Assignment of the Internal Vibrational Modes of C70 by Inelastic Neutron Scattering Spectroscopy and Periodic-DFT

The fullerene C70 may be considered as the shortest possible nanotube capped by a hemisphere of C60 at each end. Vibrational spectroscopy is a key tool in characterising fullerenes, and C70 has been studied several times and spectral assignments proposed. Unfortunately, many of the modes are either forbidden or have very low infrared or Raman intensity, even if allowed. Inelastic neutron scattering (INS) spectroscopy is not subject to selection rules, and all the modes are allowed. We have obtained a new INS spectrum from a large sample recorded at the highest resolution available. An advantage of INS spectroscopy is that it is straightforward to calculate the spectral intensity from a model. We demonstrate that all previous assignments are incorrect in at least some respects and propose a new assignment based on periodic density functional theory (DFT) that successfully reproduces the INS, infrared, and Raman spectra.


Introduction
Fullerenes cience may be said to have been born in 1990, when the route to macroscopic quantities of the materials was invented. [1] This prompted an intensea nd continuinge ffort, both experimental and theoretical, to understanda nd exploit the properties of these novel forms of carbon. Areas of interest span physics, [2] chemistry, [3] biology, [4] and astronomy. [5,6] Most of the activity has focussed on C 60 because of the iconic nature of the icosohedral symmetrya nd fort he practical reason that it is the most readily available of the fullerenes.
The second most abundantf ullerene is C 70 (Figure 1). This may be considered as the shortestp ossible nanotube capped by ah emisphere of C 60 at each end. The additional ten carbon atoms have ap rofound effect on the molecule. The idealised (gas-phase) symmetryi sr educed to D 5h from I h .T his means that there are now eight distinct types of CÀCb ond rather than the two found in C 60 .T he number of internal vibrational modes increases to 204 from 174. These are classified as: Thus, there are many more modes in C 70 than C 60 :122 vs. 46.
The reduction in symmetry does have one advantage-in C 70 , 84 modes (31 infrared and 53 Raman) are potentially observable by infrared andR aman spectroscopies, whereas only 14 (4 infrared and 10 Raman) are allowed in C 60 .
In the solid state, the consequences of the additional carbon atoms are equally striking. From the sublimation temperaturet o2 60 K, C 60 undergoes rapid rotation and the structure can be described as af ace-centred cubic (fcc) lattice of spheres,s pace group Fm3 mw ith Z = 1i nt he primitive cell. [7] Below 260 K, an orientationally ordered simple cubic phase is obtained, space group Pa3 with Z = 4i nt he primitive cell. [8] In contrast, at room temperature, the phase obtained with C 70 may be fcc or hexagonal close packed (hcp);these may partially order as the rotation about the short axis and then the long axis is quenched on cooling.T he low temperature phase has been proposed to be orthorhombic, Pbnm, [9] with Z = 4a nd also as monoclinic. [10] Whether the room temperature phase is fcc or hcp, whether there are one or two rotator phases,a nd whethert he low temperature phase is orthorhombic or monoclinic depends ensitively on sample history,c oolingr ate, and purity,p articularly the amount of C 60 present. The complex crystallography of C 70 is well summarised elsewhere. [11] The assignment of the internal modes of C 70 is largely based on solid-state spectra;t he only gas-phase data [12] identified eight infrared activef undamentals. The infrared and Raman spectra have been measureds everalt imes; [13][14][15][16][17][18][19][20] as with C 60 it was realised that all the modes are allowed in inelastic neutron The fullerene C 70 may be considered as the shortest possible nanotube capped by ah emisphere of C 60 at each end. Vibrational spectroscopy is ak ey tool in characterisingf ullerenes, and C 70 has been studied severalt imes and spectral assignments proposed. Unfortunately,m any of the modes are either forbiddeno rh ave very low infrared or Ramani ntensity,e ven if allowed.I nelastic neutrons cattering (INS) spectroscopy is not subjectt os election rules, and all the modes are allowed. We have obtained an ew INS spectrumf rom al arge sample recorded at the highest resolution available. An advantage of INS spectroscopy is that it is straightforward to calculate the spectrali ntensity from am odel.W ed emonstrate that all previous assignments are incorrect in at least some respects and propose an ew assignment based on periodic density functional theory (DFT) that successfully reproduces the INS, infrared, and Ramanspectra. scattering (INS) spectroscopy, [21] andalow-qualitys pectrum has been reported. [22] Most [19,[23][24][25] computational studies assume D 5h symmetry,t hat is, they use the isolated molecule approximation. One study [26] employed Car-Parrinello molecular dynamics so as to include solid-state effects. Ac ritical test of the reliability of ac alculation is how well it reproduces the INS spectrum. We have previously adopted this approach to completely assign the vibrational spectrum of C 60 andh ave shown that all previous assignmentswere incorrect in some respect. [27] Twop apers using different computational methods have done so for C 70 ;b oth report good agreement. [23,25] However,t he quality of the spectrum, particularly in the 800-1600 cm À1 region is insufficientt om ake this au seful comparison. In this work, we report an ew INS spectrum obtained from al arge sample (9 g, about 18 times largert han that previously used) recorded at the highest resolution available, combined with ap eriodic density functional theory (DFT) calculation that includes solid-state effects. To gether,t hesea llow as tringent test of the published assignments and an ew assignment of all of the internal modeso fC 70 .

Results and Discussion
As described in the Introduction,t here is no agreement as to the exact form of the low-temperature structure of C 70 ,w ith both orthorhombic and monoclinic structures proposed. [9][10][11] Since fivefold symmetry is incompatiblew ith long-range order, [28] it follows that the site symmetry must be simultaneously as ubgroup of D 5h and also ap ermitted site symmetry of Pbnm or am onoclinic space group. Only C s , C 2 ,a nd C 1 meet these constraints. Spectroscopically,a ll three are equivalent:a ll degeneracies are lifted and all modes are allowed in both the infrared and Ramans pectra.( The three possibilities are only distinguishable by single crystal studies with polarised radiation.) However,t he intensitiesw ill varyc onsiderably,a nd modes that are forbidden under D 5h symmetry are likely to be weak in the solid-state spectra and may be confused with combination modes, as found with C 60 [29,30] All the modes are allowed in INS spectroscopya nd all will have similarintensities, since all the atoms in the mode have the same scattering cross section and will have similara mplitudes of vibration. Figure 2 shows the INS spectra recorded with the TOSCA and MAPS instruments (ISIS Facility,R utherford Appleton Laboratory, Chilton, UK). Figure 3a showst he Raman spectrum at 20 Kr ecorded simultaneously with the TOSCA spectrum,w hile Figure 3b shows the infrared spectrum at 113K recorded by attenuated total reflectance (ATR). In comparison to previousw ork, [22] the INS spectra exhibit as ignal-to-noise ratio more than an order of magnitude better and with significantly improved resolution, particularly in the 800-1600 cm À1 region.W hile the infrared and Raman spectra show many modes across the range 200-3000 cm À1 ,t here is as harp cut-off at 1600cm À1 in the INS spectra,d emonstrating that all modes to highere nergym ust be overtones or combinations.
The need for an ew assignment is demonstrated in Figure 4, which compares our new INS spectrum ( Figure 4a)w ith the INS spectrap redicted by literature results assuming an isolated (gas phase) molecule [19] (Figure 4b)and by an ab initio molecular dynamics simulation [26] (Figure 4c). It can be seen that while the overall profile is approximately correct, both simulated spectra are wrong in detail, particularly in the 450-800 cm À1 region. For the presentw ork, we have chosen to model the system with ab initio lattice dynamics (as implemented in the DFT code,CASTEP), that wassuccessfully used for C 60 . [27] The only structure that has the atomic coordinates available is the Pbnm [9] structure with Z = 4a nd C s site symmetry,w here the horizontal mirror plane of the D 5h point group is coincident with the crystallographicm irror plane of Pbnm.W ec hose to simplify the problem by removing three molecules from the unit cell. This reduces the space-groups ymmetry to Pm but retains the point-group symmetry reductiont hat must be present in the solids tate.
Ta ble 1c ompares the available structurald ata [9,31,32] with the results of the CASTEPc alculation for the Pm structure and that of the idealised D 5h gas phase molecule calculated using GAUS-SIAN 03. For ease of comparison with other work, we have  www.chemistryopen.org adopted the same definitions of the bonds as used previously (see Figure 1). [9] It can be seen that both calculations are in good agreement with the experimental data and with each other;t he differencesr arely exceed AE 0.1 . The only discrepancy is with the experimental gas phase value for bond 8; however, in comparison to the other distances in the molecule, this appears anomalous. We notet hat the longest experimental distance in C 60 is 1.487 . [8] Ac omparison of the TOSCA INS spectrum with that calculated for the gas phase and the solid state is showni nF igure 5. It can be seen from the comparisons of the observed and calculated spectra that the present resultsp rovide ab etter description of the vibrational modes of C 70 than all previous computational studies. [19,[23][24][25] In most cases, these have imposed D 5h point-group symmetry on the calculation, that is, ag as-phase calculation, hence mode assignment is straightforward. In our case, we have a C s symmetry molecule, and relatingt he modes back to the parent D 5h symmetry is more complex. To proceed, we note that the highest subgroup of D 5h that is compatible with translational symmetry is C 2v .I mposing C 2v symmetry on the Pm C 70 structure raises the crystal symmetry to Pmm2. This structure was then geometry optimised, and the vibrational modes were calculated.T his results in an INS spectrum that has the same profile as that from the Pm structure, but slightly shifted to lower wavenumber.T hus, the modes have been calculated in the same order in both systems. Using the correlation given in Ta ble S1 in the Supporting Information, and assuming that modes that are degenerateu nder D 5h symmetry will be very close( af ew wavenumbersa tm ost), it can be seen that the modes can be readily assigned as A', E', A",a nd E".
In order to assign the modes as to whethert hey are symmetric or antisymmetric with respect to the improper rotation 2 S 5 ,( subscript1or 2, respectively), we note that eight infrared modes [12] are observed in the gas phase and 24 Raman modes in solution. [18] In both cases, D 5h symmetry is present.H ence, the infrared active modes must be either E 0 1 or A 00 2 ,a nd the Raman active modes A 0 1 , E 0 2 ,o rE 00 1 ;t hus, these limit the possibilities for some of the modes. Additionally,i nspection of Figure 5s hows that the major discrepanciesb etween the isolated molecule (D 5h )s pectra and the solid-state (C s )s pectra (observed and calculated) occur in the region4 50-800 cm À1 . Outside this region, the INS spectralp atterns are remarkably similar,s uggestingt hat the D 5h assignments are reliable below 450 cm À1 and above 800 cm À1 .I nc ombination,t he experimental infrared and Ramant ransitione nergies with the partial gasphase assignment enableacomplete set of assignments to be made.T his, for the internal modes of C 70 with C s symmetry in the Pm structure, is given in Ta ble S2 in the Supporting Information, together with the observed INS, infrared, and Raman bands.
Further support for the validity of our modeli sp rovided by comparison of the observed and calculated Ramana nd infrared spectra (Figures 6a nd 7). Our Raman spectrum wasrecorded using 785 nm excitation,w hich is well removed from the lowest-energy electronic absorption bands at 467 and 545 nm, [33] so the spectrum should not be affected by resonance or preresonance effects, [18] and the agreement is very good. The infrared spectrum (Figure 7a), wasr ecordedb yA TR and is markedly different from those recorded by transmission infrared spectroscopy. [13,15,16,19,20] Comparisonw ith the vibrational density of states (VDOS) (Figure 7b)a nd the calculated Figure 4. Comparison of the a) TOSCA INSs pectrumo fC 70 at 7Kwith spectra simulated using literature assignments:b )for the gas phaseb yG AUSSI-AN 98 (D 5h symmetry) [19] and c) by an ab initio molecular dynamicssimulation. [26] (7) [a] See Figure 1f or the definition of the bonds.
[d] Ref. [32]. www.chemistryopen.org infrared spectrum (Figure 7c)s hows much better agreement with the former.M any more modes are observed than expected, most of which are fundamentals.I ti sn ot at all clearw hy this occurs. The ATRd evice uses ac lamp to ensure good contact between the sample and the diamondA TR element; however,pressure alone cannot be the explanation, as transmission infrared measurements [20] up to 10 GPa do not result in asimilar spectrum. We (ando thers) [34] have observed as imilar effect in C 60 . Close inspection of Figure 5a,b shows that the calculated and observed INS spectra differ in minor regards,i nt hat some of the calculated transition energies are slightly (a few wavenumbers) inaccurate. We have previously shown that the calculated intensities (eigenvectors) are not very sensitive to the transition energy, [35] so it is possible to makes mallc hanges in the transition energies and use the same eigenvectors. The procedure is to shift ac alculated band to the nearest experimentalf eature of the correct intensity.I ne ssence, this amountst oi ndividual scaling factors for each of the modes and was successfully used for C 60 . [27] Figure 8s hows the result, and it can be seen that the agreement is excellent;b ased on this, Table S2 in the Supporting Information lists the internal modeso fC 70 in D 5h symmetry.T he transition energies reported are ac ombination of experimentala nd our ab initio results and represent the best availabled escription of the vibrational modes of C 70 .
There have been attempts [19,26] to identifym odes that are characteristic of the 'belt' of the additional ten atoms that distinguish C 60 from C 70 .B yg enerating INS 'spectra' where the cross section of all the atoms except those of interest are set to zero, it is possible to see whether there are any modes localised in either the belt or the caps. Figure 9s hows the result, and it is clear that for almost all the modes,b oth the belt and cap atoms are involved. There are af ew modes where there is no motion of the belt atoms (E 00 2 mode at 307 cm À1 , A 00 1 mode at 336 cm À1 , E 00 1 mode at 705 cm À1 , E 00 1 mode at 705 cm À1 ,a nd    www.chemistryopen.org the A 00 2 mode at 707 cm À1 ), and thesea re shown in Figure S1 in the Supporting Information. We find no modes that only involve motion of the belt atoms. We also disagree with the conclusion [19] that the amplitudeo fm otion of the belt atoms is larger than that of the cap:t he intensities in Figure 9a re per carbon atom, and both spectra are plotted on the same ordinate scale. Thus the amplitudes of the atomsa re similar in both cases.

Conclusions
Ac ombination of am uch better quality INS spectrum with periodic DFT calculations has allowed all the internal modes of C 70 to be assigned. Comparison of the INS spectra predicted by previousw ork with the new data demonstrates that all previous assignments were incorrect in several respects. In particular,i solatedm olecule calculations using Gaussian basis sets result in inaccurate assignments. In contrast, the periodic-DFT approachproduces assignments that are in almost quantitative agreement with the data withoutt he need for scaling for both C 60 and C 70 .T his clearly has implications for the future assignment of higherfullerenes such as C 84 .
The need for ar eliable assignment spansa stronomy, [5,6] where the fullerenes are detected by their vibrational signatures, to polymers, where fullerenes are attracting attention as fillers [36] or novel monomers [37] and solar cells. [38,39] There is considerable current interesti ns olar cells that use C 60 or C 70 derivatives as the light-harvesting element. Knowledge of the parentf ullerene's vibrational spectrum is an essential first step in understandingh ow the spectrum is modified on derivatisation and how the materials change in use.

Experimental Section
The INS [21] experiments were performed with the high-resolution time-of-flight spectrometers, TOSCA [40] and MAPS, [41] at the ISIS [42] pulsed spallation neutron source at the STFC Rutherford Appleton Laboratory,C hilton, UK. While both TOSCAa nd MAPS access the same energy transfer range, 0t o4 000 cm À1 ,t hey provide complementary data. In INS spectroscopy,o vertones and combinations are allowed transitions, whose intensity depends on Q 2n ,w here Q ( À1 )i st he momentum transfer and n is the order of the transition, n = 1f undamental, n = 2, first overtone or binary combination, and so forth. On the indirect geometry instrument, TOSCA, each energy-transfer value (w,c m À1 )i sa ssociated with au nique momentum-transfer value: Q 2 % w/16. Thus the intensity of the overtones and combinations increases rapidly with increasing energy transfer on TOSCA and dominate the spectrum at large energy transfer.T he direct-geometry instrument, MAPS, can access lowmomentum transfer at large-energy transfer,s oi te nables high energy modes to be observed that are difficultt od etect on TOSCA. On TOSCA, the resolution is~1.25 %o fthee nergy transfer across the entire energy range, while on MAPS, under the conditions used here, it is~1.5 %o ft he incident energy at the largest energy transfer and degrades with decreasing energy transfer. Thus, TOSCAp rovides excellent energy resolution at energy transfers < 1200 cm À1 ;a tl arger energy transfer MAPS provides better resolution by virtue of the access to low Q.T OSCA and MAPS are highly complementary and enable the complete range of interest, 0-2000 cm À1 ,t ob ec overed with good resolution. The spectra were recorded from a9.1 gs ample of C 70 at 20 K.
Dispersive Raman spectra were recorded simultaneously with the TOSCAs pectra with am odified Renishaw InVia spectrometer (Wotton-under-Edge, UK) using adiode laser at 785 nm as the excitation source and aP eltier-cooled CCD detector.T he system has been described in detail elsewhere. [43] The spectral resolution is determined by the laser wavelength and the dispersion of the grating and is~4cm À1 .The laser power at the sample is afew mW;this results in a1Ktemperature rise when the Raman spectrum is recorded. The Raman spectra have been corrected for the instrument response function. Weak sample fluorescence resulted in as loping baseline;t his has been approximated by ap olynomial and subtracted from the spectrum.
Attenuated total internal reflection (ATR) infrared spectra (400-4000 cm À1 )w ere recorded with aB ruker Vertex 70 spectrometer (Billerica, USA) (4 cm À1 resolution, 128 scans), with the sample held in aS pecac Low-Temperature Golden Gate ATR( Orpington, UK) with aK Br beamsplitter and ad euterated triglycerine sulfate (DTGS) detector.T he spectra were recorded over the temperature range 113-300 K. They exhibited as trongly sloping baseline;t his was approximated by ap olynomial and subtracted from the spectrum.
C 70 (99 + %, MER Corporation, Tucson, USA) was dried overnight in av acuum oven at 383 Kb efore use. The weight loss was less than 0.1 %.
Ab initio DFT calculations of the isolated molecule in D 5h symmetry were carried out with GAUSSIAN 03. [44] The rpbe1pbe functional with the 6-311g(d) basis set was used. Periodic-DFT studies of the crystalline structures were carried out using the plane-wave pseudopotential method implemented in the CASTEP code. [45,46] Exchange and correlation were approximated using the PBE functional. [47] An orm-conserving pseudopotential for carbon was generated using the kinetic-energy optimised method, [48] with core radii of 1.2 a 0 (s) and 1.54 a 0 (p) in conjunction with ap lane-wave cut-off energy of 750 eV.A ll the calculations were carried out at the Gpoint to reduce the computational cost. The equilibrium structure, an essential prerequisite for lattice dynamics calculations, was obtained by BFGS geometry optimization, after which the residual forces were converged to zero within 0.009 eV À1 .P honon frequencies were obtained by diagonalisation of dynamical matrices, computed using density-functional perturbation theory [46] (DFPT). An analysis of the resulting eigenvectors was used to map the computed modes to the corresponding irreducible representations of the point group and assign IUPAC symmetry labels. DFPT was also used to compute the dielectric response and the Born effective charges, and from these, the mode oscillator strength tensor and infrared absorptivity were calculated. The Raman activity tensors were calculated using ah ybrid finite displacement/DFPT method. [49] The calculated infrared and Raman spectra were generated using the CASTEP utility 'dos.pl'. [50] The program ACLIMAX [51] was used to produce the INS spectrum from the ab initio results. The INS spectra in Figure 4b,c were generated by using the published assignments, [19,26] but with the eigenvectors calculated in this work. As noted earlier,t he eigenvectors are not very sensitive to the calculated eigenvalue, [35] and the INS intensity per mode is almost constant (since the intensity does not vary greatly across the spectrum, and all modes are present). Only the Debye-Waller factor changes (since it depends on Q,a nd this is directly related to the energy transfer [21,40] ), and this is included in the generation of the INS spectrum by ACLIMAX.