Cyclacenes and short zigzag nanotubes with alternanting Ge―C bonds: theoretical impacts of Ge on the ground state, strain, and band gap

Cyclacenes and short zigzag nanotubes withalternanting Ge―C bonds: theoretical impactsof Ge on the ground state, strain, and band gapMaryam Koohia, Monireh Ghavamia, Bibi Narjes Haerizadea,Hasan Zandia and Mohammad Zaman Kassaeea*

[6]n Cyclacenes and short zigzag [6]n carbon nanotubes (n=5–10) have unstable singlet open-shell (Sos) ground states. We haveboosted their stability by implementing altering Ge―C bonds that acquire Scs ground states with larger band gap (ΔELUMO–HOMO)at B3LYP and BPW91 levels of theory. Fascinatingly, homodesmic calculations indicated release of almost two folds of strainenergy upon substitution of germanium for carbon. Thismay turn the green lights for synthesis of germanium–carbon cyclacenesand short zigzag nanotubes. Copyright © 2014 John Wiley & Sons, Ltd.

Keywords: closed shell; cyclacene; homodesmic; nanotube; open shell; zigzag

INTRODUCTION

Carbon is found in several forms including amorphous, graphiteand diamond. Cyclacenes, nanotubes and fullerenes areaccepted as the fourth form of solid carbon (Fig. 1).[1,2]

Cyclacene, nanotube and fullerenes provide substitution,endohedral and exohedral dopings.[1,2] Cyclacenes are a classof fused benzenoid hydrocarbons which offer a simple concep-tual framework for understanding the structures and propertiesof nanotubes or fullerene systems.[3,4] They are of interest withrespect to their conjugation, spectroscopic properties andcavities. They serve as possible structural motifs for carbon nano-tubes (CNTs).[5] Cyclacene, a class of laterally fused benzoidhydrocarbons, has offered a simple conceptual framework forunderstanding the structure and properties of nanotube or ful-lerene systems because of their remarkable similarity to thesestructures. They have remained one of the most fascinatingbut, as yet, unrealized synthetic targets. A cyclacene structurehas two types of constituents including an arenoid belt—composed of fused benzenoid rings—and two rather largeannulenic peripheral rings that are made of transoid doublebonds with either 4 k or 4 k + 2 electrons (the Hückel andMöbious types of cyclacenes, respectively) (Fig. 2).[6]

A single-walled carbon nanotube (SWCNT) is a seamlessgraphene cylinder constructed by rolling up a two-dimensional(2D) graphite layer in such a way that the end of the roll-upvector Ch is superimposed on its origin (Fig. 3).[7]

As such, the structure of a SWCNT can be uniquely defined bythe roll-up vector, Ch = na1 +ma2, and is simply designated by(n,m). By definition, three types of SWCNTs can be constructed,i.e. achiral armchair (n,n) SWCNTs, achiral zigzag (n,0) SWCNTsand chiral (n,m) SWCNTs (n>m and m≠ 0). For a chiral (n,m)SWCNT, the chiral angle, θ, is defined as the angle between theroll-up vector Ch and the (n,0) zigzag direction (Figs. 3 and 4).[8]

Some computational studies predict triplet (ferromagnetic)ground states for cyclacenes.[9] These suggest Sos radical

character with antiferromagnetically (AFM) coupled electronspins for cyclacenes and short CNTs.[10] Two obstacles are to beovercome for synthesis of cyclacenes: the bending problemand the low singlet–triplet energy gaps (ΔE(T� S)).[3,4] Oneapproach to get around these is the incorporation of eight-membered rings, while the other is substitution of heteroatoms in cyclacenes (Fig. 5).[11]

Heteroatom substitution in cyclacenes, fullerenes, peapodsand CNTs is a promising path to control their structural, mechan-ical, and especially electronic properties.[12–15] Such substitutionmay be implied by embedding heteroatoms into the open spacein the carbon network or by substituting the host atoms withsubstituents. As we had shown, nitrogen substitutions alter theelectronic ground state of some cyclacenes from Sos to Scs andappear to decrease their reactivity while increasing theirviability.[16] As pointed out, high reactivity of the radicaloid Sosground states and strain energies complicate cyclacene and shortCNT syntheses. Hence, following our previous work,[16] our theoreticalapproach to circumvent these obstacles is the substitution ofgermanium in cyclacenes and short zigzag CNTs at B3LYP andBPW91 levels of theory. This is in order tomodify the electronic groundstate of the resulting GeC-cyclacenes and GeCNTs, respectively.

COMPUTATIONAL METHODS

Full geometry optimizations are accomplished without any symmetryconstraints by means of hybrid functional B3LYP[17,18] and the 6-31+G*basis set, employing the GAMESS program package.[19] Biradicals species,

* Correspondence to: M. Z. Kassaee, Department of Chemistry, Tarbiat ModaresUniversity, P.O. Box 14115-175, Tehran, Iran.E-mail: [email protected]

a M. Koohi, M. Ghavami, B. N. Haerizade, H. Zandi, M. Z. KassaeeDepartment of Chemistry, Tarbiat Modares University, P.O. Box 14115-175,Tehran, Iran

Research Article

Received: 11 March 2014, Revised: 23 May 2014, Accepted: 11 June 2014, Published online in Wiley Online Library: 25 July 2014

(wileyonlinelibrary.com) DOI: 10.1002/poc.3333

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including singlet open-shell (Sos) and triplet (T) ground states arecomputed using unrestricted broken spin-symmetry UB3LYP/6-31G*methods. Both restricted (for singlet close-shell) and unrestricted (forsinglet open-shell and triplet states) B3LYP density functional methodsare available in the GAMESS, allowing the dynamics to be studied atdifferent levels of accuracy with the DFT results expected to providethe more accurate structural and energetic results. The NBO populationanalyses are accomplished at the above level of theory, as well as

B3PW91.[20] The harmonic vibration fre-quency analyses are also performed at thesame level of theory used for optimizationto check whether the obtained structure isa minimum on its potential energysurface.[21] The applied basis set is comprisedof Pople’s well known 6-31G* basis set.[22–25]

To investigate the doped heteroatom substi-tution effects on the cyclacenes and shortzigzag CNT aromaticity or its electron delo-calization, nucleus-independent chemicalshifts (NICS) are calculated at the ring centersof structures in order to evaluate themobilityof electrons on the ring surfaces.[26]

RESULTS AND DISCUSSION

Evaluation of stability through singletclosed-shell, singlet open-shell andtriplet energy separations: ΔE(Sos� Scs)and ΔE(T� Scs).Thermodynamic parameters of six

[6]n GeC-cyclacenes along with sixshort nanotubes, [6]n GeCNTs (wheren= 5–10), are compared with theirunsubstituted analogs, at B3LYP/6-31G*(Figs. 6 and 7, Table 1 and S1).

The calculated harmonic vibrational frequencies at B3LYP/6-31G*level confirm that all the optimized GeC-cyclacenes and GeCNTsare true minima (NIMAG=0). All of Ge substituted Scs cyclacenes,with altering ―Ge―C― bonds, turn out to be more viable thantheir corresponding unsubstituted Sos carbon analogs reportedby Chen et al.[10] Similarly, all of GeCNTs become more accessiblethan their corresponding Sos unsubstituted carbon analogs forpossessing Scs ground states (Table 1). Clearly, the ground state isScs (when n=5–10), but E(Scs) and E(Sos) have similar values foralmost all of scrutinized GeC-cyclacenes and GeC-nanotubes,giving ΔE(Sos� Scs)≈ 0. Since the spin contaminations are zero(<S2> Sos = 0), one may conclude that the Sos species relax tothe Scs ground states with nonmagnetic character.[11] Increasingthe number of fused rings from n= 5 to 10 in [6]n Ge-C-cyclacenesand [6]n GeC-nanotubes has conspicuous effects on bothΔE(Sos� Scs) and ΔE(T� Scs) energy gaps (Fig. 8). This effect ismore pronounced for the former than the latter.Germanium substitutions on [6]n cyclacenes and short zigzag

nanotubes alter their electronic ground states from unstable

Figure 1. Allotropes of carbon

Figure 2. A general structure for [6]n cyclacenes where n is the numberof fused rings[6]

Figure 3. Roll-up vector Ch (Ch = na1 +ma2) and chiral angle θ for a (n,m)SWCNT (for this special case, n=5, m=3), where a1 and a2 are theprimitive vectors of a graphene sheet[7]

Figure 4. A general structure for [6]n single-walled carbon nanotubewhere n is the number of fused rings.[8]

M. KOOHI ET AL.

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Sos to rather stable Scs. Practically speaking, for nanotubes withn= 6–9, the Sos and Scs states emerged with similar energies[ΔE(Sos� Scs)≈ 0]. Nevertheless, considering their zero spincontaminations we assume Scs ground state for them.[27] Thestabilizing effect of germanium substitution is highest for [6]nGeC-cyclacenes and vanishes upon increasing the diameter ofthe tube for [6]n GeC-cyclacenes and short zigzag single-walledgermanium-carbon nanotubes. Another word, increasing thesize leads to Sos (antiferromagnetic) ground states. Interestingly,changes of ΔE(T� Scs) against n indicates higher singlet–triplet

energy gaps for GeC-cyclacenes than their correspondingGeC-nanotubes (Fig. 8).

Band gaps : ΔΕHOMO�LUMO ¼ ΕLUMO � EHOMO

To examine chemical reactivity of [6]n GeC-cyclacenes and [6]nGeC-nanotubes, the corresponding frontier molecular orbitalenergies, and band gaps are investigated (Figure S1). The

electrons donated by a molecule in areaction should be from its HOMO,while the electrons captured by themolecule should locate on itsLUMO.[28] On this basis, HOMO–LUMOgap (ΔEHOMO–LUMO) is traditionallyassociated with chemical stabilityagainst electronic excitation, with alarger gap corresponding to a greaterstability as well as lower chemical reac-tivity. On the other hand, the conductivityof a molecule is related to its band gap,with smaller ΔEHOMO–LUMO that accordsto more conductive species. The bandgap of species is varied depending onthe type and number of substitutedatoms.The major difference between the

substituted and unsubstituted cyclacenesis the reversal of the magnitude ofΔEHOMO–LUMO in going from thesmallest species in [6]5-cyclacene, tothe higher substituted ones (Table 2).Specifically, semiconductor unsub-

stituted [6]5-cyclacene appears withthe highest band gap (1.85 eV). Hence,it is predicted to be most stableagainst electronic excitation. While, itssubstituted analog, [6]5 GeC-cyclacene,immerges with the lowest band gap(1.15 eV). Hence, it is predicted toorchestrate the highest conductivityand charge transfer, making it a possi-ble candidate for hydrogen storage.On the other extreme, unsubstituted[6]10-cyclacene has the lowest bandgap (1.00 eV). While, substituted [6]10GeC-cyclacene has a moderate bandgap of 1.84 eV. Even though, our NBO-calculated frontier molecular orbitalsfor [6]nGeC-cyclacenes show sluggishchanges of the order of band gap as afunction of the ring size and number

of Ge atom. Also, themolecular orbital analyses show a larger bandgap for each substituted [6]n GeC-cyclacene compared to itscorresponding unsubstituted [6]n-cyclacene. The larger energygap is translated into the lower aromaticity and higher chemicalreactivity of [6]n GeC-cyclacene,[29,30] which is in contrast totypical organic molecules, where kinetic stability increases as thecorresponding ΔEHOMO–LUMO increases.[31–35] Change of bandgap as a function of the ring size approaches zero with a faster ratefor only unsubstituted [6]n-cyclacene. So, in substituted [6]n GeC-cyclacenes, the lower energy gap is translated into the higher

Figure 5. Two samples of cyclacenes with less strain and wider singlet-triplet energy gaps (ΔE(T� S))[10]

Figure 6. Three-dimensional structural representatives of our [6]n GeC-cyclacenes, where n=5–10, cal-culated at B3LYP/6-31G* (see Table 1 and S1)

CYCLACENES, AND SHORT ZIGZAG NANOTUBES WITH ALTERNANTING GE-C BONDS

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number of resonance sextets, implying a greater aromaticity andstability of the molecule. This hypothesis is already applied to theband gap of nanotubes, fullerene and graphite.[36] Except for [6]5-cyclacene, another evidence for the higher viability and en-hanced stability against electronic excitations is the larger bandgaps (ΔELUMO–HOMO) observed for substituted [6]n GeC-cyclacenescompared to their corresponding unsubstituted [6]n cyclacenes.However, the substituted [6]nGeC-cyclacenes are more conductivethan the unsubstituted [6]n-cyclacenes (n=6-10). Apparently theelectronegativity difference between Ge and C causes π electronpolarization from germanium toward carbon. Such a polarization

induces the lowering of the energylevels, especially HOMOs of GeC-cyclacenes which increase the corre-sponding band gaps (Table 2). Loweringthe energy level can overcome thepairing repulsion energy of two elec-trons in one orbital and alter Scs for Sosground state. What is more, germaniumwith larger orbital size than carbon takeslarger orbital space, too. The repulsionbetween the two electrons decreaseswhile the possibility of Scs groundstate increases. Finally, the bandgaps of unsubstituted [6]n-cyclacenestend to become somewhat narrowerwith increasing the size of rings.While, on the average, the band gapof substituted [6]nGeC-cyclacenes be-comes mostly larger with increasingthe size of rings and the number of alter-nating substituted Ge heteroatoms.

Frontier molecular orbitals energy gaps:ΔE HOMO�1ð Þ� HOMOð Þ ¼ EHOMO� EHOMO�1;and ΔE LUMOð Þ� LUMOþ1ð Þ ¼ ELUMOþ1 � ELUMO

Here, we include the MO interactionscheme comparing a carbon and agermanium in order to support thestability of the proposed dopedcyclacenes (Fig. 9).As was shown previously

cyclacenes may be considered asbeing formed from two very similarconjugated rings, the so-calledtrannulenes.[37] Considering their con-stitutional and topological similarities(Fig. 10), both trannulenes should pos-sess two very similar sets of molecularorbitals.Since the number of C atoms in

such trannulenes, being part of thenormal cyclacenes, is always even,they must be alternate in the senseof Coulson’s pairing theorem.[38] Thisstatement is valid for both 4 k and4 k + 2 cyclacenes, where k is the num-ber of the six-membered rings in themolecule. Accordingly, suchtrannulenes should have an even

number of MOs that are symmetrically displaced relative to acentral energy level, i.e. the α level in the term of the HückelMO theory. However, no non-bonded MO (NBMO) exists for suchring hydrocarbons. Connection of the two trannulenes to eachother via one or more bonds causes a symmetric split in the en-ergy levels for each two similar and energetically degenerateMOs on both trannulenes (Fig. 11).Since all the bonding MOs of both trannulenes are doubly oc-

cupied with electrons, all the newly formed MOs, due to thebond formation between the two trannulenes, are doubly occu-pied as well. The change in energy is of a second-order

Figure 7. Three-dimensional structural representatives of our [6]n GeC-nanotubes, where n=5–10, cal-culated at B3LYP/6-31G* (see Table 1 and Figure S1)

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perturbation term. It causes the formation of essential singlebonds between the trannulenes. This is seen in the calculatedlonger bonds being formed, as compared with the C―C bondsof the trannulene rings in cyclacenes. As a consequence of theequal displacements of MOs from a central energy level, a factinherent within the Coulson’s pairing theorem,[38] and of theequivalent splitting in the MO energy levels (Fig. 12), the differ-ence between the newly formed HOMO and LUMO energy levelsis expected to be small. This small HOMO–LUMO energy differ-ence compels a small energy gap between the singlet and thetriplet ground states, which might be viewed in the formerly

reported ΔE(T� S) values. The smallvalues of ΔE(T� S), which apply evenfor the odd numbered [6]7-cyclacene,seem to be the cause of chemical insta-bility of cyclacenes.[3] To increase theΔE(T� S) energy difference one shouldperturb the MOs of both trannulenesthrough introduction of a new, singlyoccupied moiety to each ring and thusform a new odd-alternate ring systemthat possesses an NBMO according tothe pairing theorem. According toDewar’s perturbation molecular orbitaltheory (PMO theory) the two oddalternant hydrocarbon rings, i.e. theperturbed trannulenes, may interactwith each other forming one or moreconnection bonds, in the form of one-or multi-center first-order interactions.[39]

Only one MO of the equally displacedNBMOs of the two trannulenes is occu-pied by the two electrons of theperturbed trannulenes; the other MOremains empty. It can be shown that thesplitting in NBMO energy levels is alwaysbigger than that in other MOs of thetrannulene ring. This is due to the factthat the NBMOs are localized at thestarred, C*, atoms of the alternatingradical and have values of zero atthe unstarred, C0, atoms.[39] Due to the

smaller number of C* atoms participating in the formation ofthe NBMO, and the fact that almost all C atoms participate inthe formation of other MOs of the hydrocarbon radical, thecalculated MO coefficients of the NBMO are usually bigger inmagnitude than those of the remaining MOs. This difference inMO coefficient magnitudes causes the bigger splitting in MOenergy levels, considering the fact that the first-order as well asthe second-order interactions are always functions of a, b and β,where a and b are the coefficients at the two atoms of interactionand β is the Hückel resonance parameter. It is easily seen thenthat the energy required exciting an electron from the new

Table 1. Point groups (PGs) and energies (hartree) for singlet closed-shell, E(Scs), singlet open-shell, E(Sos), and triplet, E(T) statescalculated for the optimized [6]n GeC-cyclacenes and [6]n GeC-nanotubes, along with the corresponding spin contaminations ofsinglet open-shell (<S2> Sos) and that of triplet (<S2> T), at B3LYP/6-31G*

Structures PGs E(Scs) <S2> Scs E(Sos) <S2> Sos E(T) <S2> T

GeC -cyclacenes [6]5 C1 �21 136.738372 0.00 �21 136.7376276 1.04 �21 136.703369 2.06[6]6 C3v �25 364.161596 0.00 �25 364.160553 1.05 �25 364.137921 2.05[6]7 C1 �29 591.559487 0.00 �29 591.558502 1.11 �29 591.523380 2.04[6]8 C4v �33 818.955769 0.00 �33 818.954469 1.05 �33 818.926970 2.04[6]9 C3v �38 046.347771 0.00 �38 046.346760 1.08 �38 046.316987 2.05[6]10 C5v �42 273.740520 0.00 �42 273.740416 1.17 �42 273.713419 2.04

GeC -nanotubes [6]5 C1 �31 702.301032 0.00 �31 702.300282 1.01 �31 702.279854 2.11[6]6 C3v �38 042.826581 0.00 �38 042.826064 1.02 �38 042.809064 2.05[6]7 C1 �44 383.326499 0.00 �44 383.325972 1.09 �44 383.313853 2.05[6]8 C4v �50 723.844958 0.00 �50 723.844457 0.99 �50 723.822227 2.05[6]9 C3v �57 064.415586 0.00 �57 064.415080 1.13 �57 064.400614 3.07[6]10 C5v �63 404.875694 0.00 �63 404.875627 1.03 �63 404.864380 2.05

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5 6 7 8 9 105 6 7 8 9 10nn

Figure 8. Effects of the number of fused rings (n) on the energy gaps between (a) singlet open- andclosed-shell cyclacenes;(b) triplet and singlet closed-shell cyclacenes; (c) singlet open- and closed-shellnanotubes and (d) triplet and singlet closed-shell nanotubes



739

doubly occupiedMO to any unoccupiedMOof the formed cyclaceneshould be bigger than that in normal and unperturbed cyclacene.

Similarly, our NBO-calculated frontier molecular orbitals ap-pear consistent with the NICS analysis through confirmation ofthe geometrical parameters. Changing the number of π electronsin the peripheral pergermatrannulene circuits, from 4 k in theeven [6]n GeC-cyclacenes (n= 6, 8 and 10) to 4 k + 2 in the oddones (n= 5, 7 and 9), impinges on the aromaticity, structural pa-

rameters, as well as the energies of the corresponding frontier mo-lecular orbitals. The energy gaps between HOMO and HOMO� 1(ΔE(HOMO� 1)� (HOMO)), as well as LUMO and LUMO+1 (ΔE(LUMO)

(LUMO+1)), are near zero for the odd and even [6]n GeC-cyclacenesand [6]n GeC-nanotubes, with the exception for ΔE(LUMO)

(LUMO+1), when n is 5 and 6 (Table 6).When n is odd, the π electron number of pergermatrannulene

is 4 k + 2 and when n is even, the π electron number of

Table 2. Frontier molecular orbital energies (eV), band gaps (ΔEHOMO–LUMO/eV), and the nucleus-independent chemical shifts atthe center of the ring (NICS (0)/ppm) calculated for singlet closed-shell [6]n GeC-cyclacenes and [6]n GeC-nanotubes, at B3LYP/6-31G* and [B3PW91/6-31G*] (along with those of their carbon analogues in the parenthesis at B3LYP/6-31G*)a

Structures EHOMO (eV) ELUMO (eV) ΔELUMO–HOMO (eV) NICS (0)/ppm

GeC -cyclacenes [6]5 �4.908, [�4.879], (�4.299)a �3.757, [�3.485], (�2.449)a 1.151, [1.394], (1.850)a �10.31, (�9.6)b

[6]6 �4.666, [�4.562], (�3.782) �3.150, [�3.137], (�2.558) 1.516, [1.425], (1.224) �7.91, (�32.1)[6]7 �4.557, [�4.450], (�4.054) �2.490, [�2.256], (�2.394) 2.067, [2.194], (1.659) �8.24, (�10.4)[6]8 �4.464, [�4.377], (�3.891) �2.624, [�2.534], (�2.775) 1.840, [1.843], (1.115) �4.41, (�31.8)[6]9 �4.557, [�4.462], (�3.918) �2.490, [�2.443], (�2.775) 2.067, [2.019], (1.143) �6.62, (�6.4)[6]10 �4.464, [�4.417], (�3.918) �2.624, [�2.529], (�2.911) 1.840, [1.888], (1.006) �3.37, (�30.0)

GeC -nanotubes [6]5 �4.667 �4.122 0.545 �15.42[6]6 �4.228 �3.713 0.515 �6.19[6]7 �4.243 �3.322 0.921 �7.94[6]8 �4.058 �3.078 0.980 �4.93[6]9 �4.073 �3.067 1.006 �5.37[6]10 �4.243 �3.322 0.921 �5.37

aThe NBO-calculated energies (eV) for the frontier molecular orbitals of optimized structure from ref 7.bThe NICS (0)-calculated values (ppm) of optimized structure from ref 37d.

Figure 9. C―C versus Ge―C bond in the molecular orbital diagrams of their corresponding structures

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pergermatrannulene is 4 k. This is illustrated by the graphicalpresentations of the frontier MOs for n=9 with ΔE(HOMO

1)� (HOMO), and ΔE(LUMO)� (LUMO+1) = 0 eV, which means that theorbitals are degenerate. It is employed as a sample to show thedegeneracy of the HOMO and HOMO� 1, as well as the LUMO

and LUMO+1 for the odd [6]n GeC-cyclacenes and [6]n GeC-nanotubes (Fig. 11a).

The side-view graphical images clearly show that there is nocoupling between the HOMOs of the two separately rather aro-matic pergermatrannulene belts, containing the Hückel (4 k + 2)π electrons, in the odd [6]n GeC-cyclacenes and [6]n GeC-nanotubes (see Figure S1). As for the even [6]n GeC-cyclacenes and[6]n GeC-nanotubes, the π electron number of pergermatrannuleneis 4 k. The frontier MOs for n=8 with ΔE(HOMO� 1)� (HOMO)= 0.32,and ΔE(LUMO)� (LUMO+1) = 0.36 eV, respectively, shows that HOMOandHOMO� 1 are not degenerate, and distinct differences in theirMO shapes are apparent (Fig. 11b). The two singly occupiedSOMOs or NBMOs at the nonbonding level break the antiaromaticcharacter of the Hückel 4 k singlet π electrons, removing destabili-zation thereof by appearing as triplet pergermatrannulenes unitsin the even [6]n GeC-cyclacenes and [6]n GeC-nanotubes. To attainstabilization, the two [4 k]pergermatrannulene moieties gothrough a coupling interaction of the SOMOs, which split in twooccupied bonding MOs (HOMO and HOMO� 1) and two unoccu-pied antibonding MOs (LUMO and LUMO+1). The shape of theHOMO� 1 for n=8 shows the presence of inter-chain couplingsbetween the MOs of the two [4 k]pergermatrannulenes.

Coupling also results in relatively longer bonds d2 and d4 (thedistances between the apex atomsand fusion site atoms) of [6]n GeC-cyclacenes and [6]n GeC-nanotubescompared to d1, d3 and d5 (the dis-tance between two fusion siteatoms) (see Table 4).

Homodesmic equations : ΔEstrain¼ EAn– EAnþ1– EA1½ �

Insertion of Ge heteroatoms inthe fully optimized [6]n cyclacenesand nanotubes perturbs the stabil-ity, size and geometry of theresulting hetero nanostructures. Itis instructive to compare the totalenergy differences between [6]ncyclacenes and the strain free sub-units An, for n=6–10, through anhomodesmic equation: ΔEstrain= EAn[EAn+1� EA1] (Fig. 12 and Table 3).This gives us estimation for the

strain energy opposed by the hoopshape of GeC-cyclacenes. Plottingthe changes of strain energy(ΔEstrain) as a function of thenumber of fused benzene rings(n) in the [6]n cyclacenes indi-cates that the ΔEstrain/n valuesdecrease as a function of n forboth carbon cyclacenes andGeC-cyclacenes. ΔEstrain/n valuesappear ≈2 times lower for GeC-cyclacenes than their correspondingunsubstituted cyclacenes, indicatingrelease of strain energy upon germa-nium substitution.

Figure 11. Comparison between molecular orbital diagrams of odd and even [6]n GeC-cyclacenes: (a) the odd[n]pergermacyclacenes are made of two rather isolated [4 k+2]rings, for n=9; (b) the even ones are made oftwo interacting [4 k]pergermatrannulenes, for n=8

Figure 10. The two alternate trannulenes from which the [6]6-cyclaceneis formed



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Structural parameters, NICS and NBO atomic charge

[6]5, [6]7 GeC-cyclacenes and GeC-nanotubes which show C1symmetry, while all others appear with Cnv symmetry (Table 1).Despite the intrinsic reluctance of germanium to form doublebonds, the peripheral and the vertical Ge―C bond lengths(1.823–1.908Ǻ) fall in a range between the reported Ge¼C dou-ble bond (1.707Ǻ) and the Ge—C single bond (1.940Ǻ) (Table 4).

Such C―Ge bond lengths of optimized fused ring systems arequite close to the sum of covalent radii of C (0.70 Å) and Ge(1.25 Å) atoms. These results indicate the possibility of resonanceand aromaticity.

As a criterion for aromaticity of [6]n GeC-cyclacenes and [6]nGeC-nanotubes, we have selected the nucleus independentchemical shift values (NICS).[28] Despite the controversy sur-rounding this method for aromaticity measurements of a fewcompounds,[40] NICS is the most exploited for the approximationof aromaticity in nanorelated structures including carbon nano-tubes, cyclacenes, trannulenes and fullerene systems.[41,42] How-ever, the original NICS method is not often reliable. It iscompared to the perpendicular tensor, which is a better probefor aromaticity.[26] Total NICS merely gives information on

shielding constants (i.e. the iso-tropic value, one third of thetrace of the shielding tensors,which reflects the magnetic re-sponse properties), thus maycause loss of information on thediatropic character of aromaticmolecules and introduce spuri-ous effects arising from electronflow perpendicular to the mole-cular plane.[43] To obtain a “bet-ter” measure for characterizationof p system of the rings, wereport the NICS (0) values at thering centers. Calculations of NICS(0) of the [6]n GeC-cyclacenesand [6]n GeC-nanotubes arecarried out. They indicate thatthe aromaticity is inversely pro-portional with the size of thecyclacene and nanotubes(Table 2). On the other hand,the NICS values on thesubstituted species show aromaticcharacters (NICS=�3.4 to �15.4).These appear consistent withtheir corresponding 1,3,5-trigerminabenzenes (Ge3C3H6)which we find with an aromaticcharacter (NICS =�7.9). Subs-tituted [6]5 GeC-cyclacene and[6]10 GeC-cyclacene appear withthe highest and the lowestNICS (0) of �9.6 and �30.0 ppm,respectively. While, unsubstituted[6]5-cyclacene and [6]10-cyclaceneappear with the lowest andthe highest NICS (0) of �10.3and �3.4 ppm, respectively.Therefore, the major difference

between the substituted and unsubstituted cyclacenes is thereversal of the magnitude of NICS in going from the smallest spe-cies in [6]5-cyclacene, to the higher substituted ones (Table 2). As nincreases, those with odd values favor singlet closed-shell statesmore for substituted than unsubstituted species. On the otherhand, as n increases, those with even values favor singlet statesmore for unsubstituted than substituted species. On another

Figure 12. The stabilizing effects of Ge substitution in cyclacenes through a general homodesmic reaction (a),change of strain energy (ΔEstrain = EAn[EAn+1EA1]) as a function of the number of fused benzene rings (n=5�10)in the [6]n GeC-cyclacenes (b) and [6]n C-cyclacenes (c), at B3LYP/6-31G*

Table 3. Strain energy derived by the energy differencesbetween the cyclacenes and strain-free Subunits (An) througha general homodesmic reaction (see Fig. 9)

Structures ΔEstrain (kcal/mol) ΔEstrain/n (kcal/mol)

[6]5 114.512 (–)a 22.902 (–)a

[6]6 112.614 (225.374) 18.769 (37.562)[6]7 98.210 (200.520) 14.030 (28.645)[6]8 92.448 (163.730) 11.556 (20.466)[6]9 81.463 (160.326) 9.051 (17.814)[6]10 68.658 (–) 6.866 (–)aC-cyclacenes and polyacenes energies are taken from ref 17 and 37.

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Table 4. Structural parameters of GeC-cyclacene andnanotubes as a function of ring size after geometrical optimization bond lengths (Å) ofthe closed-shell singlet state (SCS), open-shell singlet (SOS) and triplet (T) states of [6]nGeC (n=5–10), with d2 and d4 being the distancesbetween the apex atoms and fusion site atoms, and d1, d3 and d5 showing the distance between two fusion site atoms at (U)B3LYP/6-31G*

B3LYP/6-31G* (closed-shell singlet)

Structures d1 d2 d3 d4 d5

GeC-cyclacenes (SCS) [6]5 1.853 1.908 1.840 – –[6]6 1.849 1.905 1.836 – –[6]7 1.842 1.903 1.831 – –[6]8 1.837 1.901 1.828 – –[6]9 1.835 1.900 1.827 – –[6]10 1.832 1.898 1.826 – –

GeC-nanotubes (SCS) [6]5 1.858 1.896 1.845 1.890 1.840[6]6 1.854 1.895 1.839 1.890 1.837[6]7 1.843 1.894 1.831 1.892 1.833[6]8 1.831 1.891 1.827 1.892 1.839[6]9 1.836 1.889 1.825 1.889 1.829[6]10 1.834 1.888 1.823 1.889 1.829

UB3LYP/6-31G* (open-shell singlet)GeC-cyclacenes (SOS) [6]5 1.851 1.907 1.841 – –

[6]6 1.850 1.903 1.836 – –[6]7 1.842 1.903 1.831 – –[6]8 1.838 1.901 1.829 – –[6]9 1.835 1.899 1.827 – –[6]10 1.832 1.898 1.826 – –

GeC-nanotubes (SOS) [6]5 1.850 1.891 1.849 1.895 1.846[6]6 1.849 1.905 1.836 1.844 1.775[6]7 1.833 1.892 1.831 1.895 1.844[6]8 1.831 1.891 1.827 1.892 1.840[6]9 1.837 1.889 1.825 1.889 1.830[6]10 1.833 1.887 1.823 1.888 1.828

UB3LYP/6-31G* (triplet)GeC-cyclacenes (T) [6]5 1.845 1.877 1.853 – –

[6]6 1.854 1.883 1.844 – –

(Continues)



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Table 4. (Continued)

UB3LYP/6-31G* (triplet)

Structures d1 d2 d3 d4 d5

[6]7 1.822 1.871 1.812 – –[6]8 1.821 1.883 1.849 – –[6]9 1.825 1.889 1.822 – –[6]10 1.834 1.886 1.831 – –

GeC-nanotubes (T) [6]5 1.860 1.858 1.866 1.847 1.872[6]6 1.854 1.875 1.847 1.875 1.845[6]7 1.854 1.881 1.835 1.877 1.839[6]8 1.848 1.862 1.844 1.860 1.803[6]9 1.825 1.862 1.823 1.862 1.829[6]10 1.836 1.877 1.826 1.879 1.831

Table 5. NBO atomic charges of hydrogen (H), carbon (C), germanum heteroatom (Ge), of the closed-shell singlet state (SCS), of [6]nGeC (n= 5–10), at B3LYP/6-311++G**

GeC-cyclacenes (SCS)

[6]5 [6]6 [6]7 [6]8 [6]9 [6]10

H1 +0.040 +0.041 +0.045 +0.045 +0.046 +0.046H2 +0.144 +0.143 +0.142 +0.142 +0.141 +0.141C1 �0.233 �0.212 �0.189 �0.189 �0.182 �0.176C2 �0.374 �0.370 �0.369 �0.369 �0.368 �0.366Ge1 +0.150 +0.133 +0.115 +0.116 +0.111 +0.106Ge2 +0.272 +0.264 +0.255 +0.256 +0.252 +0.249

GeC-nanotubes (SCS)H1 +0.046 +0.049 +0.052 +0.052 +0.054 +0.054H2 +0.145 +0.140 +0.137 +0.137 +0.136 +0.136C1 �0.216 �0.200 �0.194 �0.185 �0.177 �0.171C2 �0.221 �0.202 �0.186 �0.170 �0.155 �0.144C3 �0.361 �0.368 �0.374 �0.373 �0.371 �0.370Ge1 +0.115 +0.114 +0.114 +0.105 +0.097 +0.093Ge2 +0.259 +0.247 +0.237 +0.222 +0.211 +0.202Ge3 +0.237 +0.225 +0.215 +0.210 +0.205 +0.200

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extreme, substituted [6]n GeC-cyclacenes and [6]n GeC-nanotubes(n=5, 7 and 9) have relatively higher NICS (0) values, whileunsubstituted [6]n-cyclacenes and [6]n-nanotubes (n=5, 7 and 9)have lower values. Hence, all odd substituted [6]n GeC-cyclacenesand [6]n GeC-nanotubes (n=5, 7 and 9), are more aromatic thanthe even ones (n=6, 8 and 10), and all odd unsubstituted [6]n-cyclacenes and [6]n-nanotubes (n=5, 7 and 9) are less aromaticthan the even ones (n=6, 8 and 10).A sharp decrease of aromaticity (NICS absolute values) is

observed for the even substituted [6]n GeC-cyclacene and [6]nGeC-nanotube compared to their corresponding unsubstituted[6]n-cyclacene and [6]n-nanotube with n= 6, 8 and 10. Therefore,confinement of the GeC-benzenoid rings in the cyclacene andnanotube lowers their aromaticity due to the decreased planarityand expansion of π-electron density over the Ge—C network.Hence, in contrast to CNTs,[29] no differentiation is foundbetween the NICS values of the GeC-benzenoid ring centers(0). As such the strong ring currents in CNTs vanish in theircorresponding isoelectronic GeCNTs because the π-electronsare slightly shifted from Ge to C atoms and portions of valenceelectrons remain localized over the C nuclei.Finally, on the average, the NICS (0) values of both

unsubstituted and substituted [6]n-cyclacenes tend to becomelower with increasing the size of rings and the number ofalternating substituted Ge heteroatoms. Apparently, the innerring π electron density conceivably diminishes at the ring centersof the sterically hindered species (see Figure S1).Based on Froudakis’ findings on hydrogen absorption in nano-

tubes, point charges upon the material surface can improve thestorage capacity since they increase the binding energy ofhydrogen to the surface.[44] It was shown that silicon nanotubeswith alternative Si and C atoms were full of point charges, andhence they would serve as good candidates for hydrogenstorage. Hence, to probe the possible application of [6]n GeC-cyclacene and GeC-nanotube systems in hydrogen storage, theNBO population analysis is accomplished on the optimizedstructures at B3LYP/6-311++G** level. In general, since thestoichiometric ratio between carbon and the Ge heteroatoms isequal, and electronegativity (EN) decreases down a group ofperiodic table; hence, there is a considerable electron densitytransfer from doped Ge atoms to the remaining carbon atomsof scrutinized structures. Other words, germanium is moreelectropositive than carbon (Pauling ENs: Ge = 2.0, C = 2.5).Hence, Ge doping induces a contracted range of negativecharges of �0.171 to �0.370 on carbon atoms of [6]10 GeC-nanotube resulted from expanded positive charges of +0.093to +0.202 on Ge themselves (Table 5).

CONCLUSIONS

DFT calculations are carried out on the cyclacenes and shortzigzag single-walled nanotubes, and their Ge substitutedanalogs with [6]nGeC-cyclacenes and [6]n GeC-nanotubes (n= 5,6, 7, 8, 9 and 10) formulae. In all twelve cases, positive forceconstants are encountered, indicating the involvement ofminimum structures. The electronic ground states are alteredfrom Sos in unsubstituted to Scs in Ge―C species. The effect ofGe―substitution decrease with the increase of n (n= the num-ber of fused benzenoid rings). These effects are considerablymore significant in GeC-cyclacenes than GeC-nanotubes. Thestrain energy appears ≈2 times lower for GeC-cyclacenes than

their corresponding unsubstituted species. Larger band gaps(ΔELUMO–HOMO), rather viable GeC-cyclacene and short GeC-nanotubes systems may invite future experimental explorations.

Acknowledgements

We wish to express our special thanks to Dr. M. Ghambarian, Dr. H.Aref Rad for their cordial cooperation in completing this research.

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Documents

Cyclacenes and short zigzag nanotubes with alternanting Ge―C bonds: theoretical impacts of Ge on the ground state, strain, and band gap