Article pubs.acs.org/JPCA

Kinetic Stability of Non-IPR Fullerene Molecular Ions Jun-ichi Aihara,* Yuto Nakagami, and Rika Sekine Department of Chemistry, Faculty of Science, Shizuoka University, Oya, Shizuoka 422-8529, Japan S Supporting Information *

ABSTRACT: Many fullerenes that violate the isolated pentagon rule (IPR) form stable metallofullerenes. In general, a fullerene cage is kinetically stabilized by acquiring a given number of electrons. Kinetic stability of negatively charged non-IPR fullerenes, including the recently isolated endohedral metallofullerene with a heptagonal face, was rationalized in terms of bond resonance energy (BRE). Interestingly, molecular anions of conventional fullerenes found in most isolated metallofullerenes are kinetically stable with large positive BREs for all CC bonds. As we pointed out in 1993, the IPR does not apply to charged fullerenes because π-bonds shared by two five-membered rings are aromatized to varying extents.

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INTRODUCTION Many different fullerenes are detectable during the laser vaporization of graphite,1,2 but few are isolable (i.e., extractable from the fullerene soot) in macroscopic amounts. This implies that many fullerene molecules are thermally accessible but are very different in kinetic stability. Kinetic stability means the propensity of a molecule not to undergo chemical change in a reasonable period of time.3 Chemically stable or isolable fullerenes satisfy the isolated pentagon rule (IPR).4−6 This rule states that all five-membered rings must be isolated from each other in the molecules of isolable fullerenes. Many researchers believe that the IPR is associated with enhanced steric strain and resonance destabilization pertaining to the pentalene-type 8π-electron subsystems.4−8 In 1993, however, we reported that many non-IPR (i.e., IPR-violating) fullerenes are kinetically stabilized by acquiring two or more electrons.9−11 Variation of the charge on the carbon cage also affects the relative energies of fullerene isomers.9−12 In 1995 we devised the smallest or minimum bond resonance energy (min BRE) in a molecule as a measure of kinetic stability for polycyclic π-systems and fullerenes.11,13−15 BRE is defined as a contribution of a given π-bond to the topological resonance energy (TRE).16,17 Many non-IPR fullerenes have been isolated and characterized in the form of metallofullerenes.18−23 Kinetic stability of metallofullerenes could be predicted from the min BRE for the negatively charged carbon cage.24−27 We reported that carbon cages in all IPR and some non-IPR metallofullerenes so far isolated are kinetically stable with min BREs > −0.100 |β|.24−27 Popov and Dunsch noted that the relative stabilities of different M3N@C2n (M = Sc, Y; 2n = 68−98) isomers correlate well with those of the empty C2n6− cages.28 In this context, we found that the molecular structure predicted for Ti2C80 isolated in 200129 is inconsistent with our BRE-based interpretation of kinetic stability.25,26 In this complex, two Ti atoms were thought to be encapsulated in a © 2015 American Chemical Society

D5d(IPR-6)-C80 or Ih(IPR-7)-C80 cage with a total of four electrons transferred to the carbon cage. According to our BREbased way of reasoning, however, at least one of the two charged cages, the Ih(IPR-7)-C80 tetraanion, must be very unstable with the min BRE < −0.100 |β|.25,26 Later, two research groups noticed that the geometry of this metallofullerene might be totally wrong.30,31 Ti2C80 was more likely a titanium carbide endohedral metallofullerene (Ti2C2)@C78 that has a Ti2C26+ ion within the D3h(IPR-5)-C786− cage.30,31 D3h(IPR-5)-C786− exhibits the min BRE of 0.1620 |β|.25 This example shows the utility of min BRE as a primary or complementary index of kinetic stability for metallofullerenes.24−27 We have so far examined the kinetic stability of endohedral IPR fullerenes systematically.25,26 In this paper, we apply the BRE concept to a variety of non-IPR metallofullerenes and establish that not only isolable fullerenes but also isolable metallofullerenes must on an equal footing meet the requirement that min BRE for the fullerene cage must be greater than −0.100 |β|. We use the conventional numbering of the IPR fullerene isomers based on the Fowler−Manolopoulos spiral algorithm.6

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THEORY The definition of BRE13,15,24 is outlined for the benefit of general readers. A hypothetical π-system in which a CpCq bond (i.e., a π-bond formed between the pth and qth carbon atoms) interrupts cyclic conjugation can be constructed simply by multiplying βp,q by i and βq,p by −i, where βp,q and βq,p are a pair of resonance integrals for the π bond, and i is the square root of −1. In this π system, no π circulation is expected to occur along the circuits that share the CpCq π-bond in common. BRE for Received: April 10, 2015 Revised: May 23, 2015 Published: May 28, 2015 6542

DOI: 10.1021/acs.jpca.5b03468 J. Phys. Chem. A 2015, 119, 6542−6550

Article

The Journal of Physical Chemistry A the CpCq π-bond is given as a destabilization energy of this hypothetical π-system. In other words, BRE for a given π-bond represents the contribution of all circuits that share the bond to global aromaticity. This quantity was originally defined to justify the IPR for fullerenes.13 A detailed procedure for calculating BRE has been described elsewhere.32 In general, kinetic stability of a π system is determined by the reactivity of the most reactive site in the π system.13,15 For many neutral and charged polycyclic π-systems, min BRE is associated with the most reactive site(s) or π-bond(s).15 If min BRE is positive in sign, all bonds will contribute to the aromaticity of the entire π-system in varying degrees. On the contrary, if min BRE is smaller than −0.100 |β|, the π-system must be kinetically unstable and chemically very reactive.13,15,24 This simple criterion allows us to directly compare IPR and non-IPR fullerene molecular ions. For empty fullerenes and metallofullerenes, kinetic stability must be closely related to the possibility of survival in the high-temperature arc-discharge furnace in which they are formed.25,26 An energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) has long been used as a simple measure of kinetic stability.33−38 We previously reported that the T value, i.e., the HOMO−LUMO energy separation multiplied by the number of carbon atoms, is better as an index of kinetic stability for fullerenes.39−41 In general, IPR fullerenes with a large T value were found to have a large positive min BRE.42 This fact supports the utility of min BRE as an indicator of kinetic stability for fullerenes.

Table 1. Non-IPR Metallofullerenes Studied

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RESULTS AND DISCUSSION We chose as many structurally well-characterized non-IPR metallofullerenes as possible for this study. In so doing, we basically relied on the lists of metallofullerenes compiled recently by Popov et al.20 and Garcia-Borràs et al.21 Non-IPR metallofullerenes finally chosen by us are listed in Table 1. For the connectivity of carbon atoms in these fullerenes, see the references concerned.43−70 Hereafter, m/n indicates the type of each CC bond; 5/5, 5/6, and 6/6 bonds represent CC bonds shared by two five-membered rings, by five- and six-membered rings, and by two six-membered rings, respectively. Empty Non-IPR Fullerenes. The number of possible fullerene cages grows very rapidly with the number of carbon atoms, increasing from 1812 for C60 to 1 674 171 for C120.71 Most of them are non-IPR isomers. For example, C120 has only 10 774 IPR isomers (ca. 0.644% of all possible C120 fullerene cages).71 Many metallofullerenes with non-IPR carbon cages have been produced in macroscopic amounts.18−23 However, all non-IPR fullerenes employed to form these metallofullerenes are predicted to be very unstable in the neutral charge state. As listed in Tables 2−4, all the neutral species of non-IPR fullerenes studied exhibit min BREs < −0.100 |β|, which are assigned without exception to 5/5 bonds. Remember that neutral species with min BREs < −0.100 |β| are never rare even in the group of IPR fullerenes.25 Fifteen of 51 IPR fullerenes with 60−84 carbon atoms are predicted to be kinetically unstable with min BREs < −0.100 |β|. Considering that min BRE for isolable D2d(IPR-23)-C84 is as small as −0.0564 |β|, we should deal with all IPR and non-IPR fullerenes and metallofullerenes with min BRE > −0.100 |β| as possible candidates for isolable species. Charged Non-IPR Fullerenes in General. The inapplicability of the IPR to charged fullerenes is due simply to the fact

a

metallofullerenea

formal electron transfer

reference

Sc2@C2v(4059)-C66 Sc3N@D3(6140)-C68 DySc2N@D3(6140)-C68 LuSc2N@D3(6140)-C68 Lu2ScN@D3(6140)-C68 Sc3N@C2v(7854)-C70 La2@D2(10611)-C72 Ce2@D2(10611)-C72 Pr2@D2(10611)-C72 Sc2@C2(13295)-C74 La2@C2(13295)-C74 DySc2N@Cs(17490)-C76 Y3N@C2(22010)-C78 Dy3N@C2(22010)-C78 Lu3N@C2(22010)-C78 Tm3N@C2(22010)-C78 Gd3N@C2(22010)-C78 Sc3NC@ C2(22010)-C78 LaSc2@Cs(hept)-C80 Gd3N@Cs(39663)-C82 Y3N@Cs(39663)-C82 Tb3N@Cs(51365)-C84 Tm3N@Cs(51365)-C84 Gd3N@Cs(51365)-C84 Y3N@Cs(51365)-C84 Sc2C2@C2v(6073)-C68 Sc2S@C2v(6073)-C68 Sc2S@Cs(7892)-C70 Sc2S@Cs(10528)-C72 Sc2S@Cs(13333)-C74 Y2C2@C1(51383)-C84 Sm@C2v(19138)-C76

6 6

43, 44 45, 46

6 6

47 48−50

6

51

6 6

52 53−56

6 6

57 58, 59

6

59−62

4

63, 64

4 4 4 4 2

65, 66 67 68 69 70

Numbers in parentheses indicate an isomer number.

that fullerene molecules, including non-IPR ones, enhance aromatic character by acquiring negative charge.24−28 This is a characteristic of polycyclic π-systems with one or more fivemembered rings, which tend to lower the low-lying unoccupied MOs. For example, the molecular dianion of highly reactive pentalene is iso-π-electronic with naphthalene, both being highly aromatic with large positive TREs of 0.4638 and 0.3888 |β|, respectively.15 Therefore, the carbon cages found in isolable metallofullerenes are often different from the empty neutral fullerenes isolated so far. We pointed out in 2002 that, as far as IPR fullerenes are concerned, kinetically unstable fullerenes tend to form endohedral metallofullerenes.26 Valencia et al. also noted that the relative stability of the C80 IPR fullerene isomers is inverted when they form molecular tetraanions.35 The same must be true for many other IPR fullerenes. An analogous aspect of non-IPR fullerenes, however, is not discernible because all neutral non-IPR fullerenes are unstable in nature because of the presence of 5/5 bonds. Many of the non-IPR fullerenes are stabilized not only thermodynamically but also kinetically by acquiring two or more electrons.11,27 Hexaanions of Non-IPR Fullerenes. Carbon cages in many isolated non-IPR metallofullerenes bear a formal charge of −6. These metallofullerenes consist of two major groups: dimetallofullerenes (M2@Cn)43,44,48−51 and trimetallic nitride endohedral fullerenes (M3N@Cn), where M is a rare-earth 6543

DOI: 10.1021/acs.jpca.5b03468 J. Phys. Chem. A 2015, 119, 6542−6550

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The Journal of Physical Chemistry A Table 2. Min BREs for Molecular Anions of Non-IPR Fullerenes That Tend to Form Molecular Hexaanions. min BREa (|β|) species C2v(4059)-C66 D3(6140)-C68 C2v(7854)-C70 D2(10611)-C72 C2(13295)-C74 Cs(17490)-C76 C2(22010)-C78 Cs(hept)-C80 Cs(39663)-C82 Cs(51365)-C84 a

number of 5/5 bonds

number of empty bonding orbitals

neutral

dianion

tetraanion

hexaanion

3 3b 3 3 3 3 3 3 3 3

−0.1935 −0.16235/5 −0.42495/5 −0.19975/5 −0.30475/5 −0.21825/5 −0.26035/5 −0.18845/5 −0.18115/5 −0.31315/5

−0.1986 −0.21095/5 −0.07895/5 −0.21995/5 −0.13995/5 −0.23535/5 −0.13205/5 −0.18935/5 −0.18725/5 −0.11045/5

−0.2313 −0.00815/5 −0.13205/5 0.02415/6 −0.03045/6 −0.01075/6 −0.14885/5 −0.09005/5 −0.11885/5 −0.03875/6

0.16365/6 0.15545/6 0.16305/6 0.16325/6 0.14175/6 0.14765/6 0.15715/6 −0.00795/7 0.14845/6 0.15885/6

4 3 3 2 2 2 2 2 1 1

5/5

5/5

5/5

Every superscript indicates the bond type concerned. bIncluding one empty nonbonding orbital.

Table 3. Min BREs for Molecular Anions of Non-IPR Fullerenes That Tend to Form Molecular Tetraanions min BREa (|β|)

a

species

number of 5/5 bonds

number of empty bonding orbitals

neutral

dianion

tetraanion

hexaanion

C2v(6073)-C68 C2(7892)-C70 Cs(10528)-C72 Cs(13333)-C74 C1(51383)-C84

2 2 2 2 1

2 2 2 2 2

−0.26675/5 −0.34415/5 −0.25235/5 −0.33995/5 −0.31205/5

−0.06015/5 −0.01955/6 −0.10055/6 −0.25715/5 −0.23155/5

0.00595/6 0.07175/6 0.05445/6 0.03375/6 0.08095/6

0.08025/6 −0.07025/6 0.03355/6 0.09665/6 −0.00055/6

Every superscript indicates the bond type concerned.

Table 4. Min BREs for Molecular Anion of Non-IPR C2v(19138)-C76 That Tends to Form a Molecular Dianion min BREa (|β|) species C2v(19138)-C76 a

number of 5/5 bonds

number of empty bonding orbitals

1

2

neutral

dianion

−0.1965

−0.0632

5/5

5/6

tetraanion

hexaanion

5/6

0.07705/6

0.0175

Every superscript indicates the bond type concerned.

atom.45−47,52−55,58−62 Encapsulated M2 and M3N clusters necessarily bear a formal charge of +6. Sc3NC@C2(22010)C78 likewise contains a non-IPR carbon cage with a formal charge of −6.56 As can be seen from Table 2, non-IPR fullerenes that tend to acquire six electrons have three empty bonding orbitals and one to four 5/5 bonds. It seems that the number of 5/5 bonds decreases with increasing size of a fullerene cage (Table 2). Min BREs for fullerenes that tend to acquire six electrons are also listed in Table 2, together with the type of π-bonds to which the min BREs are assigned. Apart from the Cs(hept)-C80 hexaanion,57 conventional fullerene hexaanions have large positive min BREs > 0.140 |β|. These min BREs are attributable not to 5/5 bonds but to 5/6 bonds, suggesting that pentalene motifs are highly aromatized by concentrating additional electrons there. The 5/5 bonds are no longer the most reactive sites in the hexaanions. For reference, min BREs for the molecular hexaanions of D5d(IPR-6)-C80 and Ih(IPR-7)-C80, which are known to form metallofullerenes abundantly, are 0.1652 and 0.1931 |β|, respectively.25 C s (hept)-C 80 Hexaanion. The synthesis and X-ray structural characterization of LaSc2N@Cs(hept)-C80 were reported recently.57 This is the first unconventional metallofullerene with a heptagon in the carbon cage. This carbon cage consists of 1 heptagon, 13 pentagons, and 28 hexagons (Figure 1). Introduction of one heptagon increases the number of pentagons by one, while it decreases the number of hexagons by two.57,72 Density functional theory (DFT) computations

Figure 1. CC bonds near the heptagon of Cs(hept)-C80.

predicted that LaSc2N@Cs(hept)-C80 is 27 kJ/mol less stable than LaSc2N@Ih(IPR-7)-C80 but 34 kJ/mol more stable than LaSc2N@D5d(IPR-6)-C80.57 Not in line with this, LaSc2N@ Cs(hept)-C80 is isolated much less abundantly than the other two conventional isomers. Some kinetic factors may be responsible for the low yield of the LaSc2N@Cs(hept)-C80.58 The heptagon in the Cs(hept)-C80 hexaanion are flanked by as many as four pentagons. Fowler et al. predicted in 1996 that stability of neutral fullerenes with a single heptagon increases with the number of pentagon−heptagon fusions.72,73 BREs for CC bonds characteristic of Cs(hept)-C80 are summarized in Table 5. For the locations of these CC bonds, see Figure 1. Min BREs for the hexaanion (−0.0079 |β|) are 6544

DOI: 10.1021/acs.jpca.5b03468 J. Phys. Chem. A 2015, 119, 6542−6550

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The Journal of Physical Chemistry A Table 5. BREs for CC Bonds near the Heptagon in Cs(hept)C80

thought to be the dianion of C2v(4348)-C66 and the hexaanion of D3(6140)-C68, respectively. We reported that the C2v(4348)C66 dianion and the D3(6140)-C68 hexaanion are kinetically stable with min BREs > −0.100 |β|.27 Kobayashi and Nagase later suggested that the carbon cage in Sc2@C66 was not C2v(4348)-C66 but C2v(4059)-C66, which is a non-IPR fullerene isomer with two sets of unsaturated linear triquinanes.43 In 2014, they made an experimental and computational reinvestigation of Sc2@C66 and confirmed that the carbon cage is really an oval C2v(4059)-C66.44 Sc2 was not a dimeric Sc22+ dication but a pair of Sc3+ ions with a reasonable separation of 4.90 Å. Because C2v(4348)-C666− is predicted to be kinetically unstable with min BRE = −0.1338 |β|, it must not be possible to contain Sc26+ in the cage. It then follows that carbon cages in the first two non-IPR metallofullerenes45,74 bear a formal charge of −6 in common. For this reason, C2v(4059)-C66 is listed in Tables 1 and 2 in place of C2v(4348)C66. The Sc2C70 Problem. Shi et al. first isolated Sc2C70 and formulated it as (Sc3+)2(C2)2−@C2v(6073)-C684−,63 but Zheng et al. subsequently predicted based on DFT calculations that Sc2@C2v(7854)-C70 must be much more stable than Sc2C2@ C 2v (6073)-C 68 .75 They formulated Sc 2C 70 as (Sc 2+ ) 2 @ C2v(7854)-C704−. However, we feel it is difficult to accept it straightforwardly because the molecular tetraanion of C2v(7854)-C70 appears to be very unstable with a min BRE of −0.1320 |β|. Cerón et al. recently pointed out that the originally proposed Sc2C2@C2v(6073)-C68 is more congruent not only with the size-shape complementality rule but also with the maximum aromaticity rule.21−23 Considering that the tetraanion of C2v(6073)-C68 is the most aromatic isomer of the C68 tetraanion,21 it must possibly be the kinetically most stable isomer of C684−. An X-ray single-crystal structure may be required to solve these conflicting observations.22 C2v(6073)C68 is used as a carbon cage in Sc2S@C68, in which C68 behaves as a molecular tetraanion.64 For the convenience of future study, both candidate fullerenes C2v(7854)-C70 and C2v(6073)C68 are entered in Tables 1−3. C2v(7854)-C70 is the most stable isomer of C70 when it forms a molecular hexaanion, as in Sc3N@C2v(7854)-C70.47 The Non-IPR C74 Problem. Metallofullerenes based on non-IPR C74 have not been fully studied. In 2012, Zheng et al. predicted using DFT calculations that both the most stable tetraanion and hexaanion of C74 are those of C2(13333)-C74.51 They also predicted that both M2@C2(13295)-C74 and M2@ C2(13333)-C74 (M = Sc, La) exhibit thermodynamic stability with a very small energy difference.51 The structure of Sc2S@ C74 has recently been studied by Gan et al.68 At the B3LYP/631G* level, Sc2S@C2(13333)-C74 was the lowest-energy isomer with a formal transfer of four electrons.68 Thus, C2(13333)-C74 may form not only a molecular hexaanion but also a molecular tetraanion. Both C2(13295)-C74 and C2(13333)-C74 were added to Tables 1−3 for future structural study. Large Metallofullerenes. The largest non-IPR metallofullerene structurally characterized to date is M3N@ Cs(51365)-C84 (M = Tb, Tm, Gd),60,61 whereas the largest IPR metallofullerene structurally characterized is Sm2@D6(IPR822)-C104.76 It seems that large non-IPR fullerenes are rather reluctant to form metallofullerenes.22 Molecular structures of La2@D5(IPR-450)-C10077 and Sm2@D3d(IPR-822)-C10476 allow us to evaluate whether the choice of a cage isomer in a large endohedral fullerene is still determined by the stability of the empty anionic cage.38 In these metallofullerenes, C100 and C104

BRE (|β|) bonda

bond type

neutral

dianion

tetraanion

hexaanion

a b c d e f g

5/5 5/7 5/7 5/6 5/6 6/7 6/7

−0.1884 −0.0326 −0.0456 0.0001 −0.0575 0.0438 0.1114

−0.1893 −0.0787 −0.1030 −0.0458 −0.0872 0.0438 0.0706

−0.0900 −0.0031 −0.0289 0.0313 0.1559 0.1565 0.0715

0.2347 −0.0079 0.0797 0.1138 0.1651 0.1645 0.1300

a

For the location of each CC bond, see Figure 1.

those for two of the four 5/7 bonds in azulenoid substructures, which are by far smaller than the min BRE for any known conventional fullerene hexaanions although they are larger than the critical value (−0.100 |β|). This must be the main reason for the low yield of LaSc2N@Cs(hept)-C80. The remaining two 5/7 bonds exhibit 0.0797 |β|, which are also smaller than min BREs for most conventional fullerene hexaanions. Except for the two 5/7 bonds, the Cs(hept)-C80 hexaanion has no CC bonds with negative BREs. Tetraanions of Non-IPR Fullerenes. As shown in Table 3, non-IPR fullerenes that tend to acquire four electrons have two empty bonding orbitals, which is fewer by one than those in non-IPR fullerenes that encapsulate M2 or M3N clusters. They have one or two 5/5 bonds. Valencia et al. pointed out that the (LUM0-2)−(LUMO-3) gaps for fullerenes that tend to form tetraanions are in general smaller than the (LUMO-3)(LUMO-4) gaps for fullerenes that tend to form hexaanions,35 where LUMO-n is the nth LUMO of the fullerene molecule. This fact may explain the rather low stability of M2C2 endohedral fullerenes compared to M3N endohedrals.35 In our wording, min BREs for most fullerene tetraanions in Table 3 are smaller than those for fullerene hexaanions in Table 2. All these min BREs arise not from 5/5 bonds but from 5/6 bonds, suggesting that pentalene motifs are considerably aromatized even if only four electrons are added to the carbon cage. C2v(19138)-C76 Dianion. Echegoyen and co-workers stated that the nature and geometry of the endohedral cluster play an important role in the selection of the non-IPR cage.22,65 Therefore, it has been believed that any non-IPR cage seldom forms an endohedral metallofullerene with a single metal atom. To make matters worse, many non-IPR fullerenes exhibit large negative min BREs even in the molecular dianions (see Tables 2−4). Quite recently, however, the synthesis and X-ray structural characterization of Sm@C2v(19138)-C76 were reported.70 The isolation of Sm@C2v(19138)-C76 demonstrated for the first time that a stable non-IPR metallofullerene forms even with a single divalent metal ion. Fortunately, the only 5/5 bond in the C2v(19138)-C76 dianion exhibits a small positive BRE of 0.0402 |β|. The min BRE of −0.0632 |β| arises from the four equivalent 5/6 bonds. This min BRE value happened to be as small as that for the abundantly isolable D2d(IPR-23)-C84 molecule (−0.0564 |β|).25 Even so, it is interesting to note that both of the unusual non-IPR metallofullerenes,57,70 LaSc2N@ Cs(hept)-C80 and Sm@C2v(19138)-C76, exhibit very small minBREs, although they are still larger than the critical value (−0.100 |β|). The Sc2C66 Problem. In 2000, the first two non-IPR metallofullerenes were isolated in macroscopic quantities.45,74 Carbon cages in Sc2@C66 and Sc3N@C68 isolated then were 6545

DOI: 10.1021/acs.jpca.5b03468 J. Phys. Chem. A 2015, 119, 6542−6550

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The Journal of Physical Chemistry A Table 6. Min BREs for Molecular Ions of Large Fullerenes min BREa (|β|) species D5(IPR-450)-C100 D3d(IPR-822)-C104 a

number of empty bonding orbitals b

3 3

neutral

dianion

tetraanion

hexaanion

−0.1266 −0.0830

−0.0571 −0.1275

−0.0611 0.1015c

0.1807c 0.1375

All min BREs are assigned to 5/6 bonds. bIncluding one empty nonbonding orbital. cMolecular ions that form metallofullerenes.

bear formal charges of −6 and −4, respectively. As can be seen from Table 6, min BREs for the hexaanion of D5(IPR-450)-C100 and the tetraanion of D3d(IPR-822)-C104 are larger than 0.100 |β|; kinetic stability of a charged carbon cage still seems to be crucial to the formation of such large endohedral metallofullerenes. Like smaller IPR fullerenes that form stable metallofullerenes,26 these large IPR fullerenes are kinetically unstable in the neutral state. Interestingly, the tetraanion of D3d(IPR-822)-C104 exhibits a larger min BRE of 0.1015 |β| than that of any fullerene in Tables 2 and 3. Open-Shell Species. Kinetic stability of open-shell metallofullerenes, such as La@C2(10612)-C7278 and La@D3h(IPR1)-C74,79 is not a target of this paper because open-shell species are reactive in nature.18 C2(10612)-C72 is a non-IPR fullerene with one pentalene motif, whereas D3h(IPR-1)-C74 is the only IPR fullerene of C74. Neutrals of C2(10612)-C72 and D3h(IPR1)-C74 exhibit min BREs of −0.3645 and −0.0053 |β|, respectively. These fullerenes have only one empty bonding orbital in common. When they form molecular trianions, all the CC bonds are aromatized with positive BREs. Trianions of C2(10612)-C72 and D3h(IPR-1)-C74 exhibit positive min BREs of 0.0690 and 0.0116 |β|, respectively. Therefore, the kinetic instability of La@C2(10612)-C72 and La@D3h(IPR-1)-C74 should naturally be attributed to the open-shell electronic configuration. As stressed by Diener and Alford,33 the majority of radical M-metallofullerenes are not extractable from the soot. La@C2(10612)-C7278 and La@D3h(IPR-1)-C7479 were extracted from the soot with 1,2,4-trichlorobenzene as adducts with the dichlorophenyl radical (C6H3Cl2). Trianions of carbon cages in closed-shell adducts La@C2(10612)-C72(C6H3Cl2) and La@D3h(IPR-1)-C74-C72(C6H3Cl2) exhibit min BREs of 0.0698 and 0.0006 |β|, respectively. Min BRE for the latter species is very small, although it is still much larger than −0.100 |β|. Pentalene motifs are missing in these adducts. Trends in the Magnitude of TRE. TRE for any cyclic πsystem greatly varies as a function of the number of π-electrons (Nπ), where Nπ runs from zero to twice the number of conjugated atoms. We then explore the Nπ dependence of TRE for typical fullerenes and reexamine the relative stabilities of fullerene molecular polyanions that form stable metallofullerenes. We previously pointed out that the sequential line plot of TRE against Nπ for any polycyclic benzenoid hydrocarbon is very similar in appearance with that for benzene (Figure 2).80 Both line plots have the same number of major extrema, three maxima and two minima. Thus, the global aromaticity of a polycyclic benzenoid hydrocarbon molecular ion strongly reflects that of a benzene molecular ion. Likewise, the Nπ dependence of TRE for any polycyclic π-system formed by fusion of two or more pentagonal rings resembles that for cyclopentadienyl or [5]annulene (Figure 2).80 For details of the Nπ dependence of TRE, see ref 80. Let us plot TRE against Nπ for typical fullerenes. Both Ih(IPR-1)-C60 and C2v(1809)-C60 consist of 20 benzene rings and 12 pentagonal rings. The latter C60 isomer with two 5/5 bonds is the lowest-energy non-IPR isomer of C60. The plots of

Figure 2. TREs for benzene and cyclopentadienyl, each as a function of the number of π-electrons (Nπ).

TRE against Nπ for these two C60 isomers (Figure 3) are very similar in appearance to each other. At first sight, the plot for

Figure 3. TREs for two C60 isomers, each as a function of Nπ. Neutral species are denoted by asterisks. Arrows point to the peaks that correspond to the energy levels of the spherical free-electron model.

Ih(IPR-1)-C60 cannot be distinguished from that for C2v(1809)C60. These similar plots of TRE against Nπ strongly suggest that global aromaticity of these two fullerene isomers are essentially the same at any charge state. Because TRE is a kind of indicator of thermodynamic stability, kinetic stability of any molecular ion cannot be read from these plots. TREs for neutral Ih(IPR1)-C60 and C2v (1809)-C60 are 1.6427 and 1.4255 |β|, respectively. We then examine these plots in some more detail. Fullerenes are a kind of nonalternant hydrocarbons with 12 oddmembered rings; therefore, the plots of TRE against Nπ for them are not bilaterally symmetric. The plots must be intermediate in overall shape between those for benzene and cyclopentadienyl (Figure 2) because fullerene cages always consist of 12 pentagons and many hexagons. In fact, the plots for the two C60 isomers in Figure 3 resemble the plot for cyclopentadienyl rather than that for benzene, but are somewhat deformed by the presence of 20 benzene rings. Plots for two C60 isomers and cyclopentadienyl have some common features. First, major extrema at Nπ = 10−20, 30−50, 65−75, and 85−95 in the plots for C60 isomers apparently correspond to four extrema in the plot for cyclopentadienyl. All these extrema appear to be intensified by the presence of 20 benzene rings, which in particular must be responsible for the distinct shoulder near Nπ = 60. In the plots for two C60 isomers, 6546

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than −0.100 |β|, which is nothing other than the origin of kinetic instability of the neutral species. Thus, C2v(1809)-C60 has been isolated only as a chlorinated derivative, in which πbonds shared by two pentagons are missing.86 The 5/5 bonds in this C60 isomer are rapidly aromatized on going to higher molecular anions. Line plots of BRE for 5/5 bonds in two larger non-IPR fullerenes that form metallofullerenes, Cs(39663)-C82 and Cs(51365)-C84, are shown in Figure 5. These fullerenes have

the peak corresponding to the third maximum of benzene almost cancels out that corresponding to the second minimum of cyclopentadienyl. Molecular mono- to hexa-anions of these fullerenes are hence highly aromatic with large positive TREs. The π-system of a fullerene molecule can be approximated as free electrons on the surface of a sphere.81−83 Local maxima at Nπ = 2, 8, 18, 32, and 50 in the plots for two C60 isomers, indicated by arrows in Figure 3, correspond to the closed-shell structures in this free-electron model. Hirsch et al. pointed out that spherical species with these numbers of π-electrons are aromatic.82 TREs for the two C60 isomers indeed are maximized when they have 2, 8, 18, 32, or 50 π-electrons. Note, however, that C60 ions with 32 and 50 π-electrons are antiaromatic with negative TREs. We do not want to go deeply into this problem because we have fully discussed it elsewhere.84,85 Trends in the Magnitude of BRE. We next plot BRE against Nπ for typical π-bonds in some fullerenes. Ih(IPR-1)-C60 has 90 CC bonds of which 60 are equivalent 5/6 bonds and 30 are equivalent 6/6 bonds. Figure 4 shows the plots of BRE

Figure 5. BREs for 5/5 bonds in Cs(39663)-C82 and Cs(51365)-C84, each as a function of Nπ. Neutral species are denoted by asterisks.

one 5/5 bond each. General features of these plots are similar to that for the 5/5 bonds in C2v(1809)-C60. It is noteworthy that 5/5 bonds again are kinetically very unstable in the neutral species. As can been seen from Figure 5, these 5/5 bonds are no longer the origin of min BRE in the hexaanions, in which these bonds are highly aromatized. Interestingly, local maxima at Nπ = 2, 8, 18, 32, and 50, observed in Figures 3 and 4, are missing in the BRE-vs-Nπ plots in Figure 5. Finally, line plots of BRE for 5/5 and 5/7 bonds in Cs(hept)C80 are shown in Figure 6. As shown in Figure 1, this

Figure 4. BREs for typical π-bonds in two C60 isomers, each as a function of Nπ. Neutral species are denoted by asterisks. Arrows point to the peaks that correspond to the energy levels of the spherical freeelectron model.

Figure 6. BREs for one of the 5/5 bonds (bond a) and one of the 5/7 bonds (bond b) in Cs(hept)-C80, each as a function of Nπ. Neutral species are denoted by asterisks. Arrows point to the peaks that correspond to the energy levels of the spherical free-electron model.

against Nπ for typical π-bonds in two C60 isomers, i.e., 5/6 and 6/6 bonds in Ih(IPR-1)-C60 and 5/5 bonds in C2v(1809)-C60. These plots are more or less similar to those of TRE for the same fullerenes, reflecting again the fact that they consist of pentagons and hexagons. Ih(IPR-1)-C60 has exceptionally many sets of degenerate orbitals; most spikes in these plots correspond to the complete filling of these degenerate orbitals. Local maxima at Nπ = 2, 8, 18, 32, and 50 due to the closedshell structures in the free-electron model81−83 are still discernible in the three plots in Figure 4. C2v(1809)-C60 has many nonidentical 5/6 and 6/6 bonds. For the BRE-vs-Nπ plots for representative 5/6 and 6/6 bonds in this C60 isomer, see the Supporting Information. BREs for 5/6 and 6/6 bonds in Ih(IPR-1)-C60 are larger than −0.100 |β| in the range Nπ = 60−72. The hexaanion (Nπ = 66) corresponds to the closed-shell structure.81,82 BREs for the 5/5 bonds in the neutral C2v(1809)-C60 molecule are much smaller

unconventional fullerene has two 5/5 bonds and four 5/7 bonds. General features of the plots in Figure 6 are still similar to those for the 5/5 bonds in Cs(39663)-C82 and Cs(51365)C84 in Figure 5. Although Cs(hept)-C80 has one sevenmembered ring, it does not deform the plots appreciably. As in the case of porphyrins,87 a small number of large rings does not contribute significantly to the TRE and BREs for many πbonds. The 5/5 bonds are no longer the origin of min BRE in the hexaanions, in which 5/5 bonds are highly aromatized. It is interesting to see that BREs for the 5/7 bonds vary modestly as compared to those for 5/5 bonds. BREs for two of the 5/7 bonds then are slightly negative in sign in the molecular hexaanion. Local maxima corresponding to energy levels of the 6547

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free-electron model81−83 are again visible in the two BRE-vs-Nπ plots in Figure 6. As has been seen above, for all conventional fullerenes, 5/5 bonds in the molecular hexaanion are stabilized in the region where both benzene rings and cyclopentadienyl rings are more or less aromatic. The same must be true for molecular tetraanions of some fullerenes. In general, BREs for 5/6 bonds vary from species to species, depending on what charge state the species takes. Figures 3−6 strongly suggest that realistic molecular cations of any fullerene are highly antiaromatic with large negative min BREs. This constitutes the main reason why stable fullerene molecular cations have never been observed. Fowler and Ceulemans once classified conventional fullerenes as electron-deficient alkenes.88

CONCLUDING REMARKS As has been seen, the BRE model of kinetic stability can be applied without any difficulty to fullerenes and metallofullerenes, both IPR and non-IPR ones. It was fortunate that BREs can be calculated easily even for very large π-systems. Min BRE can be calculated readily for any neutral and charged fullerene. In contrast, TRE is very difficult to calculate for large fullerenes. Our BRE-based approach to metallofullerenes relies only on the intrinsic stability of the charged fullerene cage and does not take into account any interaction between the cage and the endohedral metal or metal cluster. This presumption is apparently consistent with that of the maximum aromaticity rule proposed by Garcia-Borràs et al.21,23 Very many fullerene and metallofullerene isomers indeed are formed in the gas phase, but most of them cannot be isolated in pristine form.1,2,33,89 For example, D3h(IPR-5)-C78, C2v(IPR-5)C80, and C2(IPR-5)-C82 lie only 4.5−8.2 kcal/mol higher than their respective lowest-energy fullerene isomers.90−93 They really are produced in the arc-discharge furnace but cannot be extracted from the fullerene soot89 because min BREs for them are smaller than −0.100 |β|.25 Thus, kinetic stability does not always vary in parallel with thermodynamic stability. Note that the condition that min BRE must be smaller than −0.100 |β| is only one of the necessary conditions for the isolability or extractability of fullerenes and metallofullerenes. Fullerenes and metallofullerenes with min BREs > −0.100 |β| are not always extractable. For example, D3h(IPR-1)-C74 with min BRE = −0.0053 |β| is not extractable from the fullerene soot.33,89 Even so, our BRE-based approach must be very useful for greatly decreasing the number of candidate carbon cages for isolable metallofullerenes and estimating the reasonableness of experimentally determined carbon cages. ASSOCIATED CONTENT

S Supporting Information *

Sequential line plots of BRE against the number of π-electrons for representative CC bonds in C2v(1809)-C60. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b03468.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 6548

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The Journal of Physical Chemistry A

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DOI: 10.1021/acs.jpca.5b03468 J. Phys. Chem. A 2015, 119, 6542−6550