2959 (2014).
 40.
Kohn, W. & Sham, L. J. Selfconsistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).
 41.
Cohen, A. J., MoriSánchez, P. & Yang, W. Challenges for density functional theory. Chem. Rev. 112, 289–320 (2012).
 42.
Andersson, K., Malmqvist, P. Å & Roos, B. O. Secondorder perturbation theory with a complete active space selfconsistent field reference function. J. Chem. Phys. 96, 1218–1226 (1992).
 43.
Hachmann, J., Dorando, J. J., Aviles, M. & Chan, G. K. L. The radical character of the acenes: A density matrix renormalization group study. J. Chem. Phys. 127, 134309 (2007).
 44.
Gidofalvi, G. & Mazziotti, D. A. Activespace twoelectron reduceddensitymatrix method: Complete activespace calculations without diagonalization of the \(N\)electron hamiltonian. J. Chem. Phys. 129, 134108 (2008).
 45.
Gryn'ova, G., Coote, M. L. & Corminboeuf, C. Theory and practice of uncommon molecular electronic configurations. WIREs Comput. Mol. Sci. 5, 440–459 (2015).
 46.
FossoTande, J., Nguyen, T.S., Gidofalvi, G. & DePrince, A. E. III. Largescale variational twoelectron reduceddensitymatrixdriven complete active space selfconsistent field methods. J. Chem. Theory Comput. 12, 2260–2271 (2016).
 47.
Battaglia, S., FaginasLago, N., Andrae, D., Evangelisti, S. & Leininger, T. Increasing radical character of large [\(n\)]cyclacenes unveiled by wave function theory. J. Phys. Chem. A 121, 3746–3756 (2017).
 48.
Mullinax, J. W. et al. Heterogeneous CPU + GPU algorithm for variational twoelectron reduceddensity matrixdriven complete activespace selfconsistent field theory. J. Chem. Theory Comput. 15, 6164–6178 (2019).
 49.
Chai, J.D. Density functional theory with fractional orbital occupations. J. Chem. Phys. 136, 154104 (2012).
 50.
Chai, J.D. Thermallyassistedoccupation density functional theory with generalizedgradient approximations. J. Chem. Phys. 140, 18A521 (2014).
 51.
Chai, J.D. Role of exact exchange in thermallyassistedoccupation density functional theory: A proposal of new hybrid schemes. J. Chem. Phys. 146, 044102 (2017).
 52.
Xuan, F., Chai, J.D. & Su, H. Local density approximation for the shortrange exchange free energy functional. ACS Omega 4, 7675–7683 (2019).
 53.
Lin, C.Y., Hui, K., Chung, J.H. & Chai, J.D. Selfconsistent determination of the fictitious temperature in thermallyassistedoccupation density functional theory. RSC Adv. 7, 50496–50507 (2017).
 54.
Wu, C.S. & Chai, J.D. Electronic properties of zigzag graphene nanoribbons studied by TAODFT. J. Chem. Theory Comput. 11, 2003–2011 (2015).
 55.
Yeh, C.N. & Chai, J.D. Role of Kekulé and nonKekulé structures in the radical character of alternant polycyclic aromatic hydrocarbons: A TAODFT study. Sci. Rep. 6, 30562 (2016).
 56.
Seenithurai, S. & Chai, J.D. Effect of Li adsorption on the electronic and hydrogen storage properties of acenes: A dispersioncorrected TAODFT study. Sci. Rep. 6, 33081 (2016).
 57.
Wu, C.S., Lee, P.Y. & Chai, J.D. Electronic properties of cyclacenes from TAODFT. Sci. Rep. 6, 37249 (2016).
 58.
Yeh, C.N., Wu, C., Su, H. & Chai, J.D. Electronic properties of the coronene series from thermallyassistedoccupation density functional theory. RSC Adv. 8, 34350–34358 (2018).
 59.
Seenithurai, S. & Chai, J.D. Electronic and hydrogen storage properties of Literminated linear boron chains studied by TAODFT. Sci. Rep. 8, 13538 (2018).
 60.
Chung, J.H. & Chai, J.D. Electronic properties of Möbius cyclacenes studied by thermallyassistedoccupation density functional theory. Sci. Rep. 9, 2907 (2019).
 61.
Seenithurai, S. & Chai, J.D. Electronic properties of linear and cyclic boron nanoribbons from thermallyassistedoccupation density functional theory. Sci. Rep. 9, 12139 (2019).
 62.
Deng, Q. & Chai, J.D. Electronic properties of triangleshaped graphene nanoflakes from TAODFT. ACS Omega 4, 14202–14210 (2019).
 63.
HansonHeine, M. W. D. Static correlation in vibrational frequencies studied using thermallyassistedoccupation density functional theory. Chem. Phys. Lett. 739, 137012 (2020).
 64.
PérezGuardiola, A. et al. The role of topology in organic molecules: Origin and comparison of the radical character in linear and cyclic oligoacenes and related oligomers. Phys. Chem. Chem. Phys. 20, 7112–7124 (2018).
 65.
PérezGuardiola, A. et al. From cyclic nanorings to singlewalled carbon nanotubes: Disclosing the evolution of their electronic structure with the help of theoretical methods. Phys. Chem. Chem. Phys. 21, 2547–2557 (2019).
 66.
Yoshikawa, T., Doi, T. & Nakai, H. Finitetemperaturebased linearscaling divideandconquer selfconsistent field method for static electron correlation systems. Chem. Phys. Lett. 725, 18–23 (2019).
 67.
Shao, Y. et al. Advances in molecular quantum chemistry contained in the QChem 4 program package. Mol. Phys. 113, 184–215 (2015).
 68.
Smith, M. B. & Michl, J. Singlet fission. Chem. Rev. 110, 6891–6936 (2010).
 69.
Rivero, P., JiménezHoyos, C. A. & Scuseria, G. E. Entanglement and polyradical character of polycyclic aromatic hydrocarbons predicted by projected HartreeFock theory. J. Phys. Chem. B 117, 12750–12758 (2013).
 70.
Che, Y. et al. Enhancing onedimensional charge transport through intermolecular \(\pi \)electron delocalization: Conductivity improvement for organic nanobelts. J. Am. Chem. Soc. 129, 6354–6355 (2007).
 71.
Cohen, A. J., MoriSánchez, P. & Yang, W. Development of exchangecorrelation functionals with minimal manyelectron selfinteraction error. J. Chem. Phys. 126, 191109 (2007).
 72.
Lin, Y.S., Li, G.D., Mao, S.P. & Chai, J.D. Longrange corrected hybrid density functionals with improved dispersion corrections. J. Chem. Theory Comput. 9, 263–272 (2013).
Acknowledgements
This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST1072628M002005MY3), National Taiwan University (Grant No. NTUCDP105R7818), and the National Center for Theoretical Sciences of Taiwan.
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S.S. and J.D.C. designed the project, performed the data analysis, and wrote the manuscript. S.S. carried out the calculations.
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Seenithurai, S., Chai, J. TAODFT investigation of electronic properties of linear and cyclic carbon chains. Sci Rep 10, 13133 (2020). https://doi.org/10.1038/s4159802070023z
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