Aromaticity, Coulomb repulsion, π delocalization or strain: who is who in endohedral metallofullerene stability?

Yang Wang *abc, Sergio Díaz-Tendero bcd, Manuel Alcamí bce and Fernando Martín *bdef
aSchool of Chemistry and Chemical Engineering, Yangzhou University, Yangzhou, Jiangsu 225002, China. E-mail:
bDepartamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail:
cInstitute for Advanced Research in Chemical Sciences (IAdChem), Universidad Autónoma de Madrid, 28049 Madrid, Spain
dCondensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain
eInstituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain
fDonostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain

Received 29th October 2018 , Accepted 23rd November 2018

First published on 23rd November 2018

Endohedral metallofullerenes (EMFs) synthetized in the laboratory are known to often violate the isolated pentagon and pentagon adjacency penalty rules that successfully describe the relative stability of pristine fullerene isomers. To explain these anomalies, several models have been proposed. In this work, we have systematically investigated the performance of the widely used IPSI (inverse pentagon separation index), ALA (additive local aromaticity) and CSI (charge stabilization index) models in predicting the relative stability of a large number of EMF isomers with cages ranging from C28 to C104 and charge states of 4− and 6−. By explicitly comparing with existing experiments and quantum chemistry calculations, we show that the predictive power of the ALA and CSI models is similarly good, with CSI being slightly superior though computationally much less involved. IPSI's performance is generally worse though still acceptable in a wide range of cage sizes, except for the higher charge states in the C62 to C82 size interval. From our analysis, we conclude that neither Coulomb electronic repulsion (IPSI) nor aromaticity (ALA) are the sole parameters governing the relative stability of EMF isomers. Electron delocalization in the π shell in combination with minimum strain (CSI) provides a more realistic description of the relative stabilities observed experimentally, as the former can compensate an unfavorable Coulomb repulsion and account for stabilizing binding effects that do not necessarily translate into aromaticity.

1 Introduction

Endohedral metallofullerenes (EMFs)1,2 have received a great deal of attention in recent years due to their exclusive characteristics among carbon-based compounds.3 In EMFs, the encapsulated species, usually a metal atom or a molecule, or a cluster containing one or several metal atoms, transfers electrons to the fullerene cage, which is thus negatively charged and acquires electronic and magnetic properties that are nonexistent in pristine neutral fullerenes. As a consequence, EMFs are considered good candidates for potential new applications in different fields of science,4–8 as for example organic photovoltaics and biomedicine. In the latter case, they have been proposed as possible antitumor9 and antimicrobial10 drugs, as contrast agents in magnetic resonance5 and X-ray imaging,11 and as radiotracers and radiopharmaceuticals.12

Most EMFs are synthesized in the lab in very hot environments, e.g., through arc-discharges,13 combustion14,15 or radio-frequencies.16,17 In such environments, cages are formed through carbon-clustering processes18 at temperatures typically higher than 4000 K19 and thus the final products usually correspond to the global minima of the potential energy surface. In this way, carbon cages with sizes ranging from C2820 to C13021 have been produced in very diverse isomeric forms. Negatively charged fullerene cages can also be synthesized by directly attaching electrons to pristine neutral fullerenes, either in the gas phase22–24 or in solution.25,26

Spectroscopic measurements1,2 and quantum chemistry calculations27,28 have shown that the cage structures of synthesized EMFs often violate the well established isolated pentagon (IPR)29 and pentagon adjacency penalty (PAPR)30,31 rules. These rules state that pentagonal rings (hereafter called “pentagons”) tend to separate from each other to minimize strain in the fullerene cage, which is the dominant effect that controls the relative isomer stability of pristine neutral fullerenes. Therefore, it is clear that, to understand the frequent appearance of non-IPR or PAPR-violating isomers in synthesized EMFs, apart from strain, one should take into account other factors. With this aim, many different models have been proposed in the literature.32–38

In this work, we leave apart models designed for specific EMFs and will focus on those developed to predict the most stable structures of negatively charged fullerene isomers in a wide range of cage sizes and charge states. It has been found that the relative isomer stability of EMFs generally coincides with that of the pristine fullerene anions in the appropriate charge states.1,27 Such “general” models, however, stand on quite different physical grounds. Relying on the assumption that negative charges mainly concentrate on the pentagonal rings of the EMF cage, Poblet et al.35,36 have suggested that the larger the separation between the pentagons, the larger the stability. This simple electrostatic picture has been quantified by means of the inverse pentagon separation index (IPSI), which is calculated from the equilibrium geometry of a charged fullerene cage. In contrast, Solà et al.37,38 have found that the relative isomer stability of fullerene anions is mainly governed by the total aromaticity of the carbon cage. They proposed the additive local aromaticity (ALA) index to successfully predict the most stable hosting cages of a variety of EMFs in different charge states. In practice, the ALA is evaluated on the basis of the harmonic oscillator model of aromaticity (HOMA)39 index of each ring, which can be readily calculated using the equilibrium geometry of the cage. At variance with these approaches, more recently,40,41 we have proposed that the relative stability of EMF isomers results from the interplay between the stabilizing effect of π delocalization and the destabilizing effect of σ strain. Both effects have been taken into account through the charge stabilization index (CSI), which is exclusively based on simple topological arguments, namely the carbon–carbon connectivity in the fullerene cage, which does not require the knowledge of the actual EMF geometry. The CSI model has successfully predicted the cage structures of many charged fullerenes and EMFs observed in experiments or determined from elaborate quantum chemistry calculations.

Although the IPSI, ALA and CSI models are now well established and can be routinely applied to predict the lowest-energy structures of EMFs by using available computational tools (e.g., semiempirical quantum chemistry42–44 or density functional tight-binding45 methods), we do not know yet how good they are in providing the correct stability hierarchy within a large set of EMF isomers. As these models rely on very different assumptions, this is a critical point to determine which one provides a better representation of the physical reality.46 Another important point worth exploring is the performance of the CSI model against the IPSI and ALA models, since the former, not requiring any geometry optimization, is more affordable than the other two, thus permitting exploration of the large sets of EMF isomers.

In this work, we present a systematic application of the IPSI, ALA and CSI models to a large number of EMF isomers with cages ranging from C28 to C104 and charge states of 4− and 6−. By explicitly comparing with existing experimental results and quantum chemistry calculations performed for the occasion, we show that the predictive power of the ALA and CSI models is, in general, very good in the whole range of sizes investigated, with CSI being superior in some cases. The IPSI model exhibits a similar good predictive power except for the highest charge state in the C62 to C82 size interval, where errors can be up to three times larger than for the other two methods. Our analysis shows that Coulomb repulsion and aromaticity are two partial manifestations of π delocalization occurring in the fullerene cage, and that other effects resulting from this delocalization not included in the IPSI and ALA models may be crucial in determining the stability of the EMFs in some cases. These are correctly described by the CSI model.

The paper is organized as follows. In Section 2, we briefly describe the three methods, the basic equations on which they are based and the mathematical algorithm used to calculate the corresponding prediction error. In Section 3, we present the results of this comparative study and in Section 4, we summarize the main conclusions of our work.

2 Theoretical methods

We start by briefly describing the basic assumptions and mathematical formulae for the IPSI, ALA and CSI models.

2.1 IPSI

The IPSI35,36 for a given cage isomer of fullerene anions is calculated as
image file: c8cp06707d-t1.tif(1)
where Rps is the distance between the centroids of pentagons p and s, and the summations run over all possible pairs of pentagons. The Rps values (in units of Å) are computed based on the optimized geometry of a fullerene cage. According to its definition, the larger the IPSI value, the larger the Coulomb repulsion and hence the less relative stability of a fullerene isomer. It is important to note that the comparison of IPSI with other models can only be made either in between IPR isomers, or in between non-IPR isomers containing the same number of adjacent pentagon pairs (APPs).35

2.2 ALA

The ALA index is defined as
image file: c8cp06707d-t2.tif(2)
where Ar is the local aromaticity of ring r (being either a pentagon or a hexagon), and Nring is the total number of rings in the fullerene cage. The simplest yet most effective way to evaluate the local aromaticity Ar is to use the harmonic oscillator model of aromaticity (HOMA).39 The HOMA index of a given n-membered ring is computed using the following formula:
image file: c8cp06707d-t3.tif(3)
where db is the length of the C–C bond b (in units of Å) in the ring, 1.388 is the optimal value of the C–C bond length (in Å) for an ideal system of maximum aromaticity (HOMA = 1), and 257.7 is the normalization factor that gives HOMA = 0 for a model nonaromatic system. As we can see, the calculation of the ALA index using the HOMA as local ring aromaticity also requires the equilibrium geometry of the fullerene cage.

2.3 CSI

The charge stabilization index (CSI)40 of a given negatively charged fullerene isomer i with charge q and size 2n is defined as
CSIqiXqi + 0.2NAPPi,(4)
where NAPPi is the number of APPs of cage isomer i and
image file: c8cp06707d-t4.tif(5)
where χk,i is the k-th eigenvalue of the carbon–carbon connectivity matrix, {alm}, in descending order. The alm elements are equal to 1 if carbon atoms l and m are bonded, and 0 otherwise. According to Hückel theory, Xqi represents the variation of π energy, in units of −2β, due to addition of |q| electrons to the frontier π orbitals (LUMOs) of the corresponding neutral isomer i (β is the Hückel resonance integral). Therefore, it represents the contribution of π delocalization to the stabilization of the system. The coefficient 0.2 in eqn (4) is a characteristic value of the energy penalty per APP (in units of −2β), which falls in the accepted range established in previous work.31,47 Therefore, the quantity 0.2NAPPi is an approximation to the strain energy as dictated by the PAPR rule. Hence, differences in the CSI index can be regarded as differences in the isomer energy, in units of −2β. All CSI calculations have been performed by using the FullFun software package.41

2.4 Electronic structure calculations and geometry optimizations

All initial geometries of fullerene isomers were generated by the CaGe program.49 The conventional labeling50 of isomers with their highest possible symmetries was determined using the FullFun software package,41 on the basis of the database in the Fullerene program.51

All energy calculations and geometry optimizations (no symmetry constraint imposed) were performed at the self-consistent charge density functional tight-binding (SCC-DFTB)45 level, using the DFTB+ (version 1.2) code.52 As demonstrated in previous work,40,48,53,54 the SCC-DFTB method describes reasonably well the relative isomer energies of both neutral and charged fullerenes: it predicts correctly the experimental cage structures in EMFs as the lowest-energy isomers, and also gives similar results to those obtained from DFT calculations (B3LYP using 6-31G(d) or 6-31+G(d) basis sets).48

Due to the large computational effort required, we have only considered singlet spin multiplicity in all calculations. To validate this assumption, we have compared the energies of singlet and triplet states for a considerable number of isomers of C2nq with 2n = 68–104 and q = 4−, 6−, at the B3LYP/6-31G(d) level using the Gaussian 09 package.55 It turns out48 that for all the considered systems of C2n6−, the singlet state is more stable than the triplet one. This is also true for most C2n4− fullerene cages, except for a very few cases where the triplet state has slightly (typically 2–3 kcal mol−1) lower energy. Therefore, we believe that the spin multiplicity does not play an important role here.

2.5 Prediction error

In order to compare the performance of the IPSI, ALA and CSI models in predicting the relative stability of fullerene anions, we introduce the quantity PE (prediction error), which measures how much the isomer energy ordering predicted by a given model deviates from that resulting from the SCC-DFTB calculations. The prediction error is evaluated as follows:
image file: c8cp06707d-t5.tif(6)
where Ei is the SCC-DFTB energy of isomer i, ER(i) is that of isomer R(i), R(i) is the ranking number (in ascending order of energy) for isomer i as predicted by a given indicator (IPSI, ALA or CSI), and Niso is the total number of isomers within a reasonable energy window (since highly unstable isomers are less significant and are correctly filtered out by all models). We use an energy window of EminEi ≤ 30 kcal mol−1, where Emin is the SCC-DFTB energy of the most stable isomer for a given cage size and charge state. Since we apply the three prediction models to IPR and non-IPR isomers separately,35Emin is accordingly the lowest energy among all IPR isomers or among all non-IPR isomers. This energy window typically contains a sufficiently large number of isomers for a statistically meaningful analysis (e.g., for C1024−, the number of analyzed isomers is 469). According to eqn (6), for a perfect prediction, one must have R(i) = i for all i, and therefore PE = 0. A larger value of PE indicates a larger error in the prediction of the relative stability of fullerene anions.

3 Results

We have applied the IPSI, ALA and CSI models to study the relative stability of fullerene anions C2nq with 2n = 28–104 and q = 4−, 6−. The prediction errors, evaluated as defined in eqn (6), are shown in Fig. 1. For the sake of clarity and the requirement that the ISPI indicator needs to be applied separately to IPR and non-IPR isomers, we show the results corresponding to IPR and non-IPR isomers in separate panels. Fig. 1 shows that, with very few exceptions, prediction errors for the ALA and CSI models are of the order of 5 kcal mol−1 in the whole size range. This is quite remarkable in the case of the CSI calculations, in view of the simplicity of the model. Notably, ALA performs significantly worse than CSI for C304−, C664−, C764−, C784−, C346− and C466−. Typical prediction errors for IPSI are slightly larger than for ALA and CSI, and significantly larger, up to 10 kcal mol−1 or more, for C664−, C1006− and C346−, and all fullerenes with q = −6 in the C626−–C826− size interval. Thus, the problem with IPSI seems to be more severe for medium-sized fullerenes with the highest charge. This means that, for these systems, charge distribution is not the dominant effect at play. The good performance of the ALA and CSI models in these particular cases indicates that stability is mainly the consequence of the π electronic structure, which is well described by both models at different levels of approximation (HOMA versus Hückel). This observation is in line with the connections between electron delocalization, bond length and aromaticity, as recently revealed56,57 within the context of quantum theory of atoms in molecules.58,59
image file: c8cp06707d-f1.tif
Fig. 1 Prediction error for the CSI, IPSI and ALA indicators for fullerene anions C2nq (2n = 28–104; q = 4−, 6−): (a) IPR and (b) non-IPR isomers with q = 4−; (c) IPR and (d) non-IPR isomers with q = 6−.

One may notice in Fig. 1a and c that the ALA predictions are slightly better than the CSI ones for the larger IPR cages (2n ≥ 94). This is due to the fact that in the latter model, the π and strain energies are estimated in a simple and thus rather approximate manner (based solely on the atomic connectivity), while the ALA indicator is evaluated using the actual molecular geometry (optimized by quantum chemistry calculations). In view of this, we have applied the improved CSI model recently reported in ref. 41 to the IPR cages (C2nq with 2n = 70–104 and q = − 4, −6) by including a structural motif model60 to account for the relative strain energy of IPR isomers. As shown in Fig. S1 of the ESI, the overall performance of the improved CSI model is better than that of the original CSI model, and becomes comparable with, and in some cases even better than, that of the ALA model.

Originally, the IPSI model was introduced to predict the relative isomer stability of fullerenes containing the same number of APPs.35 For the sake of simplicity, however, this distinction has not been made in Fig. 1. To prove that the above conclusion does not change when we only compare fullerene isomers containing the same number of APPs, we have also evaluated the ALA and CSI indicators for NAPP = 1, 2 and 3 separately and compared the corresponding prediction errors with those of IPSI. The results for C2n6− (2n = 66–82) with NAPP = 1, 2 and 3 are respectively shown in Fig. S2–S4 of the ESI. For isomers with NAPP = λ (λ being 1, 2 and 3), the energy window of 30 kcal mol−1 has been defined by choosing Emin as the lowest energy among all isomers with λ APPs. This more refined analysis leads essentially to the same conclusion as that obtained from Fig. 1: IPSI leads in general to a significantly larger prediction error than the CSI and ALA indicators.

To gain a deeper insight into the reasons for the different performances of the three models, we show correlations of the CSI index and its π contribution, Xqi, with the ALA and IPSI indexes Fig. 2 and 3. We have chosen to show such diagrams for a case in which all three models work reasonably well, C806−, and a case in which CSI is clearly superior to the other two models, C346−. We have considered all isomers containing at most three APPs. As can be seen, for C806− (panel a), there is a very good correlation between CSI and ALA results irrespective of the number of APPs contained in the isomer. Correlation between the π contribution of CSI and ALA is also reasonable (panel b), although here we rather see a correlation line per group of isomers according to the number of APPs. The latter behavior is the consequence of the absence of strain in the Xqi index. This comparison shows that both π delocalization and strain are reasonably described by ALA and CSI, which is the reason why the corresponding prediction errors are very similar. In contrast, similar plots comparing CSI and IPSI (panels c and d) show that correlations are only observed between groups of isomers, irrespective of whether strain is included or not in the CSI model. This suggests that Coulomb repulsion due to the charge distribution resulting from π delocalization is not the only effect that controls the relative isomer stability in this case. As a consequence, the IPSI prediction error is larger than for ALA and CSI, although still acceptable. The result also helps to understand why the IPSI model needs to be applied separately to IPR and non-IPR isomers.35

image file: c8cp06707d-f2.tif
Fig. 2 Plots of the CSI indicator and its π component, Xq, with respect to the ALA and IPSI indicators for all isomers of C806− with at most three APPs. (a) CSI and (b) Xqversus ALA. (c) CSI and (d) Xqversus IPSI.

image file: c8cp06707d-f3.tif
Fig. 3 Idem Fig. 2 for all isomers of C346− with at most 17 APPs.

Similar plots for C346− shown in Fig. 3 show no apparent correlations between CSI or its π component and the other two models. In this case, the CSI prediction error is less than 2 kcal mol−1, while that for ALA and IPSI is 15 kcal mol−1 and 22 kcal mol−1, respectively. So, again Coulomb repulsion and also total aromaticity resulting from π delocalization are not the key parameters responsible for the relative stability of C346− isomers. Both are partial aspects of π delocalization, which is a subtle effect that has other manifestations. Delocalization does not always imply aromaticity, or at least, not only aromaticity.61 Also, in order to lead to the largest possible stabilization, delocalization does not always place the charges in the most distant places in a molecule, as stabilization due to π delocalization can compensate the increase in Coulomb repulsion. In contrast with ALA and IPSI, CSI reasonably accounts for all aspects of π delocalization and its consequences, because, as described above and in ref. 40 and 41, the π contribution Xqi is mathematically equivalent to solving the Hückel equations for a π delocalized system. As is well known, although Hückel theory is not as accurate as standard quantum chemistry methods, it is still able to provide a very reasonable description of the relative energies of frontier orbitals in π delocalized systems,40,41,62,63 which are the relevant orbitals at play in negatively charged fullerenes. Even more importantly, Hückel theory allows one to easily and unambiguously compute the relative π stabilization energy, as this is the only energy considered in the model. The present results show that, in those cases where several effects due to π delocalization are at play, the CSI index provides a more equilibrated description of all these effects than the other two models.

To better illustrate that neither Coulomb repulsion nor aromaticity are the only factors determining the relative stability of fullerene anion isomers, we analyze in more detail the cases of non-IPR C786− and C304−, for which the IPSI and ALA prediction errors are, respectively, much larger than those resulting from CSI (see Fig. 1b and d). We first consider two isomers of C786−, both having two APPs. The IPSI model predicts that isomer Cs(24[thin space (1/6-em)]101) is the most stable one among all isomers with NAPP = 2, while the CSI model assigns the lowest-energy isomer to C2(22[thin space (1/6-em)]010) (see Table 1). In fact, SCC-DFTB calculations show that the former isomer is considerably higher in energy (by 48.8 kcal mol−1) than the latter, which contradicts the prediction by ISPI based on Coulomb repulsion arguments. Fig. 4a and b show the calculated charge distribution for both isomers. As can be seen, the charge is more evenly distributed in the Cs(24[thin space (1/6-em)]101) cage than in the C2(22[thin space (1/6-em)]010) one, with a standard deviation of 0.0488 and 0.0518 for the former and the latter, respectively. The charge distribution on rings also shows a similar trend (see Fig. 4c and d). Consequently, one would expect34 less Coulomb repulsion in Cs(24[thin space (1/6-em)]101) than in C2(22[thin space (1/6-em)]010), however, as mentioned above, the former structure is not the most stable one. Therefore, IPSI provides a reasonable estimate of the relative importance of Coulomb repulsion in both isomers, but clearly this is not the only criterion that explains their relative stability. In other words, a minimum Coulomb repulsion does not always correspond to maximum stability.

Table 1 Relative SCC-DFTB energy (Erel in kcal mol−1), IPSI (in Å−1) and CSI (in −2β) for two isomers of C786−, C2(22[thin space (1/6-em)]010) and Cs(24[thin space (1/6-em)]101), both having two APPs
Isomer of C786− E rel IPSI CSI
C 2(22[thin space (1/6-em)]010) 0.0 12.880 −0.248
C s(24[thin space (1/6-em)]101) 48.8 12.867 0.144

image file: c8cp06707d-f4.tif
Fig. 4 Charge distribution on the C786− cage, obtained from SCC-DFTB calculations. The upper panels show the charge distribution on carbon atoms in isomers (a) C2(22[thin space (1/6-em)]010) and (b) Cs(24[thin space (1/6-em)]101). The lower panels (c) and (d) present the charge distribution on the rings in both isomers. Standard deviation of distribution, σ, is also given in each case.

We now consider the two isomers of C304− presented in Table 2. SCC-DFTB calculations indicate that isomer C2v(3) is considerably more stable (by 22.9 kcal mol−1) than C2v(2), as correctly predicted by the CSI model. The ALA index, however, predicts the opposite. In fact, Table 2 shows that the less stable isomer C2v(2) is much more aromatic (ALA index being less negative) than isomer C2v(3). Therefore, the present result clearly demonstrates that higher aromaticity does not necessarily imply higher stability of the fullerene anion.

Table 2 Relative SCC-DFTB energy (Erel in kcal mol−1), ALA and CSI (in −2β) for two isomers of C304−, C2v(3) and C2v(2)
Isomer of C304− E rel ALA CSI
C 2v(3) 0.0 −19.593 2.922
C 2v(2) 22.9 −17.958 3.320

4 Conclusions

In this work, we have systematically analyzed the performance of the widely used IPSI, ALA and CSI models in predicting the relative stability of endohedral metallofullerene isomers. The three models link stability to different physical properties: minimum electronic repulsion (IPSI), maximum aromaticity (ALA), and maximum π delocalization in combination with minimum strain (CSI). We have shown that, with few exceptions, the performance of ALA and CSI is similarly good, with CSI being slightly superior though computationally much less involved, as it only requires the knowledge of the fullerene cage topology without the need for elaborate quantum-chemical geometry optimizations. IPSI's performance is generally worse but still acceptable in a wide range of cage sizes; however, for medium-sized fullerene cages containing six extra electrons, the corresponding prediction errors are significantly large.

From our analysis, we conclude that neither Coulomb electronic repulsion nor aromaticity are the sole concepts governing the relative stability of endohedral metallofullerene isomers. Electron delocalization in the π shell, which is a more general concept than aromaticity, in combination with minimum strain provides a more realistic description of the relative stabilities observed experimentally.40,48 Indeed, in some cases, π delocalization of the additional cage electrons compensates an unfavorable Coulomb repulsion thus leading to a larger stability. Similarly, π electron delocalization does not necessarily translate into aromaticity, as the latter is only a partial aspect of the former.

Conflicts of interest

There are no conflicts to declare.


This work has been supported by the MINECO projects FIS2016-77889-R and CTQ2016-76061-P, the ‘Severo Ochoa’ Programme for Centres of Excellence in R&D (SEV-2016-0686) and the ‘María de Maeztu’ Programme for Units of Excellence in R&D (MDM-2014-0377). We also acknowledge the generous allocation of computer time at the Centro de Computación Científica of the Universidad Autónoma de Madrid CCC-UAM. Y. W. acknowledges the Thousand Talents Plan for Young Professionals of China.

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Electronic supplementary information (ESI) available: Fig. S1–S4. See DOI: 10.1039/c8cp06707d

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