Comment on “Theoretical investigation on bond and spectrum of cyclo[18]carbon (C18) with sp-hybridized”


Recently, Shuhong Xu et al. reported theoretical calculation of molecular structure, bonding, aromaticity, electron delocalization, and electronic spectrum of cyclo[18]carbon in J. Mol. Model., 26, 111 (2020). Due to inappropriate consideration of calculation strategy, misunderstanding of some analysis methods and concepts, and some errors in the data, there are misleading statements and unconvincing conclusions in their paper. Here, we will point out inadequacies of Shuhong Xu’s paper and put forward our own views. The contents of this comment will also help those who are studying cyclo[18]carbon to better understand this system and its analogues.


The cyclo[18]carbon first observed in the condensed phase in 2019 presents a very unique geometric configuration and electronic structure [1], which has aroused widespread interest in theoretical chemistry community. Recently, Shuhong Xu et al. discussed the bonding, electronic delocalization, aromaticity, and spectral character of the cyclo[18]carbon and its derivatives from a theoretical perspective [2]. We have read this paper with great interest, but based on our theoretical knowledge of quantum chemistry, our recent research experiences on the cyclo[18]carbon [3,4,5,6,7,8], and our verification of some results reported in their work, we noticed that there are some misleading statements, inappropriate conclusions, and incorrect data in their article. In this comment, we will list and discuss these issues one by one and present our own viewpoints. We hope that the content of this comment will enable readers to better understand the novel cyclo[18]carbon molecule and realize the importance of properly designing computational strategy to study it and analogous systems.

The quantum chemistry calculations involved in this article were carried out by Gaussian 16 (A.03) program [9], and the wavefunction analyses were realized by Multiwfn 3.7 code [10].


Selection of calculation method

The authors of Ref. [2] employed B3LYP exchange-correlation functional in density functional theory (DFT) [11] and Hartree-Fock (HF) method (specifically, restricted closed-shell form) in combination with 6-31G(d,p) basis set [12] to optimize the structure of the cyclo[18]carbon. However, the geometries obtained in this way are not reliable, making all subsequent analyses unconvincing.

Due to the quite unusual electronic structure of the cyclo[18]carbon, proper choice of calculation level should be paid great attention to. In our previous works, we have discussed in detail the selection of calculation method for the cyclo[18]carbon [5, 7]. Here, we will discuss this further. We have optimized the cyclo[18]carbon using different theoretical methods and basis sets and obtained minimum structures without imaginary frequency. The bond lengths and bond angles are listed in Table 1. Owing to the instability of the restricted closed-shell HF (RHF) wavefunction at the minimum structure of the cyclo[18]carbon, the HF calculation in the form of unrestricted open-shell (UHF) was also carried out by us with symmetry-broken treatment [13], which is frequently employed in literatures to study singlet biradical and polyradical systems [14]. RCCSD and UCCSD in Table 1 correspond to the coupled-cluster singles and doubles (CCSD) calculations with RHF and UHF wavefunctions as reference states, respectively. The large expectation values of S2 operator in Table 1 imply strong spin polarization in the corresponding wavefunctions, in other words, a significant separation of alpha and beta density distributions. For all DFT functionals involved in Table 1, it is found that the stable wavefunctions of the cyclo[18]carbon at their respective minimum structures are all closed-shell, so there is no difference between the restricted and unrestricted DFT results.

Table 1 Bond length (in Å) and bond angle (in °) of cyclo[18]carbon optimized at different calculation levelsa

It is well known that CCSD is a high level, accurate, and robust theoretical method, and the def-TZVP is a basis set of triple-zeta quality with polarization functions [15] sufficient for geometry optimization. Therefore, it can be expected that the CCSD/def-TZVP level should be reliable enough for geometry optimization of the challenging cyclo[18]carbon, and the optimized structure at this level can be used as a reference to judge the rationality of cheaper calculation levels. The results of RCCSD and UCCSD in Table 1 are only marginally different; they are completely consistent with the STM/AFM experimental observations [1]; that is, the cyclo[18]carbon is alternately arranged with short and long bonds, which is also known as a polyynic structure.

We first investigate the performance of various theoretical methods under the satisfactory def-TZVP basis set. Although it can be seen from Table 1 that the structure optimized by RHF seems to agree reasonably with that of RCCSD, the RHF wavefunction is not a stable solution at its optimized structure. In fact, the stable wavefunction can be obtained by UHF calculation with symmetry-broken treatment, whose energy is as much as 101.6 kcal/mol lower than that of RHF at their respective minimum structures. Therefore, the seemingly correct description of the cyclo[18]carbon structure by RHF should be regarded as a typical “good result due to wrong reason” phenomenon. Note that the use of UHF to represent this system is also unacceptable because it optimizes the cyclo[18]carbon to a so-called cumulenic structure with exactly equal C–C bonds corresponding to the D9d point group. The spin density corresponding to the UHF wavefunction is shown in Fig. 1. As can be seen, alpha and beta unpaired electrons are alternately distributed, showing that this system is in a polyradical state. We can therefore conclude that the HF method is essentially failed to describe the cyclo[18]carbon, mostly because it does not take electron correlation effect into account [13]. Although B3LYP is a widely employed DFT functional, it incorrectly describes the cyclo[18]carbon as a cumulenic structure, indicating that it is also failed completely for representing this system. One possible reason for the failure is the well-known excessive self-interaction error problem [13] in pure DFT functionals and those with relatively low HF exchange compositions, such as the B3LYP. The ωB97XD functional, on the other hand, describes the structural characteristics of the cyclo[18]carbon very well, this is why it has always been used in our series studies on the cyclo[18]carbon [3,4,5,6,7].

Fig. 1

Spin density isosurface map (isovalue = 0.02 a.u.) of cyclo[18]carbon calculated at UHF/def-TZVP level with symmetry-broken treatment. Green and blue regions denote the positive and negative parts, respectively. The structure has also been optimized at this level

As for the selection of basis set, it can be seen from Table 1 that under RHF, UHF, and ωB97XD, the stable structure optimized based on the 6-31G(d,p) basis set is obviously inconsistent with that obtained under the better def-TZVP basis set; that is, the bond angles in the 6-31G(d,p) structure are not all equal, and the corresponding C9h symmetry is lower than the D9h point group of the actual cyclo[18]carbon structure. It was found that the D9h symmetric cyclo[18]carbon structure has imaginary frequencies under the relatively poor 6-31G(d,p) basis set. For other commonly used double-zeta basis sets, such as def2-SVP [16] and cc-pVDZ [17], the same defects as 6-31G(d,p) were also identified, while the basis sets of triple-zeta quality including 6-311G(d) as well as the more expensive def-TZVP and def2-TZVP work well. Obviously, although double-zeta basis sets are usually adequate for geometry optimization of common chemical systems, it is absolutely necessary to employ basis sets of triple-zeta quality for the cyclo[18]carbon and its analogues.

According to the above discussion, it can be recognized that it is inappropriate to use the RHF/6-31G(d,p) like that in Ref. [2] to optimize and analyze the cyclo[18]carbon and its derivatives. The ωB97XD functional with at least 6-311G(d) basis set is suggested by us to study these kinds of systems to ensure qualitatively correct results.

Analysis of bonding

In the original text [2], the bonding character of the cyclo[18]carbon optimized at RHF/6-31G(d,p) level was described as single-triple bond alternation, which is actually a common misunderstanding of the nature of the cyclo[18]carbon. According to Mayer bond order [18], Laplacian bond order [19], localized molecular orbital analysis [20], and so on, we have shown that the long C–C bonds in this system are notably stronger than a typical single bond, while the short C–C bonds are weaker than a typical triple bond, and the numbers of effectively shared electron pairs of the long and short C–C bonds are greater than one and less than three, respectively [5]. In addition, the electron localization function (ELF) [21, 22], valence electron density [23], and deformation density [23] all proved that the short C–C bonds have significantly higher strength than the long C–C bonds. Therefore, the cyclo[18]carbon should be characterized by “alternating short and long C–C bonds” or “alternating strong and weak C–C bonds,” rather than “alternating single and triple C–C bonds” as claimed by the authors.

Based on the optimized structure and wavefunction yielded at the RHF/6-31G(d,p) level, the authors conducted a natural bond orbital (NBO) [24] analysis for the cyclo[18]carbon, and they claimed that the NBO analysis supports the argument of alternating single and triple bonds, but they did not discuss this point in any detail. If the authors reached this conclusion by simply counting the number of bond (BD)-type of NBO orbitals searched by the NBO program, we would like to point out here that this judgment is not convincing. The BD-type orbitals always tend to describe bonding in a chemical system as a fully localized state even if the system actually possesses highly delocalized electrons. For example, benzene molecule is described by BD-type orbitals as containing alternating single and double bonds; thus, the six-center delocalization nature of the π electrons is completely concealed. Similarly, it is inappropriate to characterize the bonding of the cyclo[18]carbon simply based on NBO orbitals, because this system shows prominent multi-center electron delocalization character. In Ref. [2], the authors also analyzed the bonding of other carbon rings (C16 and C20) in terms of NBO orbitals, the same attention should be paid.

Discussion of aromaticity

The authors performed a nucleus-independent chemical shifts (NICS) [25] analysis on the cyclo[18]carbon and claimed that “NMR results show that C18 molecule does not have aromaticity in NICS = 0 position” [2]; in fact, this conclusion is wrong. To verify their data, we also calculated the NICS(0) of the cyclo[18]carbon at the RHF/6-31(d,p) level adopted by the authors, and we found the NICS(0) is − 3.3 ppm. This nonnegligible negative value already exhibits the aromatic nature of the cyclo[18]carbon. It is generally believed that NICS(0)ZZ is a more reasonable descriptor of molecular aromaticity than the NICS(0).[26] In Ref. [5], we calculated the NICS(0)ZZ of the cyclo[18]carbon at the reliable ωB97XD/def2-TZVP level, and the result is − 25.7 ppm, which is even much more negative than that of benzene (− 16.1 ppm) at the same level, indicating that not only the cyclo[18]carbon is aromatic, but also its aromaticity is extremely strong. The iso-chemical shielding surface (ICSS) [27] method was also adopted by us to more rigorously and intuitively characterize the unusual aromatic characteristics of the cyclo[18]carbon [5]. Recent theoretical works about the cyclo[18]carbon by other groups also demonstrated that this molecule displays obvious aromaticity [28, 29].

Discussion of electron delocalization

The conclusion “HOMO and LUMO also illustrate that there is no delocalization in C18 molecule” [2] by the authors is arbitrary and not in line with reality. It is important to recognize that electron delocalization of a practical chemical system is never only reflected by HOMO and LUMO. We note that the Fermi hole function [22, 30] is a physically very rigorous real space function used to reveal the electron delocalization, and Bader clearly pointed out that “an electron can go where its hole goes and, if the Fermi hole is localized, then so is the electron” [30]. All occupied orbitals are involved in the expression of Fermi hole; that is, electrons in all occupied orbitals actually affect electron delocalization character in a chemical system. Therefore, it is evidently incomplete to only consider HOMO in the electron delocalization analysis. The LUMO, on the other hand, has no contribution to the electron delocalization at electronic ground state, because it does not participate in the Fermi hole function at all.

It is noteworthy that the HOMO isosurface given in Fig. 1 of the original text [2] is simultaneously distributed on multiple atoms, which, in fact, is already a direct evidence of existence electron delocalization in the cyclo[18]carbon. It is also worth mentioning that the HOMO and LUMO of the cyclo[18]carbon are both doubly degenerate [6], so the two degenerate orbitals of each should be given together during discussing to avoid misleading conclusion.

Currently, there are many analysis methods available to easily and intuitively reveal electron delocalization. The localized orbital locator of π electrons (LOL-π) [31], multi-center bond index [32], induced current diagrams [33, 34], and the aforementioned ICSS have been employed by us in a recent study on the cyclo[18]carbon [5]. These analyses rigorously exhibited that the cyclo[18]carbon possesses highly delocalized π electrons, which is the basis of its remarkable aromaticity.

Calculation of ECD signal

The authors declared that electronic circular dichroism (ECD) signals of C6, C12, C16, C18, C20, and B9N9 have been found, and the simulated ECD spectra were given in Fig. 5c of their paper [2] (where C22 in the figure legend should actually be C20). However, we found that their observations are fully artificial. We optimized these molecules at the reliable ωB97XD/def2-TZVP level and then calculated a total of 50 singlet excited states by time-dependent density functional theory (TDDFT) at the same level. The point groups of the optimized structures were found to be D3h, C6h, D8h, D9h, D10h, and D9h, respectively, and the calculated rotatory strengths of all excited states are exactly zero, indicating that there is no ECD signal for these molecules. In order to try to reproduce the ECD data of the cyclo[18]carbon in Ref. [2], we also employed the same calculation levels as the authors, that is, RHF/6-31G(d,p) for geometry optimization and then time-dependent Hartree-Fock (TDHF) for excited state calculation. We still found that the rotatory strengths of all excited states are zero. As to why ECD signals were detected by the authors, a possible reason is that the authors did not pay attention to examination and utilization of molecular point groups in their calculations. Due to the limited convergence threshold of geometric optimization in Gaussian program, if the initial structure was not properly symmetrized, then the optimized geometry may be marginally deviated to the actual point group. Incorrect point groups of the geometries employed in the excited state calculations may cause artificially non-zero rotatory strengths and detection of ECD signals.

Analysis of electrical conductivity

Based on the HF/6-31G(d,p) calculation, the authors mentioned that “the bandgap of C18 molecule is 9.70 eV” [2]. According to our reproduction of their data, we found that they actually referred to HOMO-LUMO gap. Employing the term “bandgap” in this context is evidently confusing and inappropriate, because isolated systems like molecules have no band structure at all. Similarly, the term “bandgap” was misused in the context of C18 derivatives with CO. Regarding this point, we strongly recommend Ref. [35], which gives an excellent introduction and distinction to the concept of “gap.”

In addition, the HF method adopted by the authors has a known problem of seriously overestimating gap [36]. As an example, we optimized the geometry of a representative periodic conjugated polymer, polyacetylene, at HSE06/6-311G(d) level [37, 38] and then calculated its bandgap at HF/6-311G(d) level, and the result 5.63 eV is much higher than the experimental measurement of 1.5 eV [39]. So, the discussion of the electrical conductivity of the cyclo[18]carbon based on the HF/6–31(d,p) level is undoubtedly unconvincing.

Analysis of molecular stability

The authors stated that “the binding energy of C18 molecule with HF/6-31G(d,p) is − 680.998 a.u. and − 685.23 a.u. in DFT/B3LYP/Lanl2dz. Obviously, the C18 molecule has good stability in DFT/B3LYP/Lanl2dz method, which is consecutive double bonds” [2]. This argument about binding energy is misleading. According to the reported data and our calculations, we found that their so-called binding energy is actually the electronic energy of the cyclo[18]carbon, rather than the binding energy in common sense. The binding energy and electronic energy of a chemical system are quite different concepts. In the field of computational chemistry, the binding energy usually refers to the energy change in the process of binding isolated fragments to the current system, while the electronic energy is defined as the energy change in the process of combining all electrons and nuclei of the system from infinitely separated status to the current status. In addition, it is the basic knowledge of quantum chemistry that the electronic energies estimated at different calculation levels are commonly not comparable, because electronic energy of any computational level used in daily practical research is subject to significant systematic error. Therefore, comparing the energy obtained from HF/6-31G(d,p) and B3LYP/Lanl2DZ calculations is totally meaningless. Moreover, stability is an inherent property of molecules themselves, so it is obviously inappropriate to discuss stability of molecules at a certain level of calculation. We also note in passing that the Lanl2 pseudopotential and its matching valence basis set Lanl2DZ are only defined for the elements starting from Na [40]. In the Gaussian 03 program employed by the authors, according to the statement in the program manual, using the “Lanl2dz” keyword for carbon will not cause the program to report an error, but instead assigns an completely outdated D95V basis set.

The conclusion by the authors according to the aforementioned energy data, namely, “maybe C18 molecule with consecutive double bonds exists and can be obtained one day, which has good stability than that of the C18 molecule with alternating single and triple bonds,” is evidently wrong. The high-resolution atomic spectroscopy [1] and the robust CCSD calculation mentioned above have fully confirmed that the actual structure of the cyclo[18]carbon has alternating short-long bonds, while the carbon ring with equal bond length (D9d point group, assumed by the authors to be all double bonds) cannot exist stably at all. Some recent theoretical studies on the cyclo[18]carbon have proved that the cyclo[18]carbon with D9d symmetry in fact is the transition state corresponding to its automerization process [29, 41].

Structure of cyclo[24]carbon

The authors mentioned in the paper that the stable structure of cyclo[24]carbon could not be obtained at HF/6-31G(d,p) level, but they did not explain any reason [2]. In fact, we found that the minimum structure of the cyclo[24]carbon without imaginary frequency can be successfully located at this level. Under the 6-31G(d,p) basis set, we optimized the structure of the cyclo[24]carbon by RHF method. The result shows that the lengths of the short and long C–C bonds are 1.196 and 1.379 Å, respectively, and the bond angles are 163.1° and 166.9° alternately. The minimum structure obtained by UHF method with symmetry-broken treatment showed that all the bond lengths are 1.298 Å, and the bond angles are 167.4° and 162.6° alternately. In this case, an alternating distribution of alpha and beta unpaired electrons is observed, and the characteristics of the spin density are very similar to those in Fig. 1. In addition, we optimized this system at the more reliable ωB97XD/def2-TZVP level and obtained the structure with bond lengths of 1.214 and 1.354 Å alternately and bond angle of 165.0°.

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Zeyu Liu: Investigation, and writing - review and editing

Tian Lu: Conceptualization, Investigation, Software, and writing - original draft

Qinxue Chen: Validation and writing - review and editing

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Liu, Z., Lu, T. & Chen, Q. Comment on “Theoretical investigation on bond and spectrum of cyclo[18]carbon (C18) with sp-hybridized”. J Mol Model 27, 42 (2021).

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  • Cyclo[18]carbon
  • Density functional theory
  • Hartree-Fock
  • Aromaticity
  • Electron delocalization
  • Conductivity
  • Stability