Ab initio infrared vibrational modes for neutral and charged small fullerenes (C₂₀, C₂₄, C₂₆, C₂₈, C₃₀ and C₆₀)

Abstract

We calculate the infrared (IR) absorption spectra using DFT B3LYP(6–311G) for a range of small closed-cage fullerenes, C_n, n=20, 24, 26, 28, 30 and 60, in both neutral and multiple positive and negative charge states. The results are of use, notably, for direct comparison with observed IR absorption in the interstellar medium. Frequencies fall typically into two ranges, with C−C stretch modes around 1100–1500 cm⁻¹ (6.7–9.1 μm) and fullerene-specific radial motion associated with under-coordinated carbon at pentagonal sites in the range 600–800 cm⁻¹ (12.5–16.7 μm). Notably, negatively charged fullerenes show significantly stronger absorption intensities than neutral species. The results suggest that small cage fullerenes, and notably metallic endofullerenes, may be responsible for many of the unassigned interstellar IR spectral lines.

This article is part of the themed issue ‘Fullerenes: past, present and future, celebrating the 30th anniversary of Buckminster Fullerene’.

1. Introduction

The recent confirmations of the presence of C₆₀ and C₇₀ in a wide range of interstellar and circumstellar environments [1–6] have major implications for astrochemistry as well as the distribution of carbon in space. These results have been contingent on two factors: the high-quality infrared (IR) results provided by the Spitzer telescope infrared spectrograph instrument [7] and the knowledge gained by theoretical and experimental studies on the IR signatures of these fullerenes [8–12]. In addition to detection in their neutral state, fullerenes have been found to be prevalent in space in charged states. In fact, owing to their low ionization potential, it was foreseen in the late 1980s [13] that C₆₀ could easily be stripped of an electron to become , at least in the gas phase. This charged fullerene has recently been confirmed to be the origin of two diffuse interstellar bands [14] and was evaluated to represent up to 0.9% of interstellar carbon [15]. Cationic fullerenes may also exist in space, notably via charge transfer from endohedral metal atoms trapped within the fullerene cage [16]. Dunk et al. [8] have recently proposed that the Na@C₆₀ metallic endofullerene provides a mechanism for ²²Na capture during supernova events, and subsequent incorporation of its decay product ²²Ne in carbonaceaous dust in the interstellar medium.

Notwithstanding the advances made in detecting the contribution of fullerenes to the IR spectra of various stellar objects, the attribution to specific species of the remaining observable spectral features is an ongoing process. A major effort has been made to collect experimental and theoretical spectra for polycyclic aromatic hydrocarbons (PAHs) [17]. The confirmation of the existence of stable small closed-caged fullerenes [18,19] and endohedral metallofullerenes [20] extends the range of candidates for this process. These fullerenes have specificities that differ from those of their larger cousins that need to be taken into account. They display interesting motifs in their structure such as squares and paired or triple pentagons, and their synthesis is likely to be metal assisted [21]. However, until such species can be chemically isolated and their individual IR spectra measured, direct assignment to stellar IR spectral lines remains elusive.

For this reason, this paper presents calculated IR absorption spectra for the smallest closed-cage fullerenes of C_n, where n=20, 24, 26, 28 and 30, as well as of C₆₀ for comparison. We do so in both neutral and various positive and negative charge states. The positive charge state calculations are important because, as discussed above for C₆₀, under strong 13.6 eV irradiation in diffuse interstellar space, electron loss is likely and hence observed fullerenes are likely to be positively charged.

We include the negatively charged fullerene results for comparison with metallic endofullerenes [22]. For C₆₀, it was noted that Na@C₆₀ and IR spectra are effectively identical [8]. This is coherent with the ionic metal–fullerene bonding model whereby charge is transferred from the metal to the fullerene and is not highly localized. This phenomenon has been investigated by Kobayashi and co-workers [23,24] and shown to be the norm (with the exception of the system, which shows covalent characteristics). Experimentally, the vibrational spectra of M@C₈₂ (Y, La, Ce, Gd) in the IR range were measured by Lebedkin et al. [25]. The spectra were found to be virtually identical in the range above 200 cm⁻¹, consistent with a picture whereby the role of the endohedral metal is primarily one of charge transfer to the surrounding cage, with little or no other perturbation of the cage itself. We therefore hypothesize that in the case of small closed-cage fullerenes one can determine the likely IR spectra for endohedral metal fullerenes by simply negatively charging the empty fullerene cage appropriately.

2. Material and methods

Calculations were performed with density functional theory (DFT) as implemented in the Gaussian09 code [26] using B3LYP exchange correlation and a 6–311G basis set, following the method tested and successfully applied to C₆₀ and Na@C₆₀, both in a range of charge states [8]. In that paper [8], we compared the 6–31G* and 6–311G basis sets, finding a better match to experimental IR spectra for the larger 6–311G basis without the need for a frequency scaling factor. We note that the NASA AMES PAH database [17] uses a 4–31G basis set to calculate IR signatures of PAHs [27]. For comparison, we also calculated the IR spectra for C₂₀ and C₆₀ in various charge states with this basis set, obtaining near identical results to the 6–311G basis (see the electronic supplementary material, figure S1). In their comparative analysis to experimental spectra, they apply a constant 15 cm⁻¹ red shift to their calculated values [17] and a 0.986 scaling factor to all modes [27].

For each fullerene and for charge states ranging from −4 to +4, the structures are fully relaxed using a conjugate gradient algorithm or a quadratically convergent self-consistent field procedure (‘XQC optimization’) when the conjugate gradient failed, and normal frequencies are then calculated. Different multiplicities ranging from singlet to quintet have been checked and the most stable is reported (table 1). Calculations are carried out until forces on each atom are less than 0.04 eV Å⁻¹. In the calculations, the force constants are determined analytically and the vibrational frequencies are computed by determining the second derivatives of the energy with respect to the Cartesian nuclear coordinates. Spectra are generated using Gaussian broadening with 15 cm⁻¹ full-width at half-maximum (FWHM) for frequency spectra and 0.25 μm FWHM for wavelength plots. Different charge states have intensities offset on the IR plots by 40 km mol⁻¹ for improved visibility. The plots given in the paper show frequency on the x-axis; equivalent plots with wavelength on the x-axis are included in the electronic supplementary material. All vibrational frequencies are positive, confirming that the optimization has reached a minimum energy structure.

Table 1.

Small cage fullerenes considered in this study, including their symmetry group in the neutral charge state and thermodynamically most stable spin multiplicities. C₆₀ is included for comparison.

		stable multiplicity for given charge state
	symmetry	+4	+3	+2	+1	0	−1	−2	−3	−4
C20	Ih	3	2	1	2	1	4	1	2	1
C24	D6d	1	2	3	2	1	2	1	2	1
C26	D3h	3	2	5	6	5	4	3	2	1
	Cs	1	2	1	4	3	2	1	2	1
C28	Td	1	2	3	4	5	4	3	2	1
	D2	3	2	1	2	1	2	1	2	1
C30	C2v–I	5	4	3	2	1	2	1	2	1
	C2v–II	3	2	1	2	3	2	1	2	1
C60	Ih	3	4	3	2	1	2	3	4	3

3. Results

After initially considering C₆₀, we present each of C₂₀, C₂₄, C₂₆, C₂₈ and C₃₀ in sequence. Considering classical fullerene pentagon–hexagon structures, there are only a limited number of isomers for these small size fullerenes; indeed, C₂₀, C₂₄ and C₂₆ have only one available isomer, there are two isomers for C₂₈ (T_d and D₂ symmetry), and three for C₃₀. We have excluded the D_5h isomer of C₃₀ from this study because it is significantly less thermodynamically stable than the other two. In addition to these species, we have also included a C_s symmetry isomer of C₂₆ constructed from a square, 10 pentagons and 3 hexagons. Fullerenes containing square motifs have been proposed as more stable isomers for small fullerenes [28] because they are able to localize curvature more effectively and hence relieve surface strain elsewhere in the fullerene cage. As they should also have distinct IR signatures associated with distorted square motifs, we felt this was valuable to include.

We note that, for the pentagon–hexagon isomers, the relative energy of the anions and cations between different isomers matches that of the neutral species, validating the ionic model discussed above. This is also the case for the non-classical C_s−C₂₆ square-containing isomer, which we find less stable than its conventional D_3h isomer in all charge states, in agreement with the results of An et al. [29].

(a). C₆₀

The IR absorptions associated with C₆₀, and its charged species and , have been extensively explored in the literature [8–12] and serve as a useful benchmark for our calculations. C₆₀ has only four vibrational IR absorption lines due to its I_h symmetry, at 1436, 1191, 574 and 531 cm⁻¹ in the 0 K gas phase [9,10]. Removal of an electron gives , the electronic transitions of which were recently identified as the source of two strong diffuse interstellar bands at 9632.7 Åand 9577.5 Å[14] and weaker bands at 9428.5 Åand 9365.9 Å[30]. Charging the fullerene breaks its symmetry, giving rise to extensive new IR active vibrational modes [31], notably a high-frequency mode at approximately 1550 cm⁻¹ (6.45 μm) due to vibrations of pentagons at opposing ends of the fullerene cage [8]. The IR bands possibly due to have been tentatively assigned in spectra from objects Tc1, SMP SMC 16, NGC 7023, NGC 2244 and SMP LMC 02, while features possibly due to were identified in NGC 7023 [32].

Our calculated frequencies for neutral C₆₀ lie at 1453.5 cm⁻¹, 1197.2 cm⁻¹, 580.8 cm⁻¹ and 550 cm⁻¹, in good agreement with experiment (figure 1). In , the symmetry is broken and many more modes are visible; notably, we see strong peaks in the calculations at 1584, 1542, 1467, 1434, 1360, 1314, 1231, 1187, 965 (726 and 686) and 549 cm⁻¹, agreeing with experimental measurements of in a Ne matrix [31], with the exception of our peak at 1434 cm⁻¹ not reported in the experiment, presumably because of its overlap with neutral C₆₀ at 1432 cm⁻¹. was also measured in a Ne matrix [33], and again we have excellent agreement on all peak positions with the exception of peaks around 1450 cm⁻¹, which again may be due to C₆₀ signal masking. shows excellent peak-to-peak matching with experimental data in the Ne matrix [31], with the calculated (experimental) peaks as follows: 1563 (1546), 1450 (*), 1410 (1386), 1349 (1352), 1320 (1334), 1206 (1201), 1194 (1176), 974 (964), 727 (729), 681 (*), 576 (576 and 575), 542 (*) and 401 (397) cm⁻¹; asterisks indicate no experimental peak (in the case of 1450 cm⁻¹ and 542 cm⁻¹, once again probably due to neutral C₆₀ masking). The strongest peak is at 1410 cm⁻¹.

Figure 1.

Calculated IR spectra in cm⁻¹ for I_h–C₆₀ in various charge states. (Online version in colour.)

We note that the total energies are very close to each other between the doublet/quartet and the singlet/triplet (approx. 1 meV) spectra for and , respectively, with those of the triplet/quartet being the most stable. The strong peak at approximately 1580 cm⁻¹ discussed above is present in the calculated singlet/doublet spectra for and , but not for the triplet/quartet systems.

It is important to note that our calculated IR intensities for are over an order of magnitude higher than those for C₆₀. In general, the peak intensities are significantly higher for the negatively charged than for the neutral fullerene species considered here. This trend is consistent with the observed higher peak intensities in IR for metallic endofullerenes when compared with their equivalent empty cage spectra [8,16,34,35]. This suggests that metallic endofullerenes may be disproportionately represented in experimental IR spectra.

The excellent agreement between literature experimental values for C₆₀ in a range of different charge states and the calculated theoretical values presented here gives us confidence in the calculated frequencies presented below. We note that a scaling factor of approximately 0.99 in general brings all the modes even closer to experiment, independent of the fullerene charge state. Thus while no scaling factors are applied throughout this article, this factor may be useful for future spectral line identification.

(b). C₂₀, the smallest cage

Experimentally the monocyclic isomer of C₂₀ dominates and there is no experimental evidence for direct C₂₀ fullerene formation [20], although it can be (briefly) synthesized by gas-phase removal of the functional groups from C₂₀H₂₀ [36]. However, we include it here as the smallest possible classical fullerene cage, for comparison with previous literature calculations, and because of its structural motif consisting of purely fused pentagons, which should result in a distinctive IR signature. Although the perfect cage shows icosahedral I_h symmetry, it has been shown that the structure can relax through a Jahn–Teller distortion to a D_3d structure [37]).

The calculated spectrum for neutral C₂₀ (figure 2) agrees well with previous literature data (B3LYP with the 6–31++G basis [38] and 6–31G(d) basis [39]), featuring two primary peaks at 1324 cm⁻¹ (7.55 μm) and 769 cm⁻¹ (13.00 μm). Peaks in these two spectral regions are seen in many of the small fullerene spectra discussed below. The peak at 1324 cm⁻¹ corresponds to tangential carbon–carbon bond stretching (figure 3), at lower frequency than the C−C stretch mode in C₆₀ or graphene due to the longer, weaker bonds in C₂₀. The peak at 769 cm⁻¹ is fullerene cage specific and involves radial motion of the carbon atoms. Opposing pentagon corners move radially in phase, while the remaining carbon atoms move radially with alternating phase. The result is a synchronized torsional motion around the central circumference of the fullerene. The symmetric radial breathing mode where all atoms move radially towards the fullerene centre is IR inactive, and is calculated at 816 cm⁻¹ (12.25 μm). loses the strong peak at 769 cm⁻¹, while the 1324 cm⁻¹ peak is split into multiple peaks, with the primary feature at 1195 cm⁻¹ (8.37 μm). Signal intensity is significantly weaker than for the neutral, and for the signal intensity is nearly two orders of magnitude weaker than for the neutral species. There is a general tendency of downshifting frequency (increasing wavelength) of the peaks with increasing negative charge on the cage.

Figure 2.

Calculated IR spectra for I_h–C₂₀ in cm⁻¹ in various charge states, as marked. (Online version in colour.)

Figure 3.

Calculated eigenvectors for neutral I_h–C₂₀ modes at (a) 1324 cm⁻¹ (7.55 μm), primarily associated with carbon–carbon stretch, and (b) 769 cm⁻¹ (13.00 μm), involving coupled torsional motion of the fullerene cage circumference. (Online version in colour.)

(c). C₂₄, the transition from rings to fullerenes

C₂₄ is close to the thermodynamic transition between stable ring and fullerene structures [1–6,40,41], and under certain experimental conditions can be a stable product from carbon vapour condensation [42]. There is only one classical fullerene isomer of C₂₄, the highly symmetric D_6d form consisting of an opposing hexagon pair connected by two rings of six pentagons each (figure 4), although calculations suggest that a non-classical isomer incorporating two squares may be energetically competitive [28]. It is suggested that the D_6d symmetry can be lifted through a small Jahn–Teller distortion to D₆ [37].

Figure 4.

Calculated IR spectra for D_6d–C₂₄ (cm⁻¹) in various charge states, as marked. (Online version in colour.)

The high fullerene symmetry results in a dominating peak for the neutral species at 674 cm⁻¹ (14.84 μm), with secondary peaks at 903 and 1255 cm⁻¹ (11.07 and 7.97 μm), again in good agreement with literature calculations (DFT-PBEPBE/6–31G(d,p) [43]) (figure 4). The 674 cm⁻¹ peak has a similar character to the 769 cm⁻¹ mode described above for C₂₀, i.e. a coupled torsional mode around the equator of the fullerene, with in-phase motion of the capping hexagons (figure 5). Positive charging leads to a rapid decrease in IR intensities with slight shifting of all features.

Figure 5.

Calculated eigenvectors associated with the 674 cm⁻¹ (14.84 μm) mode of neutral D_6d–C₂₄. (Online version in colour.)

(d). C₂₆, exploring square motifs

The next closed fullerene cage by size is C₂₆, of which there is only one classical closed-cage isomer, with D_3h symmetry. The cage consists of opposing triply fused pentagons, bridged by a ring of three alternating hexagon and pentagon pairs. However there are other non-classical isomers, and we also consider here a C_s−C₂₆ isomer containing a single square that was found to be close in energy to the classical cage [29]. The square replaces two pentagons, and the structure also contains four hexagons, three of which are localized around the square. Both of these structures are the first we have considered with an open shell electronic structure in the neutral charge state (quintet and triplet, respectively), with the classical isomer being 5.29 kcal mol⁻¹ more stable.

There is once again good agreement with an available literature spectrum for the neutral species (B3LYP/6–31G* basis) [29] (figure 6), with primary peaks at 1180 and 726 cm⁻¹ (8.47 and 13.77 μm), and secondary peaks at 1413, 1328 and 839 cm⁻¹. The 1180 cm⁻¹ peak corresponds to pentagon–pentagon C−C stretch. The 726 cm⁻¹ peak corresponds to the two triplet pentagons at opposing ends of the cage, with the apex carbon atoms vibrating in phase along the fullerene axis, while their three surrounding neighbours vibrate radially out of phase with the apex atoms (figure 7). Thus, this mode is distinctly characteristic of triplet-pentagon groups. All primary peaks in the 600–750 cm⁻¹ range discussed thus far share the common feature of radial carbon motion at the centre of triple pentagons.

Figure 6.

Calculated IR spectra for D_3h–C₂₆ (cm⁻¹) in various charge states, as marked. (Online version in colour.)

Figure 7.

Calculated eigenvectors associated with the 726 cm⁻¹ (13.77 μm) mode of neutral D_6d–C₂₄. (Online version in colour.)

The stable spin state for the neutral species is the quintet (multiplicity of 5), and the calculated IR spectrum shows qualitative differences if the electrons are spin-flipped. Notably, the singlet structure, 10.91 kcal mol⁻¹ less stable than the quintet, shows a primary peak at approximately 1080 cm⁻¹ instead of 1180 cm⁻¹ for the quintet.

The C_s−C₂₆ isomer does not have such strongly defined IR peaks (figure 8). It is difficult to clearly identify specific peaks with the square motif as all modes are coupled around the cage. Nonetheless, the C−C stretch mode of the square is strongly active in the mode at 973 cm⁻¹, while the peak at 620 cm⁻¹ demonstrates a similar radial carbon motion as described previously for triplet pentagons, but in this case for a carbon atom forming part of the square. The cage has an IR inactive radial breathing mode at 706 cm⁻¹.

Figure 8.

Calculated IR spectra for C_s–C₂₆ (cm⁻¹) in various charge states, as marked. (Online version in colour.)

(e). C₂₈, the beginning of metal encapsulation

C₂₈ is the first of the ‘magic number’ fullerenes and, arguably, the most important of those considered in this study. The most stable classical isomer, T_d–C₂₈ is the smallest fullerene to contain isolated triple-pentagon groups, one at each corner of its tetrahedral cage structure. This results in highly localized curvature, stabilizing the fullerene sides and rendering the corners highly chemically reactive, as the central carbon atom of each pentagon triplet is essentially a sp³-coordinated dangling bond. The structure can be chemically stabilized by hydrogenating these dangling bonds to give C₂₈H₄ [44], forming a closed shell species, or through charge transfer, giving , typically through encapsulation of a 4+ metal cation.

This unique structure means that C₂₈ is the smallest fullerene cage for which metal encapsulation has been demonstrated experimentally [20]. The first metal-encapsulated C₂₈ species to be detected was U@C₂₈ [45], relatively early in the history of fullerene research, and more recently Ti@C₂₈, Zr@C₂₈, U@C₂₈ and even Hf@C₂₈ have been unambiguously identified [20]. For larger metal species, the internal cavity of C₂₈ remains too small for encapsulation, for example the smallest Th-encapsulated cage is Th@C₃₆ [20], and similarly the La metal in fullerene species was shown to be exohedral for 29<n<35, and only endohedral for 36<n<90 [46]. Thus with the possibility of endohedral metal doping, for the purposes of our calculations, C₂₈ is the first cage species for which it becomes important to determine the IR spectra at potentially high charge states up to .

C₂₈ is a common product from laser vaporization of C₆₀, suggesting that if C₆₀ is present in large quantities in interstellar environments, under irradiation, C₂₈ may also be expected to exist. Equally, C₂₈H₄ is highly stable, for example field-evaporated graphite also results in a strong signal [47]; subsequently hydrogen loss may also lead to the C₂₈ presence in space.

For neutral T_d–C₂₈, there are two primary modes at 1417 and 1188 cm⁻¹, with weaker modes at 1263, 978, 772, 724 and 809 cm⁻¹ (figure 9). However, this picture changes significantly upon charging to . Firstly, signal intensity increases by a factor of nearly five and the primary peaks now lie at 763 cm⁻¹ and 1373 cm⁻¹ (13.11 and 7.28 μm), with a secondary peak at 625 cm⁻¹ (16.0 μm) and weak peaks (absorption comparable to the neutral species) at 492, 705, 1082, 1178 and 1264 cm⁻¹. The charge state similarly has a primary absorption feature at 771 cm⁻¹ (12.97 μm) with intensity approaching that of the peaks, and weaker peaks at 908, 670 and 480 cm⁻¹. The 763 cm⁻¹ triplet mode for comes from radial breathing of a triple-pentagon carbon atom and its surrounding 10 neighbours, in phase with a radial motion of a hexagon on the opposing side of the cage. The 1373 cm⁻¹ triplet mode comes from hexagon C−C bond stretching (figure 10).

Figure 9.

Calculated IR spectra for T_d–C₂₈ (cm⁻¹) in various charge states, as marked. (Online version in colour.)

Figure 10.

Calculated eigenvectors associated with the (a) 763 cm⁻¹ (13.11 μm) and (b) 1263 cm⁻¹ (7.28 μm) mode of neutral T_d–C₂₈. (Online version in colour.)

Comparing our result with previous calculations for Ti@C₂₈ shows qualitatively good agreement [48], with strong peaks for Ti@C₂₈ reported at 1357/1396 cm⁻¹, and several bands around 780–830 cm⁻¹. This peak splitting compared with our results comes from symmetry breaking by the Ti atom, which displaces from the cage centre towards one of the triplet-pentagon centred carbon atoms. This suggests limitations to the ionic metal–fullerene bonding model for small fullerenes where hybridization with the encapsulated metal atom also takes place.

For completion, we also include here the second classical C₂₈ fullerene isomer, D₂–C₂₈. This isomer has six pentagons at each fullerene end, linked via four hexagons around the fullerene equator. As it is 30.84 kcal mol⁻¹ less stable than T_d–C₂₈, it is unlikely to exist in significant quantities. As for the previous species discussed, D₂–C₂₈ exhibits strong absorption in the 700–800 and 1200–1500 cm⁻¹ regions with precise peak positions depending on the charge state (figure 11). Again in common with previous species, IR absorption intensity is significantly higher for the negatively charged fullerenes. The primary neutral D₂–C₂₈ modes at 863 and 1212 cm⁻¹ (11.59 and 8.25 μm) correspond similarly to radial and bond stretch (tangential) carbon motion, respectively.

Figure 11.

Calculated IR spectra for D₂–C₂₈ (cm⁻¹) in various charge states, as marked. (Online version in colour.)

(f). C₃₀

There are three classical isomers of C₃₀. We find that the ground state C_2v–II–C₃₀ isomer is 6.23 kcal mol⁻¹ more stable than C_2v–I–C₃₀, which is in turn 52.98 kcal mol⁻¹ more stable than D_5h–C₃₀. Owing to the low stability of this third isomer, we consider only the two C_2v species here. While C₃₀ is experimentally much less common than C₂₈ and not a ‘magic number’ fullerene, it is nonetheless detected in laser ablation studies, and the metallic endofullerene Ti@C₃₀ has been observed, albeit in lower concentrations than Ti@C₂₈ [20]. Thus, we include it here for completion but do not expect it to be a major source of interstellar carbon.

The spectra of C_2v–II–C₃₀ nicely follow the general trend seen above with peaks in two primary spectral windows (figure 12b). The first group of peaks lie in the 1100–1500 cm⁻¹ range, peaking at 1217 cm⁻¹ (8.22 μm) for the neutral species and increasing in frequency and intensity with increasing charge state to 1434 cm⁻¹ (6.97 μm) for C_2v–II–. Strong absorptions are also seen at approximately 700–800 cm⁻¹, the most intense at 776 cm⁻¹ (12.89 μm) for C_2v–II–, notably increasing in strength with negative charging and spreading to lower frequencies. The spectra for C_2v–I–C₃₀ also show the same general trend (figure 12a).

Figure 12.

Two principal classical fullerene isomers of C₃₀: (a) C_2v–I–C₃₀ and (b) C_2v–II–C₃₀. Isomer structures and calculated IR spectra in cm⁻¹ in various charge states, as marked. (Online version in colour.)

4. Discussion and conclusion

In order to draw some general conclusions, in figure 13 we show the calculated IR spectra for all of the neutral charge state fullerenes considered here, versus frequency (cm⁻¹) and wavelength (μm). Certain common features can be observed. In general, the IR absorption bands fall into two broad spectral regions: the first in the approximately 1200–1400 cm⁻¹ range (approx. 6–8.5 μm), the second in the approximately 700–800 cm⁻¹ range (approx. 12–14.5 μm), with a few additional peaks at other frequencies. The first of these ranges correspond to C−C stretch modes, tangential to the fullerene cage surface, and overlap with vibrational modes of most sp²-carbon-based molecules. The second range however corresponds to radial motion of carbon atoms involved with shared pentagons, and can be considered ‘fullerene specific’. Peaks in this second range are notably more intense when they involve carbon atoms at the centre of triple-pentagon patches. These peaks can be understood because carbon atoms shared between pentagons are chemically under-coordinated and have dilated C−C bonds as a result, explaining why this mode is not present in isolated pentagon fullerenes C₆₀ and above.

Figure 13.

Calculated IR spectra in (a) cm⁻¹ and (b) μm for small fullerene cages C_n, n=20–30, and C₆₀, all in the neutral charge state. (Online version in colour.)

A second general observation is that calculated IR intensities are generally much higher (up to an order of magnitude) for strongly charged fullerenes, notably negatively charged. This is particularly true for T_d–C₂₈ and I_h–C₆₀. In general, this can be understood through the increased dipole moment associated with vibrating charged cages. Additionally, the IR intensity is enhanced in cases with a carbon radical (dangling bond) at the centre of triple pentagons. In these cases, excess negative charge localizes on these sites, for example in the Mulliken charge on these four symmetric corner atoms is 0.331e/C with 0.179e/C on the 12 neighbouring sites. By comparison, in the charge is distributed evenly over the cage (with some symmetry breaking), ranging from 0.004e to 0.033 e/C. This charge localization means that radial vibrations associated with these sites, such as the approximately 700 cm⁻¹ mode seen in many of the negatively charged cages examined here, have significantly stronger dipole moment and thus IR absorption. Our negatively charged cage results show generally good agreement with the previous calculations of metallic endofullerenes that were available, suggesting that they can be used as good indicators of spectral features of metallic endofullerenes. We note that the enhanced intensity for strongly charged species suggests that metallic endofullerenes will show disproportionately strong interstellar absorptions relative to their concentrations.

These closed-cage small fullerenes and their metallic endofullerene equivalents may be stable enough to be found in interstellar space, and it is our hope that the current calculated spectra can now be linked to interstellar spectral data. Figure 14 shows Spitzer data from planetary nebula of the Magellanic cloud, which have been investigated for the presence of C₆₀ and C₇₀. These are residual spectra, adapted from García-Hernandez et al. [49], for which the dust continuum emission has been subtracted. We have indicated the position of the peaks obtained from our calculations as vertical lines. Figure 14a includes our 20 strongest calculated lines from neutral and +1 species, and figure 14b includes those from the negatively charged species, which we take as indicative of potential endohedral metallofullerene absorptions (values listed in table 2).

Figure 14.

Experimental residual spectra from Spitzer measurements of planetary nebula in the Magellanic cloud with dust continuum emission subtracted (Adapted from García-Hernandez et al. [49].) Overlaid are lines showing the 20 most intense calculated IR absorption modes for C₂₀–C₃₀ and C₆₀, colour coded based on the fullerene, wavelengths scaled to align the mode at 19.011 μm. (a) Neutral and +1 charged fullerenes; (b) −1, −2, −3 and −4 charged fullerenes (indicative of metallic endofullerenes). Impulses plotted below show the intensities of each calculated mode. The black bars above each spectrum mark the spectral ranges with calculated absorption by PAHs summed from the NASA AMES PAH database [27] (see the electronic supplementary material, figure S13). Calculated values are listed in table 2.

Table 2.

The 20 most intense IR absorption modes from all calculated fullerenes in the paper for the (left) neutral and +1 species and (right) −1 to −4 charged species(as plotted in figure 14). Values taken from the Gaussian broadened data, with a 1.046 scaling factor applied to align on the experimental 19.011 μm peak.

neutral, +1 species					−1, −2, −3, −4 species
fullerene	charge	wavelength (μm)	frequency (cm−1)	intensity (km mol−1)	fullerene	charge	wavelength (μm)	frequency (cm−1)	intensity (km mol−1)
D6d–C24	0	15.52	644.1	279	Ih–C60	−3	7.59	1316.9	1815
Ih–C60	+1	7.36	1358.6	249	Ih–C60	−4	7.69	1301.1	1761
Ih–C60	+1	6.59	1518.5	211	Ih–C60	−2	7.50	1333.1	1154
Ih–C20	0	7.91	1264.9	189	Ih–C60	−1	7.40	1352.1	578
Td–C28	+1	7.30	1370.7	186	Ih–C60	−4	18.45	542.1	442
Td–C28	+1	10.69	935.1	130	D3h–C26	−1	11.12	899.0	341
Td–C28	+1	8.73	1145.0	129	Ih–C60	−3	18.30	546.3	332
C2v–II–C30	+1	8.42	1187.7	118	Td–C28	−4	13.71	729.5	318
D2–C28	0	8.58	1164.9	102	Td–C28	−4	7.62	1312.1	271
C2v–I–C30	+1	11.26	887.9	99	Ih–C60	−2	8.68	1152.2	210
Td–C28	+1	10.14	986.3	96	Ih–C60	−2	18.19	549.8	208
Td–C28	+1	8.33	1200.5	96	Td–C28	−3	13.56	737.2	207
Ih–C60	+1	8.45	1183.8	95	D6d–C24	−1	7.96	1255.6	185
D6d–C24	0	11.58	863.7	85	C2v–II–C30	−4	13.47	742.6	182
D6d–C24	+1	16.00	624.9	77	C2v–I–C30	−4	8.09	1236.4	182
Ih–C60	0	19.01	526.0	77	D3h–C26	−1	7.86	1272.5	178
Ih–C60	+1	7.80	1282.7	76	Td–C28	−4	16.70	599.0	173
D2–C28	0	7.21	1387.5	76	D6d–C24	−1	8.48	1178.9	167
Td–C28	0	8.81	1135.0	70	Ih–C60	−1	8.75	1142.7	143
Ih–C20	0	13.60	735.4	66	D2–C28	−4	7.61	1313.8	143

We emphasize that the exact position of these lines is strongly influenced by the scaling factor chosen to account for inaccuracies in the calculation and thermal effects. This is particularly true for the long wavelength modes where the use of a single scaling constant for all systems is far from ideal. For this study, we have chosen to align our data on the 19.011 μm C₆₀ peak, but these figures should only be taken as qualitative indicators of potential peak positions and do not have sufficient accuracy for quantitative peak assignment. Future calculations could make use of significantly larger basis sets to improve this. Equally, we emphasize that for clarity we have plotted only the 20 strongest lines in each case.

The black bands at the top of each plot indicate the main features from 490 summed PAH species from the NASA AMES theoretical PAH database (electronic supplementary material, figure S13). As can be seen, one of the major challenges, which might reduce detection chances, is the spectral overlap between these fullerene cages and the PAHs. Notably, they strongly overlap in the IR stretching mode region around 6–8.5 μm, making unique assignment difficult. Interstellar bands are reported at 3.3, 6.2, 7.7, 8.6, 11.2 and 12.7 μm (and weaker ones at 13.5 and 14.1 μm) and features seen at these energies are commonly identified as PAHs, although there has not yet been a conclusive identification of the carrier(s) [50–52]. More recently, effort has been devoted to analysing 15–20 μm features corresponding to C−C−C modes [53]. These modes, often (but not always) 10 times weaker in most stellar regions, are centred at around 15.8, 16.4, 17.4, 17.8 and 18.9 μm. We therefore suggest that a promising strategy to assign closed-cage fullerenes would be to focus on the 13–16 μm spectral window, which contains very few PAH-related modes and is the range associated with radial carbon motion in fullerene cage paired and triplet pentagons (figure 15). For example, neutral D_6d–C₂₄ shows an intense and sharp peak around 14.8 μm. Bernard-Salas et al. [54] conducted a survey of fullerene-rich planetary nebulae in the Milky Way and the Magellanic clouds, and reported that SMC 16 shows an asymmetric feature that peaks at 14.5 μm which is currently unexplained. Furthermore, we suggest that a better approach to attributing features in stellar spectra should be to move away from single peak fitting to fitting complete signatures. Notably, our objects display several peaks in the 5–20 μm region with varying intensity ratios which should be fitted together to ensure proper species attribution.

Figure 15.

Sum of all 478 purely carbon–hydrogen species in the NASA AMES PAH database. Each spectrum intensity is divided by the number of carbon atoms in the PAH (convolved with a Lorentzian emission profile with a FWHM of 10 cm⁻¹). As can be seen, in good agreement with literature reports of PAH interstellar bands, there is strong absorption across the 6–9 μm range, with secondary features in the 10–14 μm region, and relatively little signal in the 14–20 μm range. In comparison, similarly weighted sums of all small fullerene and C₆₀ calculations in this paper notably show extensive fullerene cage modes in the 14–18 μm range.

Clearly, this study needs extension to allow further unambiguous peak identification, both to larger fullerenes C_n, 30<n<60, and notably to include hydrogenated species. Hydrogen contamination is seen as artefacts in experimental mass spectrum data, and one study suggested that about half of their observed odd-numbered pure carbon clusters C_2n−1 had a hydrogen atom [55]. We also suggest that in future it would be useful to update the NASA AMES PAH database [17], a useful resource for IR spectral peak assignment, to include spectra for small cage fullerenes.

Data accessibility

Full calculated data for all spectra in the paper, plus total energies and multiplicities, and weighted NASA AMES data are included in an Excel spreadsheet in the article’s electronic supplementary materials.

Authors' contributions

J.-J.A. performed the calculations, A.V. analysed the literature absorption and PAH data, C.P.E. proposed and designed the project and analysed the data, all authors wrote and revised the article.

Competing interests

We have no competing interests.

Funding

C.P.E. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 642742. We thank the CCIPL, ‘Centre de Calcul Intensif de Pays de la Loire’, where these calculations were performed. J.-J.A. acknowledges support from the Marcel De Merre Prize. J.-C.C. acknowledges financial support from the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS).