Vibrational anatomy of C 90 , C 96 , and C 100 fullertubes: probing Frankenstein ’ s skeletal structures of fullerene head endcaps and nanotube belt midsection †

Fullertubes are tubular fullerenes with nanotube-like middle section and fullerene-like endcaps. To understand how this intermediate form between spherical fullerenes and nanotubes is re ﬂ ected in the vibrational modes, we performed comprehensive studies of IR and Raman spectra of fullertubes C 90 - D 5h , C 96 - D 3d , and C 100 - D 5d . An excellent agreement between experimental and DFT-computed spectra enabled a detailed vibrational assignment and allowed an analysis of the localization degree of the vibrational modes in di ﬀ erent parts of fullertubes. Projection analysis was performed to establish an exact numerical correspondence between vibrations of the belt midsection and fullerene headcaps to the modes of nanotubes and fullerene C 60 - I h . As a result, we could not only identify fullerene-like and CNT-like vibrations of fullertubes, but also trace their origin in speci ﬁ c vibrational modes of CNT and C 60 - I h . IR spectra were found to be dominated by vibrations of fullerene-like caps resembling IR-active modes of C 60 - I h , whereas in Raman spectra both caps and belt vibrations are found to be equally active. Unlike the resonance Raman spectra of CNTs, in which only two single-phonon bands are detected, the Raman spectra of fullertubes exhibit several CNT-like vibrations and thus provide additional information on nano-tube phonons.


Introduction
The discovery of arc-discharge synthesis of fullerenes 1 brought new attention to carbonaceous materials leading soon to refocusing on carbon nanotubes (CNTs) and later on graphene.Fullerene and CNT research went hand-in-hand from the start, as both feature curved C-sp 2 network, and it was natural to think about CNTs as of strongly elongated, tubular fullerenes.Although ideal CNT is infinitely long, real nanotubes must have an end, and a seamless termination of the tubes can be achieved with fullerene-like caps.This imaginary is also supported from the fullerene side.By the Euler's theorem, all classical fullerenes have 12 pentagons, and the growth of fullerenes size inevitably means increase of the number of hexagons.The latter can be arranged in many ways, from quasispherical structures with uniform distribution of pentagons to tubular fullerenes with curved caps and CNT-like belts.Spherical fullerenes tend to be more thermodynamically stable, and high-resolution trapped ion mobility spectrometry suggested spherical shapes of large fullerenes, such as C 110 -C 150 . 2 Nevertheless, tubular fullerenes are also well documented.C 70 -D 5h can be considered as the first tubular fullerene, and its structure can be obtained from that of C 60 -I h by insertion of a belt of 10 carbon atoms between two C 30 hemispheres and rotation of one of the hemispheres by 36°.Consequent addition of C 10 fragments into the belt and 36°rotation of one cap creates a family of C 60+10n tubular fullerenes with alternating D 5h and D 5d symmetry and a growing (5, 5) CNT fragment.The next after C 70 -D 5h is C 80 -D 5d (1) isolated in 2000, 3 the minor isomer of C 80 seconds to the main C 80 -D 2 (2). 4 Then follows C 90 -D 5h (1), first characterized in 2010 in a pristine form 5 as well as chlorofullerenes C 90 Cl 10,12 . 6These works proved formation of C 90 -D 5h (1) in the arc-discharge process despite the low relative stability of this isomer.Likewise, computational studies of C 100 isomers showed that the tubular structure C 100 -D 5d (1) is not among the most stable ones. 7,8owever, this isomer was captured as a chloride C 100 Cl 12 , 9 and a molecular structure of the non-derivatized C 100 -D 5d (1) was recently proved by single-crystal X-ray diffraction of its co-crystal with decapyrrylcorannulene. 10Another structurally characterized tubular fullerene is C 96 -D 3d (1). 11In this molecule, the caps are not resembling C 60 hemispheres and have hexagon in the base, while the belt is a fragment of a zigzag (9,0) CNT.
The number of possible fullerene isomers grows dramatically with the fullerene size, and the separation of multiple similar isomeric structures turns into a complex and tedious procedure.Fortuitously, the studies of tubular fullerenes, aka fullertubes, gained the new boost with the discovery of the isolation route based on their reduced chemical reactivity. 10,12hile fullerenes usually readily react with amino-alcohols, tubular fullerenes appeared to be significantly less reactive, which allowed their facile separation from other fullerenes.The main fullertubes obtained this way are C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d (Fig. 1), but a possibility to isolate even larger tubular fullerenes was demonstrated in ref. 10, opening the way to their systematic exploration.C 96 -D 3d was found to be an efficient O 2 -reduction electrocatalyst in the recent study. 13he increasing availability of such tubular structures, which can be seen as intermediates between spherical fullerenes and 1D nanotubes, raises a question of whether fullertubes exhibit the properties of fullerenes, CNTs, or should be treated as a unique phenomenon.On the other hand, the systematic study of fullertubes may help to pinpoint a transition between discrete molecular properties to periodic 1D behavior.There is hardly a universal answer to this question because different properties have different degree of locality.For instance, the low chemical reactivity of fullertubes with amino-alcohols and selective chlorination of cap regions 6,9 seems to indicate that belt regions may be similar in their reactivity to CNTs.On the hand, the study of the electronic properties of fullertubes with (5, 5) and (9, 0) CNTs fragments concluded that the convergence is far from reach in realistic fullerene sizes and requires much longer tubes. 14,158][19] Force constants usually vanish over several bonds and therefore vibrations can be considered as more local than electronic excitations.Thus, the transition between confined and periodic properties may happen on a smaller length scale.Here we combine IR and Raman spectroscopy with DFT computations to obtain comprehensive information on vibrational modes of fullertubes C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d , and then use projection technique to establish genetic relationships between vibrations of C 100 -D 5d , (5, 5) CNT and C 60 -I h .

Experimental and computational details
The synthesis and characterization of fullertubes C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d were described in ref. 10.In brief, the fullerene-containing soot produced by arc-discharge synthesis was extracted with xylene.The fullerene mixture was then reacted with 3-amino-1-propanol.While most fullerenes react readily with 3-amino-1-propanol and form products soluble in water, fullertubes are much less reactive in these conditions and remain in the organic phase.Individual compounds were then isolated by HPLC.
For vibrational spectroscopic studies, the samples were drop-casted from toluene solution onto KBr single-crystalline substrates and dried under vacuum.IR measurements were performed at room temperature in transmission mode using a Hyperion FTIR microscope attached to Vertex 80 spectrometer (Bruker).Raman measurements were performed with T64000 spectrometer (Horiba).The samples on KBr substrates were cooled down to 78 K, and the spectra were measured with laser excitation at 532 nm (Nd:YAG laser Torus by Laser Quantum), 620 nm and 656 nm (Matisse dye laser by Sirah Lasertechnik).The spectra were also excited with 785 nm laser (BrixX diode laser by Omicron Laserage).For the latter, the samples were drop-casted on gold SERS substrates (Metrohm DropSens DRP-C220BT) and immersed in water to improve the heat transfer, and the measurements were performed at room temperature using immersion objective.
DFT calculations of fullerenes molecules were performed with PBE density functional 20 using two DFT codes.Vibrational frequencies, IR and off-resonance Raman intensities were calculated with molecular code Priroda 21,22   basis set 23 with {4,3,2,1}/{12s,8p,4d,2f} contraction scheme.Vibrational calculations were also performed with periodic code VASP 5.4.4, 24-26using recommended pseudopotentials and energy cut-offs for projector-augmented wave (PAW) scheme and 6 Å of vacuum layer to prevent interaction between periodic images.Importantly, calculations with molecular and periodic codes gave very similar vibrational frequencies, ensuring the use of balanced wavefunction description in both codes.In calculations of (5, 5) CNT, the Γ-centered sampling of the Brillouin zone along the periodic axis used 4 Monkhorst-Pack grid points per unit cell.With an accurate grid option, the tube was optimized to a mean gradient of 10 −5 eV Å −1 .The Hessian matrix was calculated using density-functional-perturbation theory for a supercell of four primitive unit cells.Phonopy libraries 27 and in-house python scripts were used to analyze phonon spectra and dynamic matrix operation/analysis at different q-points.Vibrational symmetry analysis was performed using DISP/SYMM package. 28

Results and discussion
Fig. 2 shows UV-vis absorption spectra of C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d measured in toluene solution.Although (5, 5)  and (9, 0) CNTs are metallic, the CNT fragments in fullertubes are obviously not long enough to develop CNT-like electronic properties and close the band gap. 14Thus, all studied fullertubes have considerable gaps exceeding at least 1.5-2 eV.The spectra closely resemble those reported in the literature 5,10,11,15 thus confirming the structural identity of studied molecules.

Experimental vibrational spectra
Vibrational modes of fullertubes span the following irreducible representations of their point-symmetry groups: ð1:bÞ ð1:cÞ where R and IR in parentheses denote Raman and IR activity, respectively.Despite rather high molecular symmetries, a large number of optically active modes are expected for all three compounds.Vibrational density of states of these molecules is thus quasi-continuous, and reliable vibrational assignment based only on computed frequencies and symmetry analysis is impossible.Therefore, we have to consider not only frequencies, but also computed IR and Raman intensities.
Experimental IR spectra of fullertubes are compared to the computed ones in Fig. 3. To benchmark the computational method, Fig. 3 also shows the spectra of well-known fullerene C 70 -D 5h .Based on the latter, we can conclude that the computational method gives a very good match of experimental vibrational frequencies.The difference between experiment and theory is usually less than 5 cm −1 and increases up to  10 cm −1 only at the highest frequencies.At the same time, computed IR intensities are less satisfactory.Relative intensities of the bands deviate significantly from experimental counterparts even for vibrations of the same type.Besides, theory tends to strongly overestimate the relative intensity of tangential modes at frequencies above 1300 cm −1 .These caveats notwithstanding, computations provide a reasonable guide for the interpretation of the IR spectra of fullertubes.The full list of experimental and computed frequencies for C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d can be found in ESI (Tables S1-S3 †).Table 1 lists the assignment of the IR spectrum of C 100 -D 5d .The spectrum of C 90 -D 5h obtained this work closely resembles the data reported in a recent study of vibrational spectra of C 90-D 5h under high pressure. 29aman spectra of C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d are shown in Fig. 4, 5, and 6, respectively.To obtain the most detailed information, the spectra were excited with several laser lines.From Fig. 2, comparing laser wavelengths in relation to the absorption spectra of fullerenes, we find that the 785 nm laser should produce a non-resonant Raman scattering, whereas other laser lines appear close to or overlap with the fullerene absorptions and may therefore induce resonant effects.Indeed, the computed spectra agree well with the experimental spectra recorded with 785 nm excitation.At the same time, the spectra measured with shorter laser wavelengths demonstrate considerable redistribution of intensity pointing to their preresonance character.A particular strong difference from the non-resonance scattering is found for the green laser (532 nm), which strongly enhances the intensity of the highfrequency tangential modes.Overall, the spectral information obtained with a combined set of spectra is very rich with 60-70 peaks detected for each compound.The assignment of the spectra is facilitated by a good agreement between experiment and theory.Using computed intensities as a guide, we could first identify the peaks in the off-resonance spectra with a high degree of certainty and then continue with the remaining peaks detected only in resonant conditions.The procedure allowed determination of almost all fully symmetric modes (A 1 ′ or A 1g types), which as a rule are more prominent in the spectra, as well as a large part of E-type modes (E 1 ″/E 2 ′, E g , or E 1g /E 2g types, see eqn (1)).1][32][33][34] A complete list of experimental Raman features and computed frequencies are given in ESI, † whereas Table 2 lists all A 1g and selected E 1g and E 2g modes of C 100 -D 5d .

Projection analysis and symmetry considerations
Having established the correspondence between experimental spectral features and computed normal modes, we can proceed to the main goal of this work, the analysis of the fullertube vibrations in terms of their fullerene and CNT fragments.For the sake of convenience, the analysis will be performed for C 100 -D 5d as its belt region comprises two CNT periods, while two caps build up fullerene C 60 -I h if the belt is removed (Fig. 1).Establishing a resemblance of vibrations of different molecules is straightforward with projection analysis used earlier by some of us for fullerene derivatives [35][36][37] and endohedral fullerenes. 38The method is based on the fact that normal modes in mass-weighed Cartesian coordinates form an orthonormal basis, and vibrations of one molecule (C 100 -D 5d in this work) can be projected onto the space of vibrational eigenvectors of another molecule (C 60 -I h or CNT) using scalar products: where Q C 100 ,i is ith vibrational mode of the fullertube C 100 -D 5d , Q frag,i is the jth mode of the fragment of interest (C 60 -I h or CNT), and Q C 100 ,i vector includes only a subset of atomic displacements corresponding to this fragment.Consequently, the square a frag,ij 2 gives the contributions of the jth fragment mode to the ith mode of C 100 -D 5d , and the sum of a frag,ij 2 over all fragment modes amounts to the contribution of a given fragment to the ith vibrational mode of the whole C 100 -D 5d molecule: In fact, d frag,i can be also obtained from vibrational eigenvectors in cartesian coordinates without the projection, and we will use that for C 90 -D 5h and C 96 -D 3d .It is also important to note that in case of a uniform distribution of a given vibration over the whole fullertube molecule, the CNT-like belt will contribute 33.3% in C 90 -D 5h , 18.8% in C 96 -D 3d , and 40% in C 100 -D 5d .A predominant localization of the vibration on the belt region can be concluded when d CNT considerably exceeds these values.
To summarize, using DFT-computed vibrational eigenvectors, we can establish exact correspondence between the vibrations of C 100 -D 5d caps and the modes of C 60 -I h , as well as between vibrations of the C 100 -D 5d belt and the modes of (5, 5) CNT.When one particular fragment mode has dominant contribution (a frag,ij 2 is close to 1 or at least to d frag,i ), there is a close resemblance of the corresponding vibrations of C 100 -D 5d and C 60 -I h or CNT.When two or more fragment modes have comparable contributions, it is said that these modes are mixing.The full list of DFT-computed vibrational frequencies of C 60 -I h and (5, 5) CNT can be found in ESI, Tables S4 and  S5.† For C 60 -I h , Table S4 † also compares the frequencies with complete set of fundamentals, including silent modes, determined in ref. 39.It is also useful to establish a between symmetry types of the vibrations as the symmetry restricts the optical activity and a possibility of the mode mixing.Since D 5d is a subgroup of I h , the connection between C 60 -I h modes and vibrations of C 100 -D 5d caps is straightforward (Table 3).All gerade (g-type) modes of C 60 -I h become Raman active in the D 5d group as they include either A 1g , E 1g , or E 2g representations.However, only Raman-active A g and H g modes of C 60 -I h have A 1g component in the D 5d group, and as mentioned above, the vast majority of strong Raman modes of C 100 -D 5d are of A 1g symmetry.As for ungerade modes, C 60 -I h has only 4 IR-active vibration of F 1u type, but all its degenerate u-type modes become IR active in the D 5d symmetry as they have either A 2u or E 1u representations.Only A u mode of C 60 -I h remains silent.
The situation with the CNT is more complex and requires a deeper discussion.(5, 5) CNT is described by the T 1 10 D 5h linear group, which is isogonal to the D 10h point symmetry group. 40In the following, we will mainly use the irreducible representations of the D 10h group to label vibrational modes of the (5, 5) CNT.As with any other armchair CNT, it has 8 Raman active modes (2A 1g + 2E 1g + 4E 2g ) and 5 IR-active modes (A 2u + 4E 1u , of which A 2u and one E 1u are acoustic modes with zero frequency in Γ-point). 41In C 100 , the symmetry is reduced to D 5d .Irreducible representations for the CNT in T 1 10 D 5h , D 10h , and D 5d groups are compared in Table 4. Furthermore, the CNT-like belt of C 100 -D 5d has 40 atoms, in which 20-atomic unit cell of the (5, 5) CNT is repeated twice.For this reason, it is insufficient to consider only Γ-point modes of the CNT when comparing its vibrations to those of C 100 -D 5d .Vibrations of the double-cell fragment can be formally described by two sets of the unit cell modes, one with both unit cells vibrating in phase and corresponding to Γpoint, and one with unit cells vibrating in anti-phase to each other and thus corresponding to X-point on the edge of the Brillouin zone.Fig. 7a and b shows DFT-computed dispersions of CNT phonons and variation of the composition of selected modes with k obtained by projecting k-point eigenvectors on Γpoint eigenvectors.Fig. 7 shows that both frequencies and composition of the modes can change considerably in k-space along going from Γ to X.For instance, whereas B 2u (1) and A 1g (2) modes tend to retain their shape at k ≠ 0, A 2u (1) mode is Computed Raman intensity is for off-resonance conditions.d frag and mode compositions are given in %, contributions of less than 8-9% are omitted.Experimental intensity scale: vw < w < w + < m < s < vs, where wweak, mmedium, sstrong, vvery, shshoulder.Furthermore, in the X-point, the modes become degenerate and therefore exhibit an additional mixing.The X-to-Γ projection matrix plotted in Fig. 7c has substantially non-diagonal pattern.
From this analysis we can infer that each CNT vibrational mode will contribute to vibrations of two symmetry types in C 100 -D 5d , one of the same type (in D 5d symmetry) and one of its conjugate of the opposite parity reducing to the same irreducible representation in the C 5v subgroup.This leads to the situation when, for instance, A 1g mode of C 100 -D 5d may have contributions from A 1g and A 2u CNT modes considered in D 5d symmetry, or more precisely from A 1g , B 1g , A 2u , and B 2u modes of the CNT in its rigorous D 10h symmetry (Table 4).

Normal mode analysis for C 100 -D 5d
For a hypothetical situation in which all fullertube vibrations are exclusively localized either on the caps or the belt with each fullertube mode exhibiting one-to-one resemblance to a certain vibrational mode of C 60 or CNT, the projection matrices {a frag,ij 2 } would be quasi-diagonal with a handful of a frag,ij 2 elements equal 1 and most of others being 0.
Visualization of {a C 60 ,ij 2 } and {a CNT,ij 2 } projection matrices for C 100 in Fig. 8 shows that it is not the case (numerical values of d frag,i and leading a frag,ij 2 terms can be found in Tables 1, 2 and Table S3 †).showing that a certain similarity of vibrations is indeed present.This resemblance is more evident for vibrations of C 60 -like caps (Fig. 8a), whereas the belt vibrations have a more pronounced mixed character (Fig. 8b).The latter can be explained by the symmetry factor (there are more symmetry types of CNT modes which can mix in C 100 -D 5d ) as well as the a R and IR in parentheses denote Raman and IR-active modes b Note that E 1u and A 2u are acoustic modes with zero frequency in Γ-point, leaving 3E 1u optical IR-active mode in armchair CNTs; one of the A 2g modes corresponds to the rotation of CNT around its axis and also has zero frequency in Γ-point.We also tried perform projection analysis using CNT modes computed at X-point, but this resulted in a much more pronounced mixing, so we will stay with Γ-point modes of the (5, 5) CNT in the following discussion.With all these caveats, the analysis of C 100 -D 5d mode composition in terms of C 60 and CNT fragments is still quite instructive.In the IR spectrum, all strong bands have predominant contributions from C 100 caps and can be traced back to IR-active F 1u modes of C 60 .For instance, the strong peak at 1430 cm −1 is assigned to the E 1u ( 26) mode of C 100 with 51% contribution from F 1u (4) (1429 cm −1 in C 60 ).Another very strong IR peak of C 100 at 548 cm −1 corresponds to the E 1u (8)  mode with 65% of F 1u (2) (576 cm −1 in C 60 ).Finally, the peak at 480 cm −1 can be assigned to two C 100 modes with close frequencies and large weights of F 1u (1) (526 cm −1 ) and F 1u (2) C 60 modes.Vibrations of the CNT-like belt have lower IR intensities.The only range where they have some prominence is around 600-850 cm −1 , where several weak absorptions can be seen.Particularly, a weak band at 818 cm −1 is caused by the C 100 vibration resembling the E 1u (3) CNT mode, which in due turn originates from the IR-active phonon of graphene detectable in graphite at 868 cm −1 .Further details of IR spectral assignment can be found in Table 1.
In Raman spectra, C 60 -like and CNT-like vibrations are represented more uniformly as can be deduced from the data in Table 2 and Fig. 9.The majority of prominent Raman features in the spectra of C 100 are assigned to vibrations of A 1g symmetry type (see Table 2), and according to the symmetry analysis (Table 3), the A 1g modes with predominant localization on caps can be traced to A g or H g modes of C 60 .For instance, the Raman feature of C 100 at 243 cm −1 is related to the H g (1) mode of C 60 at 272 cm −1 , the peak at 427 cm −1 corresponds to the H g (2) mode (432 cm −1 ), the strong peak at 1232 cm −1 is related to the H g (6) mode (1248 cm −1 ), whereas the peak at 1575 cm −1 corresponds to the H g (8) mode (1574 cm −1 in C 60 ).
Considerable components of two A g modes of C 60 are found in 4 modes of C 100 .The strongest peak in the non-resonant spectrum of C 100 at 379 cm −1 is assigned to A 1g (3), which has 26% belt and 74% cap contributions, including 43% A g (1) mode of C 60 (495 cm −1 ).The latter is known as the fullerene breathing mode.Interestingly, the CNT part in this vibration is mainly represented by the radial breathing mode (RBM), A 1g (1), detected in the experimental spectrum of (5, 5) CNT at 338 cm −1 (ref.42; our DFT-computed frequency for RBM is 332 cm −1 ).Note that the study of fullerene vibrations based on the isotropic spherical shell model demonstrated that the breathing mode frequency scales as M −1/2 , where M is the fullerene mass, and this scaling works very well even for tubular fullerenes. 43The shift of the breathing mode frequency from 495 cm −1 in C 60 to 379 cm −1 in C 100 -D 5d agrees well with this scaling.The breathing mode of C 60 also contributes to the A 1g (6) mode of C 100 at 599 cm −1 , in which it is mixed with H g (4) (772 cm −1 ).
The second totally symmetric vibration of C 60 , A g (2) (1468 cm −1 ) known as the pentagonal pinch mode, mixes with H g (7) (1422 cm −1 ) in the A 1g (14) mode of C 100 .In the experimental Raman spectrum it is detected as the medium-intensity feature at 1447 cm −1 .Another C 100 vibration with the large  weight of the pentagonal pinch mode is 1g (15) predicted at 1493 cm −1 .Here the cap and the belt contribute equally, each represented by a totally symmetric mode, A g (2) for C 60 and A 1g (2) for CNT.However, this vibration has low predicted Raman intensity, and we cannot assign it with an acceptable degree of certaintythe only plausible experimental signal in the corresponding frequency range is a very weak and broad feature at 1482 cm −1 .
On the CNT side, several totally symmetric modes of C 100 -D 5d with considerable Raman activity can be pointed out.The vibration resembling the CNT RBM mode is found at 325 cm −1 , not far from the actual RBM mode frequency in the (5, 5) CNT at 338 cm −1 (ref.42).The relative intensity of this vibration in C 100 is rather low in pre-resonant conditions, when the spectra are excited with 620 nm or 532 nm lasers.Quite interesting is the A 1g (5) mode of C 100 at 551 cm −1 , which resembles the radial B 2u (1) CNT mode predicted at 625 cm −1 .Its intensity is strongly enhanced under 620 nm excitation so that it becomes the strongest feature in the spectrum.A 1g (10)  at 1195 cm −1 is another medium-intensity Raman feature of C 100 largely originating from ungerade CNT vibration, this time B 2u (2) predicted at 1319 cm −1 .This CNT mode, mixed with B 1g (1) ( predicted at 1377 cm −1 ), also contributes to the second strongest non-resonant Raman feature of C 100 , A 1g (12)  at 1301 cm −1 .In fact, the latter can be described as the Kekule vibration of the belt hexagons.Note that Kekule vibrations are usually rather strong in Raman spectra of aromatic compounds.
Only a handful of E 1g and E 2g modes of C 100 have considerable Raman intensity.Those of them with enhanced cap contribution can be traced back to H g modes of C 60 , similar to A 1g vibrations discussed above (Table 2).A specific example of the belt-localized mode worth highlighting here is the lowest-frequency E 2g (1) vibration at 144 cm −1 with the strong non-resonance Raman intensity.It shows considerable temperature dependence and is shifted to 154 cm −1 at 78 K.This mode is traced back to the squashing CNT vibration, also of E 2g (1) type, predicted at 69 cm −1 in the (5, 5) CNT.Note that in the aforementioned treatment of fullerene vibrations as that of an elastic sphere, 43,44 this vibration can be identified as the component of the 5-fold degenerate quadrupolar mode.Other components are cap-based vibrations at 220-245 cm −1 with a large weight of the H g (1) C 60 mode.Ref. 44 showed that the average frequency of the quadrupolar mode also scales with the fullerene mass roughly as M −1/2 , whereas the splitting degree in non-spherical fullerenes was found to be a function of the cage form, being the largest for elongated molecular shapes.
Another CNT-like vibration of C 100 of E-type, E 1g (28) at 1508 cm −1 , corresponds to the E 1g (2) mode of the CNT with longitudinal oscillations of carbon atoms along the tube axis.This is one of the CNT modes, whose origin comes from the Raman-active phonon of graphene at 1582 cm −1 known as the G-band, but it is not active in the Raman spectrum of the (5, 5)  CNT. 42t is quite remarkable that the Raman spectra of C 100 have several pronounced features of CNT-like vibrations.Vibrational studies of CNTs usually rely on strong resonance enhancement in Raman spectra when the laser matches transition between van-Hove singularities.In such resonance spectra, only two single-phonon bands are observed, the lower-frequency RBM mode and the high-energy G-band, which in CNTs includes several modes originated from the graphene optical phonon at 1582 cm −1 .In armchair (n, n) CNTs, three modes of this type are Raman active, E 2g (4) + A 1g (2) + E 1g (2). 416][47][48][49] In the (5, 5) CNT, such G + feature was detected at 1573 cm −1 , whereas the RBM mode is found at 338 cm −1 (ref.42).While the resonance enhancement is extremely useful for CNTs as it enables the measurement of tiny sample amounts, down to a single nanotube, it also imposes a limitation on the information obtained from such spectra as it only attests to enhanced modes.Other IR and Raman active vibration usually remain obscure, and even though some studies focusing on such vibration were reported, 17,50-53 their assignment is rather unspecific because bulk samples of individual CNTs were not available.
The situation is quite different for fullertubes.As they have no van-Hove singularities and therefore specific resonance conditions typical for CNTs are absent, there is no physical reason for the belt vibrations resembling A 1g modes of CNT to show enhanced Raman intensity.Indeed, the RBM-like vibration of C 100 has considerable relative intensity only in non-resonance conditions and becomes weaker when the spectra are excited at 656, 620, or 532 nm (Fig. 6), while the fullertube vibration derived from the G + -A 1g (2) CNT mode is so weak that we cannot reliably identify it in our spectra.It may certainly be that these vibrations show higher intensity at shorter excitation wavelengths, as the (5, 5) CNT has its E 11 M transition at 412 nm, 48 but such a short wavelength laser is not available to us at this moment.On the other hand, we have found several other belt-localized vibrations of C 100 , which can be traced to CNT vibrations that cannot be observed in the spectra of CNT itself.

Vibrational spectra of C 90 -D 5h and C 96 -D 3d
As C 90 -D 5h also has the belt derived from the (5, 5) CNT and two C 60 -like caps, albeit rotated at a different angle than in C 100 -D 5d , vibrational features of C 90 -D 5h and C 100 -D 5d are rather similar.The strongest IR bands in C 90 -D 5h are found near 500-600 cm −1 , as in many other fullerenes.Among the CNTlike vibrations, the most prominent is the band at 815 cm −1 with rather high IR intensity.It has 72% belt contribution and can be assigned to the IR-active phonon of graphene (868 cm −1 in graphite).Similar vibration in C 100 -D 5d is found at 818 cm −1 , but with a much lower intensity.In Raman spectra, the prominent peaks with strong CNT contribution are found at 165 cm −1 (CNT squashing mode), 338 cm −1 (RBM), 1216 cm −1 and 1329 cm −1 (Kekule vibration).
In C 96 -D 3d , the belt has only one unit cell of the (9, 0) CNT with 18 carbon atoms, and thus the contribution of the belt to vibrations is on average rather low.Yet, several Raman modes with enhanced contribution still can be pointed out.The squashing mode with 43% of the belt involvement is found at 158 cm −1 .Vibration with a pronounced RBM character occurs at 353 cm −1 .We are not aware of vibrational spectroscopic studies of isolated (9, 0) CNTs, but when identified as internal tubes in double-walled and triple-walled CNTs, their RBM mode was found at 324-325 cm −1 (ref.54 and 55).The strongest non-resonance Raman line at 1327 cm −1 also has noteworthy belt contribution of 40%.Finally, the highest-frequency vibration at 1599 cm −1 may have a considerable belt contribution resembling A 1g (2) mode with longitudinal displacement of carbon atoms.The assignment is however rather ambiguous since another vibration of C 96 -D 3d with only 2% of the belt has a similar DFT-predicted frequency.In the IR spectrum, the strong belt participation is found for the mode at 829 cm −1 .The rest of IR bands are caused by fullerene-like vibrations.In Raman spectra, the prominent fullerenes modes are the squashing modes at 229 and 243 cm −1 as well as the radial breathing mode at 392 cm −1 .Assignment of other lines is less straightforward since the caps are not resembling of C 60 shape, and we cannot use projection analysis here.
Variation of the low-frequency Raman modes in the C 90 -C 96 -C 100 series can be also rationalized with the elastic shell model 43,44 already discussed above in application to C 100 .The radial breathing mode frequency in the series decreases systematically from 397 cm −1 (C 90 -D 5h ) to 392 cm −1 (C 96 -D 3d ) to 379 cm −1 (C 100 -D 5d ) following the mass increase.The quadrupolar mode in these fullerenes is split into 3 components, two two-fold degenerate and one totally symmetric with the highest frequency: 165/234/257 cm −1 (C 90 -D 5h ), 158/229/ 243 cm −1 (C 96 -D 3d ), and 144/223/243 cm −1 (C 100 -D 5d ).The averaged frequency again decreases gradually with fullerene mass from 211 cm −1 in C 90 -D 5h to 203 cm −1 in C 96 -D 3d to 195 cm −1 in C 100 -D 5d , but the degree of splitting has a different order: the largest one of 99 cm −1 is found in C 100 -D 5d , followed by 92 cm −1 in C 90 -D 5h and 85 cm −1 in C 96 -D 3d .This sequence is in line with the shape variation among the three, and goes from the most elongated C 100 -D 5d to the least tubular C 96 -D 3d .

Conclusions
Detailed IR and Raman study of tubular fullerenes augmented with computational analysis allowed to address the problem of the resemblance between vibrational modes of fullertubes, nanotubes, and fullerenes.Using C 100 -D 5d (1) for a case study, we employed projection analysis to establish precise numerical correspondence between vibrations of fullertube belt and caps and normal modes of (5, 5) CNT and fullerene C 60 .This analysis showed that IR spectra are dominated by vibrations of the caps, which can be traced back to IR-active F 1u modes of C 60 .The only belt vibrations with some, although still rather low IR intensity are found near 820 cm −1 and can be traced back to CNT modes related to the IR-active phonon of graphene.In the Raman spectra, C 60 -like and CNT-like vibrations are presented more uniformly.For caps-localized modes, we again observed a high degree of transferability from C 60 as the most intense Raman features could be associated with Raman-active A g and H g modes of C 60 .Vibrations with significant belt contributions showed more unusual pattern.Raman spectra of CNTs have strong resonance character and are dominated by two single-phonon bands, radial breathing mode and components of the G-mode.But in the Raman spectra of fullertubes, these vibrations have modest intensity.We could identify RBMderived vibrations close to their position in CNTs, but the modes originating from the G-type CNT modes are hard to identify in fullertubes.On the other hand, we found several intense Raman features corresponding to CNT vibrations, which are not observed in the spectra of CNTs.This finding shows that fullertubes may provide additional information on CNTs, which is hard to obtain otherwise.

Fig. 1
Fig. 1 Molecular structures of fullertubes C 90 -D 5h , C 100 -D 5d , and C 96 -D 3d as well as the fullerene C 60 -I h and (5, 5) CNT.In each fullertube, fullerene-like caps are highlighted in pink with shaded pentagons, whereas CNT-like belts are colored blue.In the (5, 5) CNT fragment, a unit cell of 20 carbon atoms is also highlighted in blue.

Fig. 3
Fig. 3 Experimental (dark blue) and calculated ( pink, PBE/Λ2 level) IR spectra of C 70 -D 5h , C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d .The peak of the most intense band in the calculated spectrum of C 70 at 1430 cm −1 is cut for a better presentation of other parts of the spectrum.

Fig. 2
Fig. 2 UV-vis absorption spectra of C 90 -D 5h , C 96 -D 3d , and C 100 -D 5d in toluene solution.Vertical dashed lines mark laser wavelengths used in Raman measurements.

Fig. 4
Fig. 4 Experimental Raman spectra of C 90 -D 5h compared to the calculations for off-resonance conditions (dark blue curve, PBE/Λ2 level).

Fig. 5
Fig. 5 Experimental Raman spectra of C 96 -D 3d compared to the calculations for off-resonance conditions (dark blue curve, PBE/Λ2 level).

Fig. 6
Fig. 6 Experimental Raman spectra of C 100 -D 5d compared to the calculations for off-resonance scattering (dark blue curve, PBE/Λ2 level).
First, C 100 -D 5d modes rarely localize on either the belt or the caps, and more often are distributed over the whole fullertube.This obviously results in a frag,ij 2 values being considerably smaller than 1.Second, C 100 -D 5d modes often have several contributing fragment modes, which in other words means that fragment modes mix in C 100 .Nevertheless, {a frag,ij 2 } matrices are still rather sparse, and there is a clear tendency of the largest a frag,ij 2 values to cluster near diagonal,

Fig. 7
Fig. 7 (a) Phonon dispersion of (5, 5) CNT; the branches originating in Γ-point from A 1g , B 1g , A 2u , and B 2u modes (corresponding to A 1g and A 2u in D 5d group) are shown in thick color.(b) Changes in the composition of these modes on going from Γ to X point in k-space; colored circles visualize contributions of the Γ-point modes, with their radius scaling as the contribution of a given Γ-point mode to the mode at a given k-value.(c) Projection matrix of X-point modes onto Γ-point modes (a ij

Fig. 8
Fig. 8 Visualization of projection matrices (a frag,ij 2 coefficients) for C 100 -D 5d modes onto the spaces of (a) modes of C 60 -I h , and (b) modes of (5, 5) CNT in Γ-point.In (b), horizontal lines denote A 1g modes of CNT.The modes are numbered in the order of increasing frequency, CNT modes also include acoustic modes.Degenerate modes are counted once, giving 176 modes for C 100 -D 5d , 46 modes for C 60 -I h , and 36 modes for (5, 5) CNT.

Fig. 9
Fig. 9 Comparison of the Raman spectra of C 100 -D 5d (λ ex = 785 nm) and C 60 -I h (λ ex = 1064 nm) and the correspondence of the most intense C 100 Raman lines to the vibrational modes of C 60 and (5, 5) CNT.Large contributions of A g and H g modes of C 60 -I h to C 100 -D 5d vibrations are denoted with gray dashed lines, C 100 -D 5d vibrations with large contribution of CNT modes are marked with cyan arrows and labels of leading CNT vibrations.

Table 2
All A 1g and selected E 1g and E 2g Raman-active vibrational modes of C 100 -D 5d with their computed frequencies, intensities, experimental assignment, and description in terms of(5, 5)CNT and C 60 -I h modes

Table 3
Correlation between symmetry types of vibrational modes in C 60 -I h and C 100 -D 5d a same A 1 type and can mix.Similarly, mixing occurs in A 2g /A 1u , E 1g /E 1u , and E 2g /E 2u pairs as they reduce to A 2 , E 1 , and E 2 symmetry types of the C 5v group, respectively.
a R and IR in parentheses denote Raman and IR-active modes.mixingwithA1g(1),and B (1) is mixing with B 1g(1).When k ≠ 0, the point symmetry is effectively lowered to C 5v , in which A 1g and A 2u irreducible representations of the D 5d group reduce to the

Table 4
Correlation between symmetry types of vibrational modes in(5, 5)CNT and C 100 -D 5d