# Methodological Investigation for Hydrogen Addition to Small Cage Carbon Fullerenes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{n}, n < 60) are of interest as potential astrochemical species, and as intermediates in hydrogen-catalysed fullerene growth. However, the computational identification of key stable species is difficult due to the vast configurationally space of structures. In this study, we explored routes to predict stable hydrogenated small fullerenes. We showed that neither local fullerene geometry nor local electronic structure analysis was able to correctly predict subsequent low-energy hydrogenation sites, and sequential stable addition searches also sometimes failed to identify most stable hydrogenated fullerene isomers. Of the empirical and semi-empirical methods tested, GFN2-xTB consistently gave highly accurate energy correlations (r > 0.99) to full DFT-LDA calculations at a fraction of the computational cost. This allowed identification of the most stable hydrogenated fullerenes up to 4H for four fullerenes, namely two isomers of C

_{28}and C

_{40}, via “brute force” systematic testing of all symmetry-inequivalent combinations. The approach shows promise for wider systematic studies of smaller hydrogenated fullerenes.

## 1. Introduction

_{60}

^{+}is the source of some of the strong diffuse interstellar bands (DIBs) [5,11] suggests that fullerenes may play a key astrochemical role. However, the formation mechanisms of fullerenes in such extreme environments are still a debated question. Interestingly, while one might intuitively expect hydrogen to stabilise dangling bonds at edge sites and hence promote the formation of polyaromatic hydrocarbons rather than fullerenes, astrochemists have recently found that C

_{60}and C

_{70}are abundant in hydrogen-containing stars [8,9,12,13]. This suggests that hydrogen may play a role in fullerene growth. As such, stable hydrogenated small fullerenes are potentially of interest in astrochemistry, both as species in their own right and/or as degradation products from polyaromatic hydrocarbons [14], and as intermediates in the formation processes of larger fullerenes.

_{60}, Buckminster fullerene. Fullerenes smaller than C

_{60}are increasingly unstable due to steric strain originating within the non-IPR structures, and bond frustration in areas of high nonplanarity. One route to potentially stabilise local strain and under-coordination is chemical functionalisation of the most reactive carbon sites. For example, while pure C

_{50}appears unstable, C

_{50}Cl

_{10}has been isolated and characterised [16], and the stability and lifetime of C

_{20}is greatly enhanced when fully hydrogenated to C

_{20}H

_{20}[17]. This suggests that the relative stability of smaller fullerenes may be modified in the presence of hydrogen. While there has been extensive experimental [13,18,19,20,21] and theoretical [13,18,22] studies of hydrogenated C

_{60}, less attention has been given to smaller hydrogenated fullerenes. It was theoretically proposed that T

_{d}-C

_{28}H

_{4}, a tetrahedral fullerene with triply fused pentagons on each corner of the tetrahedron, might be stable [15] and indeed has since been experimentally identified [23]. For smaller fullerenes, an in silico investigation suggested that fused-pentagon sites are preferred in general for single hydrogen addition [24].

_{60}, there is only 1 isomer for each of C

_{20}, C

_{24}, and C

_{26}, and 2 for C

_{28}. However, this rapidly increases, reaching 40 isomers for C

_{40}, and 437 for C

_{52}. In total, there are 3958 C

_{x}pentagon-hexagon fullerenes with x < 60 [25].

_{n}, there are n possible addition sites for a first hydrogen atom. However, this increases rapidly as more sites are hydrogenated. For two hydrogen atoms, there are n(n−1)/2 arrangements (divided by two as the hydrogens are interchangeable), and in general, for m hydrogen atoms, there are n!/(n−m)!m! arrangements. Thus, for C

_{28}H

_{5}, this gives 98,280 possible structures, and for larger hydrogenated species such as C

_{50}H

_{4}, there are 230,300 possibilities. While symmetry can help reduce these numbers, they remain highly challenging for standard density functional theory (DFT) calculations, where it can take several minutes on a state-of-the-art desktop PC to geometrically optimise even the smallest fullerene C

_{20}. As we do not, in principle, know how many hydrogen atoms are likely to bind to a given isomer, we would be required to start with single hydrogenation and then increase the number of hydrogen atoms sequentially until the hydrogen binding energy becomes too low.

## 2. Method

_{0}= 15.875 Å to ensure negligible inter-fullerene interactions. Structures were geometrically optimised until the system energy was converged within 10

^{−5}Ha, and positions to within 10

^{−4}a

_{0}.

^{−10}convergence threshold (ratio of energy change to total energy magnitude), at zero Kelvin. The reactive empirical bond order (REBO) is a potential developed by Brenner [39]. REBO is handled by a Tersoff-type potential [42,43], which deals with the formation and deformation of covalent bonds during a calculation and was developed for hydrogen [44] and carbon [44,45]-containing systems. The adaptive intermolecular reactive empirical bond order (AIREBO) potential [40] was developed for carbon-hydrogen systems. It uses REBO to describe nonbonded interactions. In this study, we included its torsion term and the Lennard–Jones (LJ) cut-off radius set at 2.5 (in σ scale factor). AIREBO-M is a hybrid potential of AIREBO and Morse [41]. In this potential, a LJ term in AIREBO is replaced with a Morse potential parameterised to Møller–Plesset method (MP2) calculations [46,47]. In this study, we included its torsion term and the Morse cutoff radius set at 3.0 (in σ scale factor).

_{60}, xTB has already shown excellent structure–energy agreement with DFT results with Pearson’s correlation coefficient r = 0.998 [55]. Again, we used default settings including for convergence cut-offs.

## 3. Results

#### 3.1. Pure Carbon Fullerenes

_{x}fullerenes from x = 28 to x = 52, a total of 1246 structures. Figure 1 shows on the y-axis the energy difference between a given isomer and the most stable isomer for the same number of carbon atoms. On the x-axis, we give the number of atoms in the isomer neighbouring three pentagons. This plot confirms the general tendency that the stability of a given fullerene is negatively proportional to the number of triple-fused pentagons, although there are a number of interesting exceptions.

_{28}(1-T

_{d}and 2-D

_{2}, index of Yoshida’s library [26]-symmetry symbol) and C

_{40}(38-D

_{2}and 40-T

_{d}) (see Figure 2). These were chosen as the T

_{d}tetrahedral structures have four, the maximum number, triple-fused pentagons, and the other structures are the lowest-energy pure-carbon structures in each isomer group.

#### 3.2. First Hypothesis—Sequential Hydrogen Addition

_{3}groups [58,59]. This reduces the number of calculations substantially. To determine the stable structure of C

_{n}H

_{m}, we first geometrically optimised the system C

_{n}H with hydrogen in each of the n possible sites. We took the most stable solution and fixed the first hydrogen at this site, tested the n − 1 possible sites for the next hydrogen, and so on. As an example, for C

_{28}H

_{5}, this reduces the number of calculations from 98,280 to 130, and for C

_{50}H

_{4}from 230,300 to 194, with the additional benefit that the stable structures for C

_{n}H, C

_{n}H

_{2}, etc. are automatically determined as part of the process.

_{28}H

_{5}and C

_{40}H

_{5}.

_{28}H

_{5}and C

_{40}H

_{5}for the two T

_{d}and D

_{2}isomers. The energy axis shows the total binding energy of the hydrogen, calculated as

_{binding}= E(C

_{n}H

_{m}) − E(C

_{n}′) −

^{m}/

_{2}E(H

_{2})

_{n}is the fullerene in question and C

_{n}′ is the energetically most stable isomer fullerene. The projected Schlegel diagrams are marked numerically with the hydrogen addition sequence. T

_{d}-C

_{28}is sequentially hydrogenated at each triple pentagon on the tetrahedral corner sites, with the fifth hydrogenation occurring with a lower binding energy visible as a change in the total energy gradient. At each stage, this is the stable C

_{28}isomer. In contrast, the stable C

_{40}isomer is the lower-symmetry D

_{2}-C

_{40}. Subsequent hydrogenation occurs around the “caps” of this isomer. When the binding energies are normalised by the number of hydrogen atoms added, we see a general convergence towards hydrogen binding of ~1.6 eV for the D

_{2}-C

_{40}isomer, with higher binding (~2.0 eV) for the T

_{d}-C

_{28}, consistent with its smaller size and lower aromaticity.

_{d}-C

_{40}isomer than the D

_{2}-C

_{40}(although it does not, this time, adopt the triple-pentagon tetrahedrally symmetric sites). This lowers the relative energy difference between the isomers until, at C

_{40}H

_{5}, there is a transition and the T

_{d}-C

_{40}H

_{5}isomer becomes the most stable. This demonstrates that the most stable pure carbon isomer cannot be taken as a guide in general for which the hydrogenated isomer will be stable, i.e., wider isomer testing is required.

#### 3.3. Second Hypothesis—Predicting Reactive Sites via Local Geometry

_{28}H

_{5}, for example, from 130 to simply 5.

_{z}-orbital of the carbon atom is able to form a strong π-bond with one or more of its neighbours, typically through some measure of local curvature. In the following, we selected one geometric criterion and calculated its value for all nonhydrogenated sites on a given fullerene cage. The most extreme value (minimum or maximum, as discussed below) was taken as the next site to hydrogenate. The new structure was then geometrically optimised using DFT, and the process repeated. In this way, a hydrogenation sequence can be built up with geometric site selection at each step, and this sequence can be compared to the “lowest-energy” sequence derived in Section 3.2.

^{x}p

^{y}character in the system. For both of these parameters, we calculated the values for all potential binding sites and selected the maximum for hydrogenation at each step.

_{d}-C

_{40}H and T

_{d}-C

_{40}H

_{2}, in general, none of these geometric methods correctly select the lowest-energy hydrogenation sites, that is, none are capable of predicting the DFT-predicted most-stable structures.

#### 3.4. Third Hypothesis—Predicting Reactive Sites via Local Electronic Structure

_{z}-orbital character will release the most energy when hydrogenated. Alternatively, in cases where there is charge redistribution (notably on partially hydrogenated cages), sites showing excessive charging may also be targets for hydrogenation. We tested a number of electronic possibilities to select the next site for hydrogenation.

_{2}-C

_{28}in more detail, as it reveals another important result. It can be seen that when considering the lowest Frontier orbital atoms for hydrogenation, in the case of adding only 0.1e, for D

_{2}-C

_{28}H

_{3}and D

_{2}-C

_{28}H

_{5}, the addition sequence actually predicts hydrogenated isomers that are more stable than those obtained in Section 3.2. This is important as it invalidates our first hypothesis, namely that sequential hydrogen addition always generates the lowest-energy hydrogenated fullerenes. The implication is that if hydrogen can rearrange on the fullerene surface as further hydrogen atoms are added, this has the potential to lower the system energy. It is difficult to say whether kinetic barriers will limit this process under laboratory conditions; while calculations for single hydrogen migration suggest relatively high barriers [24], in practice, a range of barriers will exist depending on local geometry, degree of hydrogen surface coverage, and local environment. In interstellar environments where UV-photon absorption is the primary interaction event [14], structural rearrangement is likely.

#### 3.5. Second Strategy, Semi-Empirical and Empirical Routes

_{60}isomers, with Pearson’s correlation coefficient r = 0.998 [55].

_{40}H

_{5}in Figure 7, along with the corresponding Pearson’s correlation coefficient. Plots for two other fullerenes (C

_{28}H

_{5}and C

_{40}H) are shown in Figures S1 and S2.

_{40}H

_{5}, using PM7, PM6 + D3, and RM1, the structure predicted by DFT-LDA to be the most stable structure is only 20th on a list of 72). Thus, even if these methods were used to pre-screen suitable structures and a cut-off were applied (with full DFT-LDA geometric optimisation of only the most stable RM1 structures), the cut-off would necessarily have to be quite high, at least 30% of structures, in order to ensure that the lowest-energy DFT-LDA structure was included.

_{28}H

_{n}and C

_{40}H

_{n}, n = 1…5, calculated using DFT-LDA and GFN2-xTB. In each case, GFN2-xTB predicts the same structure ordering amongst the isomers tested, with very close energy differences (standard deviation of 0.087 eV).

_{20}on a 16-core desktop PC is summarised in Table 2 (along with the corresponding r-values from Figure 7). As expected, the empirical potentials are fastest but show no correlation to the DFT-LDA total energies. Of the semi-empirical methods, GFN2-xTB shows by far the best run time, as well as gives the best correlation to DFT-LDA. These results show that we can safely switch to using GFN2-xTB, as it is able to successfully generate DFT-LDA accuracy in total energies of hydrogenated smaller fullerenes, but at over 600 times less computational time.

#### 3.6. All Combination of Hydrogenation Sites Testing by xTB

_{d}-C

_{28}, D

_{2}-C

_{28}, T

_{d}-C

_{40}, and D

_{2}-C

_{40}. This brute force approach calculates the total energy for all hydrogen isomers of fullerenes C

_{n}H

_{m}, unlike the sequential test described in Section 3.2. As such, it provides a definitive lowest-energy hydrogenation sequence for C

_{n}H

_{m}, m = 0…4.

_{28}H

_{4}, the most stable structure found is the T

_{d}-C

_{28}H

_{4}shown in Figure 8 (Schlegel projection shown in Figure S1). This has tetrahedral symmetry, and corresponds to the structure found in the literature [23]. This is also the most stable structure found through sequential addition (using DFT-LDA, GFN2-xTB, PM7, PM6 + D3, and RM1). Energies of all the possible isomers are summarised in Figure 8. The single data point at the lowest energy lies significantly below the others, i.e., for C

_{28}H

_{4}, there is a single stable hydrogenated structure. The D

_{2}-C

_{28}isomer is significantly less stable; within the D

_{2}-C

_{28}H

_{4}subset of results, the most stable isomer consists of two hydrogen pairs at opposing ends of the molecule. This can be understood in terms of a localisation of strain, allowing a maximum flattening out and aromaticity in the two hexagon pairs along the fullerene sides. This structure is different, and more stable, than that found for D

_{2}-C

_{28}H

_{4}by sequential addition in Section 3.2 above (confirmed with DFT-LDA to be more stable by 0.21 eV).

_{n}H

_{m}structures for H = 1…4 for the two isomers are shown in Figure 9. The T

_{d}sequence for C

_{28}is largely sequential functionalisation of the triple pentagons, the exception being C

_{28}H

_{3}. In this case, the sequential structure has only a single unfunctionalised triple pentagon, which means the radical is highly localised, and hence, it is more stable to redistribute the hydrogen atoms. In contrast, the D

_{2}-C

_{28}lowest-energy series is not at all sequential, with different hydrogenation combinations balancing the relief of localised curvature at the fullerene “ends” with radical redistribution. This flux of different structures is also presumably indicative of the relative instability of this isomer compared to the T

_{d}.

_{40}is somewhat different. In this case, the most stable structures match exactly the series found through sequential addition in Section 3.2. For the T

_{d}-isomer, this is presumably because, in this case, the curvature at the triple pentagon is less localised than in C

_{28}due to the larger cage size, allowing more delocalisation of the radical for C

_{40}H

_{3}. For the D

_{2}isomer, hydrogenation localises the curvature at the fullerene ends, resulting in a cylindrical structure somewhat resembling a small-diameter capped carbon nanotube. The D

_{2}isomer is the most stable for all species up to C

_{40}H

_{4}. The relative energies show that hydrogenation is increasingly stabilising the T

_{d}isomer (and indeed, as shown above, at C

_{40}H

_{5}, there is an inversion in stability and the T

_{d}is more favoured). Further testing will be needed to determine whether, in larger fullerenes such as C

_{40}and above, the lowest-energy hydrogenated fullerenes always match the sequential addition patterns. If so, this would represent a significant computational benefit.

## 4. Discussion

^{2}value of ~1 (see Supplementary Figure S4). This also validates a choice of either LDA or GGA-Dx for the energetics of these systems. When we compare against hybrid functionals (B3LYP using CAM-B3LYP with the Def2TZVP basis set and GD3BJ dispersion corrections [68]) for the most and second-most stable isomers of T

_{d}-C

_{28}H

_{4}and D

_{2}-C

_{28}H

_{4}, we find the same qualitative energy ordering, but a slight drop in the order of 0.1 eV in the absolute energy differences (see Table S1 in Supplementary Materials). We have chosen to ignore thermal corrections to the energies in the current study as fullerene formation conditions in nonterrestrial environments are far from clear, as well as due to the high additional associated computational cost for vibrational analysis. Calculating thermal contributions with the hybrid functionals for these four fullerenes show that the thermal contribution to the relative stability between isomers is also of the order of ~0.1 eV, and again does not change the qualitative stability ordering of the isomers.

_{28}, the lowest-energy hydrogenated structure is T

_{d}-C

_{28}H

_{4}with tetrahedral hydrogenation; each triple pentagon is hydrogenated. This relaxes the angle strain localised on the tetrahedral corners and is consistent with literature observations. For the D

_{2}-C

_{28}isomer, there are two six-fused pentagons along its sides, and the curvature localised at the caps is partially relieved by pairwise hydrogenation at each cap. Nonetheless, the four hydrogen atoms appear insufficient to relax it and the structure is much less stable than the tetrahedral solution is.

_{40}, the T

_{d}isomer is less stable than the D

_{2}isomer. Instead, the D

_{2}isomer, in which there are no triple-fused pentagons but two six-pentagon chains, shows the lowest energy. When we look at the structure of C

_{40}H

_{4}, fused pentagon (pentagon chains) sites are indeed hydrogenated [3], not only on the triple pentagons. We suppose that for the larger fullerene cages, pentagon chain structures may play an important role to stabilise the whole structure of non-IPR species.

_{n}H

_{m}structures for other small fullerenes, with the eventual intention of mapping hydrogen-catalysed fullerene growth. For such large-scale studies, other methods beyond brute-force testing, such as genetic algorithms and Monte-Carlo-driven methods, will be required.

## Supplementary Materials

_{28}H

_{4}structures obtained by (a) GFN2-xTB all-combination test, PM7, PM6 + D3, RM1, GFN2-xTB, and sequential all-site testing by DFT, (b) hybridisation value and pyrA, (c) 2-bond-lengths sum and 3-bond-lengths sum, (d) Mulliken value, (e) minimum frontier value with 0.1e addition, (f) minimum frontier orbital value with 1e addition, (g) maximum frontier value with 0.1e addition, (h) maximum frontier value with 1e addition, (i) AIREBO, (j) REBO, and (k) AIREBO-M. Figure S2: Total energy correlation plots for isomers of C

_{28}H

_{5}. Energy obtained by DFT-LDA is plotted along the x-axis and by empirical and semi-empirical methods along the y-axes as labelled. Figure S3: Total energy correlation plots for isomers of C

_{40}H. Energy obtained by DFT-LDA is plotted along the x-axis and by empirical and semi-empirical methods along the y-axes as labelled. Figure S4. Correlation plot of total energy calculated using DFT-LDA against DFT-GGA for 741 hydrogenated fullerene species. Actual r = 0.99999987. Table S1: Comparison of the calculated total energy (eV) for the most stable C

_{28}H

_{4}isomer with the second-most stable T

_{d}-C

_{28}and most stable D

_{2}-C

_{28}-based C

_{28}H

_{4}isomer using GFN2-xTB and CAM-B3LYP.

## Author Contributions

## Funding

## Data Availability Statement

_{28}H

_{n}and C

_{40}H

_{n}structures and a complete library of GFN2-xTB optimised xyz files for all fullerene isomers from C20 to C80.

## Conflicts of Interest

## References

- García-Hernández, D.A.; Manchado, A.; García-Lario, P.; Stanghellini, L.; Villaver, E.; Shaw, R.A.; Szczerba, R.; Perea-Calderón, J.V. Formation of Fullerenes in H-Containing Planetary Nebulae. Astrophys. J. Lett.
**2010**, 724, L39. [Google Scholar] [CrossRef][Green Version] - García-Hernndez, D.A.; Iglesias-Groth, S.; Acosta-Pulido, J.A.; Manchado, A.; García-Lario, P.; Stanghellini, L.; Villaver, E.; Shaw, R.A.; Cataldo, F. The Formation of Fullerenes: Clues from New C
_{60}, C_{70}, and (Possible) Planar C_{24}Detections in Magellanic Cloud Planetary Nebulae. Astrophys. J. Lett.**2011**, 737, L30. [Google Scholar] [CrossRef][Green Version] - Bernard-Salas, J.; Cami, J.; Peeters, E.; Jones, A.P.; Micelotta, E.R.; Groenewegen, M.A.T. On the excitation and formation of circumstellar fullerenes. Astrophys. J.
**2012**, 757, 41. [Google Scholar] [CrossRef] - Otsuka, M.; Kemper, F.; Cami, J.; Peeters, E.; Bernard-Salas, J. Physical Properties of Fullerene-Containing Galactic Planetary Nebulae. Mon. Not. R. Astron. Soc.
**2014**, 437, 2577–2593. [Google Scholar] [CrossRef] - Campbell, E.K.; Holz, M.; Gerlich, D.; Maier, J.P. Laboratory Confirmation of C
_{60}(+) as the Carrier of Two Diffuse Interstellar Bands. Nature**2015**, 523, 322–323. [Google Scholar] [CrossRef] - Sellgren, K.; Werner, M.W.; Ingalls, J.G.; Smith, J.D.T.; Carleton, T.M.; Joblin, C. C
_{60}in Reflection Nebulae. Astrophys. J. Lett.**2010**, 722, L54. [Google Scholar] [CrossRef][Green Version] - Berné, O.; Cox, N.L.J.; Mulas, G.; Joblin, C. Detection of Buckminsterfullerene Emission in the Diffuse Interstellar Medium. Astron. Astrophys.
**2017**, 605, L1. [Google Scholar] [CrossRef] - Berné, O.; Tielens, A.G.G.M. Formation of Buckminsterfullerene (C
_{60}) in Interstellar Space. Proc. Natl. Acad. Sci. USA**2012**, 109, 401–406. [Google Scholar] [CrossRef] [PubMed][Green Version] - Candian, A.; Gomes Rachid, M.; MacIsaac, H.; Staroverov, V.N.; Peeters, E.; Cami, J. Searching for Stable Fullerenes in Space with Computational Chemistry. Mon. Not. R. Astron. Soc.
**2019**, 485, 1137–1146. [Google Scholar] [CrossRef] - Clayton, G.C.; Marco, O.D.; Whitney, B.A.; Babler, B.; Gallagher, J.S.; Nordhaus, J.; Speck, A.K.; Wolff, M.J.; Freeman, W.R.; Camp, K.A.; et al. The dust properties of two hot r coronae borealis stars and a wolf-rayet central star of a planetary nebula: In search of a possible link. Astron. J.
**2011**, 142, 54. [Google Scholar] [CrossRef] - Foing, B.H.; Ehrenfreund, P. Detection of Two Interstellar Absorption Bands Coincident with Spectral Features of C60+. Nature
**1994**, 369, 296–298. [Google Scholar] [CrossRef] - García-Hernndez, D.A.; Rao, N.K.; Lambert, D.L. Are C60 Molecules Detectable in Circumstellar Shells of R Coronae Borealis Stars? Astrophys. J.
**2011**, 729, 126. [Google Scholar] [CrossRef][Green Version] - Palotás, J.; Martens, J.; Berden, G.; Oomens, J. The Infrared Spectrum of Protonated Buckminsterfullerene C60H+. Nat. Astron.
**2020**, 4, 240–245. [Google Scholar] [CrossRef] - Omont, A.; Bettinger, H.F. Intermediate-Size Fullerenes as Degradation Products of Interstellar Polycyclic Aromatic Hydrocarbons. Astron. Astrophys.
**2021**, 650, A193. [Google Scholar] [CrossRef] - Kroto, H.W. The Stability of the Fullerenes C n, with n = 24, 28, 32, 36, 50, 60 and 70. Nature
**1987**, 329, 529–531. [Google Scholar] [CrossRef] - Xie, S.-Y.; Gao, F.; Lu, X.; Huang, R.-B.; Wang, C.-R.; Zhang, X.; Liu, M.-L.; Deng, S.-L.; Zheng, L.-S. Capturing the Labile Fullerene[50] as C
_{50}Cl_{10}. Science**2004**, 304, 699. [Google Scholar] [CrossRef] - Prinzbach, H.; Weiler, A.; Landenberger, P.; Wahl, F.; Wörth, J.; Scott, L.T.; Gelmont, M.; Olevano, D.; Issendorff, B.V. Gas-Phase Production and Photoelectron Spectroscopy of the Smallest Fullerene, C
_{20}. Nature**2000**, 407, 60–63. [Google Scholar] [CrossRef] - Hall, L.E.; McKenzie, D.R.; Attalla, M.I.; Vassallo, A.M.; Davis, R.L.; Dunlop, J.B.; Cockayne, D.J.H. The Structure of Hydrogenated Fullerene (C60H36). J. Phys. Chem.
**1993**, 97, 5741–5744. [Google Scholar] [CrossRef] - Cataldo, F. Fullerane, the Hydrogenated C60 Fullerene: Properties and Astrochemical Considerations. Fuller. Nanotub. Carbon Nanostruct.
**2003**, 11, 295–316. [Google Scholar] [CrossRef] - Iglesias-Groth, S.; García-Hernández, D.A.; Cataldo, F.; Manchado, A. Infrared Spectroscopy of Hydrogenated Fullerenes (Fulleranes) at Extreme Temperatures. Mon. Not. R. Astron. Soc.
**2012**, 423, 2868–2878. [Google Scholar] [CrossRef][Green Version] - Goldshleger, N.F.; Moravsky, A.P. Fullerene Hydrides: Synthesis, Properties, and Structure. Russ. Chem. Rev.
**1997**, 66, 323. [Google Scholar] [CrossRef] - Zhang, Y.; Sadjadi, S.; Hsia, C.-H.; Kwok, S. Search for Hydrogenated C
_{60}(Fulleranes) in Circumstellar Envelopes. Astrophys. J.**2017**, 845, 76. [Google Scholar] [CrossRef][Green Version] - Veljković, M.; Nešković, O.; Djerić, A.; Veličković, S.; Šipka, V. Hypervalent Molecular Cluster: C
_{28}H_{4}. Mater. Sci. Forum**2005**, 494, 181–186. [Google Scholar] [CrossRef] - EL-Barbary, A.A. Hydrogenation Mechanism of Small Fullerene Cages. Int. J. Hydrog. Energy
**2016**, 41, 375–383. [Google Scholar] [CrossRef] - Fowler, P.W.; Manolopoulos, D.E. An Atlas of Fullerenes; Clarendon Press: Oxford, UK, 1995; ISBN 0-19-855787-6. [Google Scholar]
- Yoshida, M. Yoshida’s Fullerene Library. Available online: http://www.jcrystal.com/steffenweber/gallery/Fullerenes/Fullerenes.html (accessed on 18 November 2020).
- Sabalot-Cuzzubbo, J.; Salvato-Vallverdu, G.; Bégué, D.; Cresson, J. Relating the Molecular Topology and Local Geometry: Haddon’s Pyramidalization Angle and the Gaussian Curvature. J. Chem. Phys.
**2020**, 152, 244310. [Google Scholar] [CrossRef] - Rayson, M.J.; Briddon, P.R. Highly Efficient Method for Kohn-Sham Density Functional Calculations of 500–10000 Atom Systems. Phys. Rev. B Condens. Matter Mater. Phys.
**2009**, 80, 205104. [Google Scholar] [CrossRef] - Rayson, M.J. Rapid Filtration Algorithm to Construct a Minimal Basis on the Fly from a Primitive Gaussian Basis. Comput. Phys. Commun.
**2010**, 181, 1051–1056. [Google Scholar] [CrossRef] - Briddon, P.R.; Rayson, M.J. Accurate Kohn–Sham DFT with the Speed of Tight Binding: Current Techniques and Future Directions in Materials Modelling. Phys. Status Solidi
**2011**, 248, 1309–1318. [Google Scholar] [CrossRef] - Hartwigsen, C.; Goedecker, S.; Hutter, J. Relativistic Separable Dual-Space Gaussian Pseudopotentials from H to Rn. Phys. Rev. B
**1998**, 58, 3641–3662. [Google Scholar] [CrossRef][Green Version] - Kohn, W.; Sham, L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev.
**1965**. [Google Scholar] [CrossRef][Green Version] - Briddon, P.R.; Jones, R. LDA Calculations Using a Basis of Gaussian Orbitals. Phys. Status Solidi
**2000**, 217, 131–171. [Google Scholar] [CrossRef] - Latham, C.D.; Heggie, M.I.; Alatalo, M.; Öberg, S.; Briddon, P.R. The Contribution Made by Lattice Vacancies to the Wigner Effect in Radiation-Damaged Graphite. J. Phys. Condens. Matter
**2013**, 25, 135403. [Google Scholar] [CrossRef] - Nekovee, M.; Foulkes, W.M.C.; Needs, R.J. Quantum Monte Carlo Investigations of Density Functional Theory of the Strongly Inhomogeneous Electron Gas. Phys. Rev. B
**2003**, 68, 235108. [Google Scholar] [CrossRef][Green Version] - Björkman, T.; Gulans, A.; Krasheninnikov, A.V.; Nieminen, R.M. Are We van Der Waals Ready? J. Phys. Condens. Matter
**2012**, 24, 424218. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hof, F.; Impellizzeri, A.; Picheau, E.; Che, X.; Pénicaud, A.; Ewels, C.P. Chainlike Structure Formed in Iodine Monochloride Graphite Intercalation Compounds. J. Phys. Chem. C
**2021**. [Google Scholar] [CrossRef] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef][Green Version] - Brenner, D.W.; Shenderova, O.A.; Harrison, J.A.; Stuart, S.J.; Ni, B.; Sinnott, S.B. A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons. J. Phys. Condens. Matter
**2002**, 14, 783–802. [Google Scholar] [CrossRef] - Stuart, S.J.; Tutein, A.B.; Harrison, J.A. A Reactive Potential for Hydrocarbons with Intermolecular Interactions. J. Chem. Phys.
**2000**, 112, 6472–6486. [Google Scholar] [CrossRef][Green Version] - O’Connor, T.C.; Andzelm, J.; Robbins, M.O. AIREBO-M: A Reactive Model for Hydrocarbons at Extreme Pressures. J. Chem. Phys.
**2015**, 142, 024903. [Google Scholar] [CrossRef] [PubMed] - Tersoff, J. New Empirical Approach for the Structure and Energy of Covalent Systems. Phys. Rev. B
**1988**, 37, 6991–7000. [Google Scholar] [CrossRef] - Tersoff, J. Modeling Solid-State Chemistry: Interatomic Potentials for Multicomponent Systems. Phys. Rev. B
**1989**, 39, 5566–5568. [Google Scholar] [CrossRef] - Brenner, D.W. Tersoff-Type Potentials for Carbon, Hydrogen and Oxygen. MRS Online Proc. Libr. (OPL)
**1988**, 141, 59–64. [Google Scholar] [CrossRef] - Tersoff, J. Empirical Interatomic Potential for Carbon, with Applications to Amorphous Carbon. Phys. Rev. Lett.
**1988**, 61, 2879–2882. [Google Scholar] [CrossRef] [PubMed][Green Version] - Møller, C.; Plesset, M.S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev.
**1934**, 46, 618–622. [Google Scholar] [CrossRef][Green Version] - Dunning, T.H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys.
**1989**, 90, 1007–1023. [Google Scholar] [CrossRef] - Stewart, J.J.P. MOPAC2016; Stewart Computational Chemistry: Colorado Springs, CO, USA, 2016. [Google Scholar]
- Stewart, J.J.P. Optimization of Parameters for Semiempirical Methods V: Modification of NDDO Approximations and Application to 70 Elements. J. Mol. Modeling
**2007**, 13, 1173–1213. [Google Scholar] [CrossRef][Green Version] - Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys.
**2010**, 132, 154104. [Google Scholar] [CrossRef][Green Version] - Stewart, J.J.P. Optimization of Parameters for Semiempirical Methods VI: More Modifications to the NDDO Approximations and Re-Optimization of Parameters. J. Mol. Modeling
**2013**, 19, 1–32. [Google Scholar] [CrossRef][Green Version] - Rocha, G.B.; Freire, R.O.; Simas, A.M.; Stewart, J.J.P. RM1: A Reparameterization of AM1 for H, C, N, O, P, S, F, Cl, Br, and I. J. Comput. Chem.
**2006**, 27, 1101–1111. [Google Scholar] [CrossRef] - Dewar, M.J.S.; Zoebisch, E.G.; Healy, E.F.; Stewart, J.J.P. Development and Use of Quantum Mechanical Molecular Models. 76. AM1: A New General Purpose Quantum Mechanical Molecular Model. J. Am. Chem. Soc.
**1985**, 107, 3902–3909. [Google Scholar] [CrossRef] - Bannwarth, C.; Caldeweyher, E.; Ehlert, S.; Hansen, A.; Pracht, P.; Seibert, J.; Spicher, S.; Grimme, S. Extended Tight-Binding Quantum Chemistry Methods. Wiley Interdiscip. Rev. Comput. Mol. Sci.
**2021**, 11, e1493. [Google Scholar] [CrossRef] - Sure, R.; Hansen, A.; Schwerdtfeger, P.; Grimme, S. Comprehensive Theoretical Study of All 1812 C60 Isomers. Phys. Chem. Chem. Phys.
**2017**, 19, 14296–14305. [Google Scholar] [CrossRef] - Van Lier, G.; Cases, M.; Ewels, C.P.; Taylor, R.; Geerlings, P. Theoretical Study of the Addition Patterns of C60 Fluorination: C60Fn (n = 1−60). J. Org. Chem.
**2005**, 70, 1565–1579. [Google Scholar] [CrossRef] - Vlandas, A.; Ewels, C.P.; Lier, G.V. Controlling Fullerene Addition Sequences, Regioselectivity and Magic Numbers via Metal Encapsulation. Chem. Commun.
**2011**, 47, 7051–7053. [Google Scholar] [CrossRef] [PubMed] - Nakagawa, A.; Nishino, M.; Niwa, H.; Ishino, K.; Wang, Z.; Omachi, H.; Furukawa, K.; Yamaguchi, T.; Kato, T.; Bandow, S.; et al. Crystalline Functionalized Endohedral C60 Metallofullerides. Nat. Commun.
**2018**, 9, 3073. [Google Scholar] [CrossRef] - Ewels, C.; Rio, J.; Niwa, H.; Omachi, H.; Shinohara, H.; Rayson, M.; Briddon, P. Determining Addition Pathways and Stable Isomers for CF
_{3}Functionalization of Endohedral [email protected]_{60}. R. Soc. Open Sci.**2018**, 5, 180588. [Google Scholar] [CrossRef][Green Version] - Haddon, R.C. C
_{60}: Sphere or Polyhedron? J. Am. Chem. Soc.**1997**, 119, 1797–1798. [Google Scholar] [CrossRef] - Haddon, R.C.; Scott, L.T. π-Orbital conjugation and rehybridization in bridged annulenes and deformed molecules in general: π-orbital axis vector analysis. Pure Appl. Chem.
**1986**, 58, 137–142. [Google Scholar] [CrossRef] - Haddon, R.C. Hybridization and the Orientation and Alignment of .Pi.-Orbitals in Nonplanar Conjugated Organic Molecules: .Pi.-Orbital Axis Vector Analysis (POAV2). J. Am. Chem. Soc.
**1986**, 108, 2837–2842. [Google Scholar] [CrossRef] - Haddon, R.C. Comment on the Relationship of the Pyramidalization Angle at a Conjugated Carbon Atom to the σ Bond Angles. J. Phys. Chem. A
**2001**, 105, 4164–4165. [Google Scholar] [CrossRef] - Mulliken, R.S. Electronic Population Analysis on LCAO–MO Molecular Wave Functions. I. J. Chem. Phys.
**1955**, 23, 1833–1840. [Google Scholar] [CrossRef][Green Version] - Fukui, K.; Yonezawa, T.; Shingu, H. A Molecular Orbital Theory of Reactivity in Aromatic Hydrocarbons. J. Chem. Phys.
**1952**, 20, 722–725. [Google Scholar] [CrossRef] - von Ragué Schleyer, P.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N.J.R. Nucleus-Independent Chemical Shifts: A Simple and Efficient Aromaticity Probe. J. Am. Chem. Soc.
**1996**, 118, 6317–6318. [Google Scholar] [CrossRef] [PubMed] - van Lier, G.; De Proft, F.; Geerlings, P. Ab Initio Study of the Aromaticity of Hydrogenated Fullerenes. Phys. Solid State
**2002**, 44, 588–592. [Google Scholar] [CrossRef] - Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Petersson, G.A.; Nakatsuji, H.; et al. Gaussian 16, Revision C.01; Gaussian, Inc.: Wallingford, CT, USA, 2016. [Google Scholar]

**Figure 1.**Plot of the number of atoms neighboured by triple pentagons versus energy difference compared to the ground state isomer. The structure instability is, in general, proportional to the number of triple pentagons. For the optimal lowest energy isomer of C

_{x}, ΔE = 0.

**Figure 2.**Structures for all site test. (

**a**) C

_{28}(1-T

_{d}), (

**b**) C

_{28}(2-D

_{2}), (

**c**) C

_{40}(38-D

_{2}), and (

**d**) C

_{40}(40-T

_{d}).

**Figure 3.**DFT-LDA hydrogen binding energy relative to the most stable isomer and isolated H

_{2}(eV) for (

**a**) C

_{28}and (

**b**) C

_{40}obtained by sequential hydrogenation. The T

_{d}(D

_{2}) isomer is given in blue (orange); symmetry notation refers to nonhydrogenated fullerene cage. (

**c**–

**f**) 3D molecular structures and Schlegel diagrams for C

_{n}H

_{5}. Grey and white represent carbon and hydrogenated carbon atoms, respectively. In the Schlegel diagrams, white circles indicate hydrogenated sites, and hexagons are shaded in grey. Numbering in Schlegel diagrams indicates the hydrogenation order.

**Figure 4.**Schematic diagram of pyrA, defined as θ-π/2. POAV is the vector along the atom p

_{z}orbital and is defined as having a constant angle with all three carbon-carbon bonds. Blue spheres represent carbon atoms.

**Figure 5.**Plots of the difference in calculated total energy via DFT-LDA between hydrogenated fullerene structures, with hydrogen site selection based on geometric parameters (pyrA and hybridisation, and sum of two- or three-bond lengths), compared to the lowest-energy hydrogenated isomer found through sequential addition from Section 3.2 Structures are C

_{28}(1-T

_{d}and 2-D

_{2}) and C

_{40}(38-D

_{2}and 40-T

_{d}).

**Figure 6.**Plots of the difference in calculated total energy via DFT-LDA between hydrogenated fullerene structures, with hydrogenation sites selected based on the electronic parameters (Mulliken charges or Frontier orbital values) compared to the hydrogenated structures obtained in Section 3.2, for C

_{28}(1-T

_{d}and 2-D

_{2}) and C

_{40}(38-D

_{2}and 40-T

_{d}).

**Figure 7.**Total energy correlation plots for isomers of C

_{40}H

_{5}. Energy obtained by DFT-LDA is plotted along the x-axis and by empirical and semi-empirical methods along the y-axes as labelled. Linear correlation in each model is shown as r in each plot.

**Figure 8.**Calculated total energy of all the possible (

**a**) C

_{28}H

_{4}and (

**b**) C

_{40}H

_{4}for T

_{d}and D

_{2}isomers obtained by GFN2-xTB, with lowest-energy structures labelled.

**Figure 9.**The lowest-energy C

_{n}H

_{m}structures for n = 28 and 40, and m = 1…4 for the T

_{d}and D

_{2}isomers calculated using GFN2-xTB, testing all symmetry-inequivalent possibilities. Grey and white in molecular structures represent carbon and hydrogen atoms, respectively. Blue arrow indicates hydrogenation in a sequential route, while orange arrow indicates nonsequential hydrogenation.

**Table 1.**Comparison of energy difference between lowest energy and second-lowest energy isomer in the calculated series, for C

_{28}H

_{n}and C

_{40}H

_{n}, n = 1…5, calculated at DFT-LDA and GFN2-xTB levels of theory (eV). The difference between these two values and the associated standard deviation is also given.

DFT-LDA [eV] | GFN2-xTB [eV] | Difference [eV] | |
---|---|---|---|

C_{28}H | 0.3494 | 0.4033 | 0.0539 |

C_{28}H_{2} | 0.3570 | 0.4026 | 0.0456 |

C_{28}H_{3} | 0.3617 | 0.4553 | 0.0936 |

C_{28}H_{4} | 0.3319 | 0.5459 | 0.2140 |

C_{28}H_{5} | 0.3210 | 0.2832 | −0.0378 |

C_{40}H | 0.1137 | 0.0834 | −0.0303 |

C_{40}H_{2} | 0.4171 | 0.4094 | −0.0078 |

C_{40}H_{3} | 0.0375 | 0.0247 | −0.0127 |

C_{40}H_{4} | 0.4699 | 0.3674 | −0.1025 |

C_{40}H_{5} | 0.0008 | 0.0000 | −0.0008 |

Standard Deviation = | 0.0869 |

**Table 2.**Comparison of empirical and semi-empirical methods with DFT. Run time is the average of 10 runs for pure C

_{20}. r is Pearson’s correlation coefficient from.

Method | AIREBO | REBO | AIREBO-M | PM7 | PM6 + D3 | RM1 | xTB | DFT |
---|---|---|---|---|---|---|---|---|

Run time | 0.039 s | 0.039 s | 0.043 s | 1.33 s | 0.95 s | 0.91 s | 0.34 s | 210 s |

r | −0.1379 | 0.3760 | 0.3453 | 0.7500 | 0.7517 | 0.7661 | 0.9753 | - |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tanuma, Y.; Maekawa, T.; Ewels, C. Methodological Investigation for Hydrogen Addition to Small Cage Carbon Fullerenes. *Crystals* **2021**, *11*, 1334.
https://doi.org/10.3390/cryst11111334

**AMA Style**

Tanuma Y, Maekawa T, Ewels C. Methodological Investigation for Hydrogen Addition to Small Cage Carbon Fullerenes. *Crystals*. 2021; 11(11):1334.
https://doi.org/10.3390/cryst11111334

**Chicago/Turabian Style**

Tanuma, Yuri, Toru Maekawa, and Chris Ewels. 2021. "Methodological Investigation for Hydrogen Addition to Small Cage Carbon Fullerenes" *Crystals* 11, no. 11: 1334.
https://doi.org/10.3390/cryst11111334