Yang
Wang
*^{abc},
Sergio
Díaz-Tendero
^{bcd},
Manuel
Alcamí
^{bce} and
Fernando
Martín
*^{bdef}
^{a}School of Chemistry and Chemical Engineering, Yangzhou University, Yangzhou, Jiangsu 225002, China. E-mail: yangwang@yzu.edu.cn
^{b}Departamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: fernando.martin@uam.es
^{c}Institute for Advanced Research in Chemical Sciences (IAdChem), Universidad Autónoma de Madrid, 28049 Madrid, Spain
^{d}Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain
^{e}Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain
^{f}Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain
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 C_{28} to C_{104} 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 C_{62} to C_{82} 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.
Most EMFs are synthesized in the lab in very hot environments, e.g., through arc-discharges,^{13} combustion^{14,15} or radio-frequencies.^{16,17} In such environments, cages are formed through carbon-clustering processes^{18} at temperatures typically higher than 4000 K^{19} 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 C_{28}^{20} to C_{130}^{21} 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 phase^{22–24} or in solution.^{25,26}
Spectroscopic measurements^{1,2} and quantum chemistry calculations^{27,28} 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 chemistry^{42–44} or density functional tight-binding^{45} 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 C_{28} to C_{104} 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 C_{62} to C_{82} 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.
(1) |
(2) |
(3) |
CSI^{q}_{i} ≡ X^{q}_{i} + 0.2NAPP_{i}, | (4) |
(5) |
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 C_{2n}^{q} with 2n = 68–104 and q = 4−, 6−, at the B3LYP/6-31G(d) level using the Gaussian 09 package.^{55} It turns out^{48} that for all the considered systems of C_{2n}^{6−}, the singlet state is more stable than the triplet one. This is also true for most C_{2n}^{4−} 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.
(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 (C_{2n}^{q} with 2n = 70–104 and q = − 4, −6) by including a structural motif model^{60} 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 C_{2n}^{6−} (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 E_{min} 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, X^{q}_{i}, 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, C_{80}^{6−}, and a case in which CSI is clearly superior to the other two models, C_{34}^{6−}. We have considered all isomers containing at most three APPs. As can be seen, for C_{80}^{6−} (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 X^{q}_{i} 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}
Fig. 3 Idem Fig. 2 for all isomers of C_{34}^{6−} with at most 17 APPs. |
Similar plots for C_{34}^{6−} 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 C_{34}^{6−} 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 X^{q}_{i} 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 C_{78}^{6−} and C_{30}^{4−}, 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 C_{78}^{6−}, both having two APPs. The IPSI model predicts that isomer C_{s}(24101) is the most stable one among all isomers with NAPP = 2, while the CSI model assigns the lowest-energy isomer to C_{2}(22010) (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 C_{s}(24101) cage than in the C_{2}(22010) 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 expect^{34} less Coulomb repulsion in C_{s}(24101) than in C_{2}(22010), 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.
Isomer of C_{78}^{6−} | E _{rel} | IPSI | CSI |
---|---|---|---|
C _{2}(22010) | 0.0 | 12.880 | −0.248 |
C _{s}(24101) | 48.8 | 12.867 | 0.144 |
We now consider the two isomers of C_{30}^{4−} presented in Table 2. SCC-DFTB calculations indicate that isomer C_{2v}(3) is considerably more stable (by 22.9 kcal mol^{−1}) than C_{2v}(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 C_{2v}(2) is much more aromatic (ALA index being less negative) than isomer C_{2v}(3). Therefore, the present result clearly demonstrates that higher aromaticity does not necessarily imply higher stability of the fullerene anion.
Isomer of C_{30}^{4−} | E _{rel} | ALA | CSI |
---|---|---|---|
C _{2v}(3) | 0.0 | −19.593 | 2.922 |
C _{2v}(2) | 22.9 | −17.958 | 3.320 |
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.
Footnote |
† Electronic supplementary information (ESI) available: Fig. S1–S4. See DOI: 10.1039/c8cp06707d |
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