TAO-DFT investigation of electronic properties of linear and cyclic carbon chains

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 (l-CC[n]) and cyclic carbon chains (c-CC[n]) with n = 10–100 carbon atoms, using thermally-assisted-occupation density functional theory (TAO-DFT). For all the cases investigated, l-CC[n]/c-CC[n] are ground-state singlets, and c-CC[n] are energetically more stable than l-CC[n]. The electronic properties of l-CC[n]/c-CC[n] reveal certain oscillation patterns for smaller n, followed by monotonic changes for larger n. For the smaller carbon chains, odd-numbered l-CC[n] are more stable than the adjacent even-numbered ones; c-CC[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4m+2$$\end{document}4m+2]/c-CC[4m] are more/less stable than the adjacent odd-numbered ones, where m are positive integers. As n increases, l-CC[n]/c-CC[n] possess increasing polyradical nature in their ground states, where the active orbitals are delocalized over the entire length of l-CC[n] or the whole circumference of c-CC[n].

Scientific RepoRtS | (2020) 10:13133 | https://doi.org/10.1038/s41598-020-70023-z www.nature.com/scientificreports/ the serendipitous discovery of C 60 1 . In attempting to synthesize l-CC[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 multi-walled 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 large-scale syntheses of linear carbon chains inside double-walled CNTs by using the confined space inside the tube as a nanoreactor to grow ultra-long 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 single-walled CNTs can modify the electronic properties of pristine single-walled 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 .
In general, it remains extremely difficult to synthesize both l-CC[n] and c-CC [n]. Accordingly, theoretical studies are complementary, and may provide immense information on the properties of l-CC[n] and c-CC [n]. In a recent theoretical study, it has been reported that very small l-CC[n] (n = 5-10) with even n values possess diradical nature in their ground states 16 . Therefore, it can be anticipated that the larger l-CC[n] and c-CC[n] could also possess radical character due to the low dimensionality of l-CC[n] and c-CC[n], respectively 39 . Despite its success in describing some ground-state properties, Kohn-Sham density functional theory (KS-DFT) 40 can yield unreliable results for systems with radical nature 41 , when the conventional XC density functionals are employed. Typically, multi-reference (MR) electronic structure methods, such as the complete-active-space self-consistentfield (CASSCF), complete-active-space second-order 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 l-CC[n] and c-CC [n].
Recently, thermally-assisted-occupation density functional theory (TAO-DFT) 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 θ and orbital occupation numbers) can approximately describe strong static correlation even when the simplest local density approximation (LDA) XC density functional is employed. In TAO-DFT, one can also adopt more sophisticated XC density functionals, such as semilocal 50 , global hybrid 51 , and range-separated hybrid 51,52 XC density functionals. Besides, aiming to improve the accuracy of TAO-DFT for a wide range of applications, a self-consistent scheme determining the fictitious temperature θ in TAO-DFT has been recently proposed 53 . Since TAO-DFT is a computationally efficient electronic structure method, a number of strongly correlated electron systems at the nanoscale have been studied using TAO-DFT in recent years 16,[54][55][56][57][58][59][60][61][62] . Besides, TAO-DFT has been recently shown to be useful in describing the vibrational spectra of molecules with radical nature 63 . In addition, TAO-DFT 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 linear-scaling divide-and-conquer approach for the study of large systems with strong static correlation effects 66 .

computational details
All geometry optimizations and other calculations are performed with Q-Chem 4.4 67 , using the 6-31G(d) basis set (i.e., a valence double-zeta polarized basis set) with a numerical quadrature that consists of 75 points in the Euler-Maclaurin radial grid and 302 points in the Lebedev angular grid. We carry out calculations using TAO-LDA 49 , which is TAO-DFT employing the LDA XC and θ-dependent density functionals, with the recommended fictitious temperature θ = 7 mhartree 49 .
In several recent studies 46,48,49,54,55 , the orbital occupation numbers obtained from TAO-LDA (with θ = 7 mhartree) have been found to be qualitatively similar to the natural orbital occupation numbers obtained from the variational two-electron reduced-density-matrix-driven CASSCF (v2RDM-CASSCF) method (i.e., an accurate MR electronic structure method), leading to a similar trend for the radical nature of several systems.

Results and discussion
Singlet-triplet energy gap and singlet-quintet energy gap. Aiming   (3) using spin-unrestricted TAO-LDA. Here, the occupation number f i,σ of the i th σ-spin orbital (i.e., up-spin orbital or down-spin orbital) obtained with spin-unrestricted TAO-LDA, which ranges from 0 to 1, is closely related to the occupation number of the i th σ-spin natural orbital [49][50][51] . For a molecule with nonradical nature, the occupation numbers of all spin-orbitals should be close to either 0 or 1, leading to insignificant contributions to the corresponding S vN value. However, for a molecule with pronounced radical nature, the occupation numbers of active spin-orbitals (i.e., the spin-orbitals 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 vN value is expected to increase when the occupation numbers of active spin-orbitals become closer to 0.5 and/or the number of active spin-orbitals increases. As shown in Fig. 4d    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   and sizable singlet-triplet 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 c-CC [10], c-CC [14], c-CC [18], and c-CC [22], are likely to be synthesized in the near future. Note that among them, c-CC [18] have been recently synthesized 26 .
While the method adopted (i.e., TAO-LDA with the fictitious temperature θ = 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 θ-dependent density functionals in TAO-DFT, TAO-LDA can yield the self-interaction error 41,49,51 , and hence bond length alternation (BLA) can be suppressed 30 . Second, the present θ value is system-independent, 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 long-range corrected hybrid XC functionals 71,72 in TAO-DFT 51,52 for an improved description of nonlocal exchange effects, and the second issue can be greatly resolved by using the respective self-consistent scheme for determining the θ 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 singlet-triplet energy gap, singlet-quintet energy gap, fundamental gap, and active orbital occupation numbers) of l-CC[n]/c-CC[n], it is essential to employ accurate multi-reference methods, such as

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.