Computational chemistry of isomeric fullerenes and endofullerenes

q 2005 Elsevier B.V. All rights reserved. Theory and Applications of Computational Chemistry: The First Forty Years Edited by C. Dykstra et al.

891

CHAPTER 32

Computational chemistry of isomeric fullerenes and endofullerenes Zdeneˇk Slanina1,2 and Shigeru Nagase1 1

Department of Theoretical Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Aichi, Japan 2 Institute of Chemistry, Academia Sinica, 128 Yen-Chiu-Yuan Rd., Sec. 2, Nankang, Taipei 11529, Taiwan, ROC

Abstract Fullerenes and metallofullerenes were for the first time observed in the gas phase about 20 years ago and then prepared in crystalline form about 15 years ago. An enormous amount of observed and computed data has been obtained during the period. The chapter surveys various computational aspects of fullerene science including rich isomerism and the enthalpy –entropy interplay both of which represent essential features of fullerene and metallofullerene formations.

32.1 INTRODUCTION Fullerenes—or cage compounds built exclusively from carbon atoms—and their metalcontaining derivatives, metallofullerenes, were first observed in the gas phase by Kroto et al. [1,2] less than 20 years ago and prepared in crystalline form by Kra¨tschmer et al. [3] less than 15 years ago. Still, an enormous amount of observed and computed data has been obtained during this time (see, e.g. recent surveys on fullerenes [4 –8] and endohedral metallofullerenes [9,10]). In addition to spheroidal fullerene cages, fullerene science also deals with other objects like elongated cylindrical bodies known as nanotubes, prepared by Iijima [11] soon after mastering the fullerene synthesis, nanocones [12] or peapods [13]. All the species exhibit a substantial application potential, especially for molecular electronics [14]. Fullerenes and metallofullerenes have attracted a considerable interest from experiment [15] as well as computations and theory [16,17]. Their experimental characterization is mostly based [18 –20] on 13C NMR spectroscopy and more recently References pp. 908 – 917

892

Chapter 32

also on 3He NMR spectroscopy. Fullerenes are commonly defined as polyhedral cages containing only carbon atoms arranged into five- and six-membered rings. Quasifullerenes [21] are cages containing other types of cycles, too, like (topological) squares and heptagons. The very early history [22 – 26] of carbon clusters starts with a mass-spectrometric observation of clusters up to C15 by Hahn and his co-workers [27] 60 years ago. In the 1950s and 1960s, experiments [28 –30] could expand up to C33. At this stage, initial simple computations had also been performed on small carbon clusters by Pitzer and Clementi [31,32], Hoffmann [33] and others [34,35]. Moreover, various qualitative estimations for larger carbon cages including C60 were presented [36 –40]. At the time of the C60 discovery [1], computations could support fullerene [16,17] and metallofullerene [41] research with increasingly reliable data. It is of historic interest that the four IR bands computationally predicted [42 –47] for C60 were used [48] in the analysis of the carbon-arc C60 synthesis. Fullerene research [49,50] has indeed been characterized by a close theory– experiment cooperation and application of largescale computations [51,52]. Although numerous surveys are available [53 –80], the sole computations of fullerenes and metallofullerenes are reviewed relatively rarely [16,17,41,81 –90]. Structural elucidations for the known higher fullerenes Cn [19,91 – 93] have been based on the so-called isolated pentagon rule (IPR), which claims [94,95] that especially stable fullerenes should have all pentagons surrounded just by hexagons. Connectivity patterns for such cages can be generated by various topological schemes [96 –99] and in fact the cages can be quite numerous. Beyond n ¼ 76 isomerism of the IPR structures should generally play a role in observations. Several such mixtures of empty fullerene isomers have indeed been computed and an agreement with experiments found: C76 [100 –110], C78 [111 –114], C80 [115 –119], C82 [120 – 125], C84 [126 – 132], C86 [91,133– 136], C88 [91,134– 136], C90 [91,137,138], C92 [139], C94 [140] and C96 [141], while there is still only a computational prediction [142] for C98. A similar isomeric interplay has also been described for smaller fullerene systems like [143 – 146] C32 or C36. Various metallofullerenes can coexist [41] in several isomeric forms as well, e.g. Ca@C72 [147 –150], Mg@C72 [151,152], Ca@C74 [147,149,153,154], Ca@C82 [155 – 161], La@C82 [162 – 166], Tm@C82 [167], Ti2@C80 [168], Sc2@C84 [169], Ti2@C84 [170] or Sc2@C76 [171]. Still other interesting isomeric sets can be created [172 – 176] by derivatives of fullerenes or nanotubes. Although the inter-isomeric separation energies are important, they alone cannot predict the relative stabilities of the isomers. Owing to very high temperatures, entropy contributions can even over-compensate the enthalpy terms. Hence, the enthalpy– entropy interplay represents an essential feature of fullerenes and metallofullerenes. 32.2 RELATIVE STABILITIES OF ISOMERS The present quantum-chemical calculations of fullerenes deal with the optimized geometries [177 – 181] obtained at semiempirical, ab initio Hartree – Fock SelfConsistent Field (HF SCF) or density functional theory (DFT) level while ab initio

Computational chemistry of isomeric fullerenes and endofullerenes

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correlated treatments are very rare. The most common semiempirical methods used in fullerene research are [182 – 185] MNDO, AM1, PM3 and SAM1 implemented in the MOPAC and AMPAC programs [186 –188]. Various ab initio procedures are mostly applied using GAUSSIAN [189,190] and SPARTAN [191] program packages. Stability of the SCF wave function [192,193] can be an issue for some fullerenic structures. The geometry optimizations are frequently followed by harmonic vibrational analysis to check the nature of stationary points found and also to construct vibrational partition functions. Vibrational frequency scaling would be relevant for the spectral simulations though it is not important for the high-temperature partition functions [194]. The geometry optimizations will end with a set of m isomeric structures. Their relative concentrations can be expressed as their mole fractions, wi ; using the isomeric partition o functions qi : In terms of qi and the ground-state energy changes DH0;i the mole fractions are given [195 –197]: o qi exp½2DH0;i =ðRTÞ wi ¼ X ð1Þ m o qj exp½2DH0;j =ðRTÞ j¼1

where R stands for the gas constant and T for the absolute temperature. Eq. (1) is an exact formula that can be directly derived [198] from the standard Gibbs energies of the isomers, supposing the conditions of the inter-isomeric thermodynamic equilibrium. Although the partition functions are to be constructed within the rigid-rotor and harmonic-oscillator (RRHO) approximation, Eq. (1) itself is essentially exact. The partition functions qi reflect the rotational, vibrational, electronic, symmetry and chirality [199] features of the individual isomers. If the partition functions are neglected and the vibrational zero-point energy is o extracted from DH0;i ; the relative potential energies DEr;i remain, giving the simple Boltzmann factors: o exp½2DEr;i =ðRTÞ w0i ¼ X m o exp½2DEr;j =ðRTÞ

ð2Þ

j¼1

entirely based only on the potential energy terms with no reference to entropy contributions. However, the simple Boltzmann factors can never cross with a temperature change.

32.3 ENERGETICS AND THERMODYNAMICS OF CARBON CLUSTERS The conventional fullerenes are understood [21,97] as cages built from three-coordinated (sp2) carbon atoms arranged into two types of rings, pentagons and hexagons. As any other polyhedra, fullerenes have to obey Euler’s polyhedral theorem [200,201] (more precisely, the theorem for convex polyhedra): V þF ¼Eþ2 References pp. 908 – 917

ð3Þ

894

Chapter 32

where V denotes the number of vertexes (atoms), F the number of faces (rings) and E the number of edges (bonds), e.g. for C60: V ¼ 60; F ¼ 32; E ¼ 90: As only pentagons and hexagons are allowed, their numbers, n5 and n6 ; give the total count of faces: F ¼ n5 þ n6

ð4Þ

As all atoms are three-coordinated, it must hold for the number of edges: E¼

3V 2

ð5Þ

as each bond is accounted twice. One can also count the edges through rings: E¼

5n5 þ 6n6 2

ð6Þ

V¼

5n5 þ 6n6 3

ð7Þ

Combining Eqs. (5) and (6) yields

Eq. (5) can be placed in Euler’s theorem: 3V þ2 2

ð8Þ

5n5 þ 6n6 þ2 6

ð9Þ

V þF ¼ and combination with Eqs. (7) and (4) gives n5 þ n6 ¼ which can immediately be reduced to:

n5 ¼ 12

ð10Þ

In other words, in any conventional fullerene Cn the number of five-membered rings must be equal to 12. The number of six-membered rings is variable; however, Eq. (7) gives a relationship to the number of carbon atoms: n6 ¼

n 2 20 2

ð11Þ

According to Eq. (11), the smallest possible fullerene has the stoichiometry C20 (12 pentagons, no hexagons). Among all possible fullerenes, a special position is given to the IPR cages [94,95] (that have all the pentagons surrounded just by hexagons) as the arrangements should be especially stable. In other words, there are no pentagon – pentagon junctions in the IPR fullerenes. As there are always 12 pentagons, then the smallest cage that could be of the IPR type should have 12 £ 5 carbon atoms. Indeed, 60 carbon atoms can create an IPR cage (which geometrically is a truncated icosahedron), and the structure is the well-known buckminsterfullerene C60. Already the MINDO/2 computations [34,202] of small carbon clusters Cn pointed out a o =n on the number of simple, smooth dependency of the relative heats of formation DHf;298 carbons n: Later on, the curve was extended into the fullerene domain [51,86,203 –206]

Computational chemistry of isomeric fullerenes and endofullerenes

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and it was still a smoothly decreasing dependency. Xu and Scuseria [206] could find such behavior even with carbon clusters up to C8640. There is a simple way to rationalize the finding. Let us limit our reasoning to the IPR fullerenes. Then, we deal with two types of bonds, frequently called as the 5/6 (between pentagons and hexagons) and 6/6 bonds (shared by two hexagons). Let us suppose moreover that those two types of bonds can be represented [207] by some uniform dissociation energies, H5=6 and H6=6 : In a general IPR fullerene Cn we have always sixty 5/6 bonds, while the number of the 6/6 bonds is variable, ð3n=2Þ 2 60: Now, we can readily write for the atomization heat of the considered Cn: DHat ¼ 60H5=6 þ

3n 2 60 H6=6 2

ð12Þ

The atomization and formation heats for carbon aggregates are linked by the heat of o vaporization of carbon, DHvap : DHfo DHat o ¼2 þ DHvap n n

ð13Þ

or simply in terms of the bond energies: DHfo 3 60 o ðH6=6 2 H5=6 Þ ¼ 2 H6=6 þ DHvap þ 2 n n

ð14Þ

This functional dependency can formally be expressed by: DHfo B ¼Aþ n n

ð15Þ

where B is a positive constant and A a relatively small number. Hence, Eq. (15) is indeed a smoothly decreasing curve. From such a curve, one cannot see a particular stability of C60 or C70. Just the opposite—it might suggest a straightforward rule that the stability would increase with the dimension which is, however, not observed. In its most general form, the fullerene synthesis could be treated as a complex kinetic scheme described by a huge number of kinetic differential equations. The equilibrium composition comes as the limiting case for infinite time. If we treat the problem from a thermodynamic point of view, we should realize that the conventional standard pressure of 1 atm is considerably different from the actual fullerene synthesis conditions. We should expect lower cluster pressures in the carbon-arc synthesis. The actual entropy and Gibbs free energy change with pressure as can be demonstrated [208 –212] on the C60 and C70 cases based on computed or observed [213] data. For example, the equilibrium constant K60=70 for an interconversion between the two clusters, expressed in partial pressures p; offers a deeper insight into the problem [208 – 212]: K60=70 ¼

References pp. 908 – 917

6=7

p70 ð1 2 x60 Þ6=7 21=7 ¼ P p60 x60

ð16Þ

896

Chapter 32

where P stands for the total pressure of the two clusters and x60 is the mole fraction of buckminsterfullerene. At higher pressures, C70 is more populated than C60, but at the conditions of a saturated carbon vapor (simulated simply as a mixture of seven cluster at present) the stability order is reversed in favor of C60 so that an agreement with experiment is obtained. Although the isomeric stability problem is much better understood [198,214– 216] at present than the relative stabilities of non-isomeric carbon clusters, there are some interesting results available for the non-isomeric situation also. The numerical simulations [208 –212] pointed out a temperature increase of the clustering degree under the saturation conditions. This somewhat surprising result can in fact be easily rationalized [217]. While the equilibrium constants for cluster formation decrease with temperature, the saturated pressure increases. It is just the competition between these two terms which decides the final temperature behavior. The finding actually has a more general validity as it was also reported for other vapors [218 – 221].

32.4 SMALL CARBON CLUSTERS The relative stabilities of small carbon clusters can be treated in a similar way as those of fullerenes and thus they represent a useful model system. They have been studied by both theoretical [222 – 240] and experimental [241 – 264] techniques. Raghavachari and Binkley [223] concluded that linear isomers exist for both even- and odd-numbered species (the even clusters being in the triplet electronic state) while planar cyclic isomers exist for the even species. Ion chromatography observations by von Helden et al. [261, 263,264] showed that starting from about n ¼ 7 cyclic rings always exist in addition to the linear forms. Consequently, C7 was computed [236,265] at the MP2/6-31Gp level and found to be about 2 kcal/mol above the linear form. A non-planar cyclic structure was found for C8 [237] and C9 [266] at the same level so that an inversion is possible [237, 267]. The non-planar C8 cycle was, however, not found in the coupled-cluster computations by Martin and Taylor [240]. The MP2/6-31Gp calculations are also available [268 – 270] for C10, C11, C12 and C13. Overall [271], the computations agree with the findings of ion chromatography [261,263,264]. C11 should be the first species for which the cyclic structure becomes dominant. Switch from cyclic to polyhedral species is expected [272] around n ¼ 45: The linear and cyclic (rhombic, bicyclic) C4 isomers were frequently computed. The cyclic vs. linear difference varied considerably; however, newer estimates [273 –277] suggest only a small separation between the linear and cyclic forms. Then, the entropy contributions can lead to one dominant species at higher temperatures [245,278 –280], namely the linear isomer in agreement with ion chromatography [261]. Role of entropy is also important for C6. Raghavachari et al. [222] found the cyclic ground state for C6. Still, the linear isomer is computed [214,240,281] to form more than 95% of the equilibrium mixture at high temperatures in agreement with ion chromatography [261] (cf. Refs. [282,283]).

Computational chemistry of isomeric fullerenes and endofullerenes

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32.5 GENERATION OF CAGES Exhaustive topological enumerations and generations of fullerene cages represent a crucial step, systematically studied by Fowler et al. [96,97,284– 302] and other groups as well [95,98,99,303– 331]. Several concepts are useful in the treatments, e.g. Goldberg polyhedra [286], leapfrog transformation [296], topological duals [306], Stone – Wales transformation [304]. Stone and Wales [304] introduced processes for ring rearrangements in fullerene cages. The topological concept can further be generalized [332] by inclusion of other types of rings. The kinetic processes can physically be realized [333 – 335] by means of catalysis or autocatalysis. The Stone – Wales transformation can in any case be applied as a formal topological structure generator [132,143,296]. A versatile enumeration technique for fullerene cages was developed by Fowler and Manolopoulos [96,97,308]. They represent fullerenes by a ring spiral as fullerenes can be peeled like an orange—each face, after the first, borders its immediate predecessor, so a single continuous spiral is created. Some exceptional cases are, however, possible [302] like a tetrahedral fullerene with 380 atoms. In the spiral algorithm, the bonding topology is reconstructed from the sequence of rings in the spiral. Thus, all possible Cn fullerene graphs can be generated by considering all the ways in which 12 pentagons and ððn=2Þ 2 10Þ hexagons can be combined into a spiral: n þ2 ! 2 Sn ¼ n 12! 2 10 ! 2

ð17Þ

A uniqueness test is based on the eigenvalues of the adjacency matrix. It was found [308 – 313] that, e.g. the number of C60 isomers is 1812. Odd-numbered cages are usually not considered in the enumerations though they are known [314,336, 337]. Among all the possible isomers, the IPR structures are particularly important species. The second smallest IPR fullerene after C60 is C70. It is still a unique structure like those for n ¼ 72 and 74. For any higher carbon atom content, we already deal with IPR isomers, e.g. [96,97], for n ¼ 76; 78, 80, 82, 84, 86, 88 and 90 there are 2, 5, 7, 9, 24, 19, 35 and 46 IPR structures, respectively. Other enumeration algorithms were developed [98,99,132,309,310,320]. There are also interesting applications of Po´lya’s enumeration theorem [198,338]. The theorem was, e.g., applied [321,324,330,331] to enumerations of isomers for various substituted fullerenes. 32.6 SMALLER FULLERENES Although fullerene science is focused at n ¼ 60 and above, smaller fullerenes have also been studied. von Helden et al. [261,272] observed gas-phase fullerene-like structures around C30 (5% for C32, 1% for C30). Even the smallest fullerene C20 is now available [339 – 342] (for its computations, see, e.g. Refs. [216,343 – 346]). C36 fullerene was also isolated [347], computed [144,145,348,349] and linked to narrow nanotubes [350 –353]. References pp. 908 – 917

898

Chapter 32

An interesting illustrative example is supplied with computations [143] of C32. There are just six conventional C32 cages built from pentagons and hexagons [348]. As no IPR structures are possible for C32, a pentagon/pentagon junction will always be present. Their number can be reduced by introduction of four-membered and/or seven-membered rings. Euler’s network closure requirement reads then: 2n4 þ n5 2 n7 ¼ 12

ð18Þ

where ni is the number of rings with i vertices. An exhaustive generation [143] of all C32 cages built from four-, five-, six- and/or seven-membered rings with a limitation, 2n4 þ n7 # 4; gives 199 cages. All the 199 structures were computed [143] and a D4d structure with two four-membered rings (and no heptagons) was found as the ground state. Among five structures lowest in energy, only two are conventional fullerenes, the other cages contain at least one square. However, the entropy contributions reverse [143] the stability order. The structure lowest in energy is the most populated species till a temperature of about 2500 K. Beyond that point it is surpassed by a conventional fullerene. At still higher temperatures, another quasi-fullerene becomes a leading isomer. An interesting enthalpy– entropy interplay was also computed [144 –146] for C36. 32.7 HIGHER FULLERENES Studies of isomerism of the IPR cages from C76 till C98 are available [100 –142]. In fact, C72 is the last member among higher fullerenes [19,92] C60 –C96 that has not been isolated in condensed phase. C74 has also been counted as a ‘missing fullerene’, however, some access to C74 is already possible [354,355]. A low solubility [356] in conventional solvents may be one reason for the difficulties. Hence, as C72 could only be recorded in gas phase [357,358], its structure is not known. Nevertheless, its computations can still serve as an interesting illustrative example. C72 can also act as a host cage for some metallofullerenes [41,147,148] like Ca@C72 or La2@C72. There is just one IPR-satisfying structure for C72, namely with D6d symmetry [97]. It was, however, pointed out in conjunction with the Ca@C72 computations [148] that a nonIPR (i.e. IPR-violating) structure with one pentagon – pentagon junction is by a few kcal/ mol lower in energy than the IPR cage. Moreover, it was demonstrated [359] on the Si60 case that the IPR/non-IPR stability order can be reversed by the entropy factor. Hence, the C72 system was investigated at semiempirical [360] and DFT levels [361]. At the DFT level, the geometry optimizations were carried out [361] using the B3LYP/3-21G treatment. In the optimized B3LYP/3-21G geometries the harmonic vibrational analysis was carried out and also the B3LYP/6-31Gp separation energies were evaluated. The electronic excitation energies were evaluated by means of the ZINDO method [362,363]. The following structures were considered [361]: the IPR cage (a), two non-IPR cages [148] with one pentagon –pentagon junction (b) and (c), a structure [148] with one heptagon (d), a cage [149] with two heptagons (e) and two structures [356] each with two pentagon – pentagon junctions (f) and (g). Fig. 32.1 presents the B3LYP/3-21G optimized structures of the seven C72 isomers. The (c) structure of C2v symmetry with just one pentagon/pentagon fusion represents the lowest energy isomer, being followed

Computational chemistry of isomeric fullerenes and endofullerenes

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Fig. 32.1. B3LYP/3-21G optimized structures of C72 isomers: (a) IPR, (b) 5/5 pair, (c) 5/5 pair, (d) sevenmembered ring, (e) two seven-membered rings, (f) two 5/5 pairs, (g) two 5/5 pairs. (Reproduced from Ref. [361] with permission of Elsevier BV.)

by the IPR structure (a). The other structure with just one pentagon – pentagon junction, (b) isomer of C2 symmetry, comes as the third lowest species. The B3LYP/6-31Gp// B3LYP/3-21G relative energies are 0.0, 11.3, 18.4, 31.1, 34.9, 35.5 and 43.7 kcal/mol for the (c), (a), (b), (e), (g), (d) and (f) structures, respectively. References pp. 908 – 917

900

Chapter 32

Fig. 32.2 presents the DFT computed temperature development of the relative concentrations of the seven C72 isomers in a high-temperature region. The lowest energy structure (c) is the most populated species at any temperature. On the other hand, the IPR structure (a) is always negligible. Its elimination cannot be ascribed only to the potential energy. In fact, several structures higher in potential energy than the (a) isomer are more populated at high temperatures. This is in particular true for the (b) structure with one pentagon – pentagon junction and the (e) structure with two heptagons (the (e) cage 100

(a) D6d (b) C2 (c) C2v (d) Cs (e) Cs (f) D2 (g) Ds

80

(c) C 2v

xi (%)

60

40

(b) C2 20

(e) Cs (d) Cs (f) D2

0

500

1000

1500

2000

2500

3000

3500

4000

4500

T (K) Fig. 32.2. Relative concentrations of the C72 isomers based on the B3LYP/6-31Gp energetics and the B3LYP/3-21G and ZINDO entropy. (Reproduced from Ref. [361] with permission of Elsevier BV.)

Computational chemistry of isomeric fullerenes and endofullerenes

901

is located in the B3LYP/6-31Gp scale about 20 kcal/mol above the (a) isomer). Clearly enough, the IPR structure is suppressed not only by its energy but also by its unfavorable entropy term (especially by the vibrational entropy). The C74 case is, however, different. Recently, Shinohara and co-workers [355] recorded electronic spectrum of C74 anion and suggested that the cage could have D3h symmetry (i.e. the only one available IPR structure [97]). This interesting finding prompted DFT calculations [364] that treated a set of six isomers, five of them being nonIPR species. The computational evaluation, however, shows that the IPR structure prevails at any relevant temperature in agreement with the experimental finding [355].

32.8 ENDOHEDRAL METALLOFULLERENES The combined quantum-chemical and statistical-mechanical stability computations have been applied to a few metallofullerenes also, especially to Ca@C72, Ca@C74, Ca@C82 or La@C82. Ca@C72 was isolated [147] though its observed structure is not yet available. It follows from its very first computations [148,149] that there are four isomers especially low in potential energy (though C72 has only one [97] IPR structure). The endohedral Ca@C72 species created by putting Ca inside the sole IPR cage has been labeled [148] by (a). The other three Ca@C72 isomers considered in Ref. [148] are related to two non-IPR C72 cages (b) and (c), and to a C72 structure with one heptagon (d)—see Fig. 32.3. The extended computations [150] started from the four optimized structures [148] derived using ab initio HF treatment with a combined basis set: 3-21G basis for C atoms and a dz basis set [365] with the effective core potential (ECP) on Ca (for the sake of simplicity, the treatment is coded by HF/3-21G , dz). The structures [148] were reoptimized at the B3LYP/3-21G , dz level. The B3LYP/6-31G , dz//B3LYP/321G , dz relative energies for the (b), (c), (d) and (a) structures are 0.0, 0.8, 18.9 and 35.3 kcal/mol, respectively. In the optimized B3LYP/3-21G , dz geometries the harmonic vibrational analysis was carried out with the analytical force-constant matrix. In the same geometries single-point energy calculations were also performed at the B3LYP/6-31Gp level. The electronic excitation energies were evaluated by means of time-dependent DFT response theory [366] at the B3LYP/3-21G , dz level. Fig. 32.4 presents [150] the temperature development of the relative concentrations of the four Ca@C72 isomers in a high-temperature region. At very low temperatures (not o shown in Fig. 32.4) the structure lowest in the DH0;i scale must be prevailing. However, already at a temperature of 226 K (though such a low temperature is of no importance for the fullerene synthesis) the relative concentrations of the (c) and (b) structures are interchanged and beyond the point the (b) structure is always somewhat more populated. Even more interesting is the behavior of the IPR-satisfying (a) structure. As the structure is the highest in the potential energy, it must be the least populated species at low temperatures. However, later on the entropy contributions (low symmetry, some lower vibrational frequencies and some lower electronic excitation energies) elevate the (a) isomer into the status of a minor isomer that could also be observed. On the other hand, the (d) isomer has the least chances to be detected. Interestingly enough, References pp. 908 – 917

902

Chapter 32

(a)

(b)

(c)

(d)

Fig. 32.3. B3LYP/3-21G , dz optimized structures of Ca@C72 isomers: (a) IPR, (b) 5/5 pair, (c) 5/5 pair, (d) seven-membered ring. (Reproduced from Ref. [150] with permission of Elsevier BV.)

the concentration order at high temperatures for Ca@C72 is quite similar to that previously computed [151] for Mg@C72. In contrast to Ca@C72, Ca@C74 was not only isolated [147] but its structure was also determined [153]. According to the 13C NMR spectra recorded by Achiba and co-workers [153], Ca@C74 exhibits D3h symmetry of its cage. There is only one IPR structure possible [97] for the empty C74 and the sole C74 IPR cage has D3h symmetry. A set of altogether six Ca@C74 isomers was subjected to the stability computations [154]. It turns out, in agreement with the experiment, that the encapsulate with the IPR cage has not only favorable enthalpy but also entropy term and thus all the remaining isomers can act as minor species at best. The third illustrative system, Ca@C82, exhibits the richest isomerism among the Ca endohedrals [147,155,157,366 –369]. Shinohara and co-workers [155] isolated four isomers of Ca@C82 and labeled the isomers (I), (II), (III) and (IV). Dennis and Shinohara concluded [158,370] from the 13C NMR spectra of Ca@C82 (III) its symmetry as C2 : The ultraviolet photoelectron spectra measured by Hino et al. [159] support the finding; a similarity with Tm@C82 (II) was also noted [167]. Very recently, Achiba and co-workers

Computational chemistry of isomeric fullerenes and endofullerenes

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100

80

(b)

x i(%)

60

40

(c)

(a)

20

(d)

0

500 1000 1500 2000 2500 3000 3500 4000 4500 T (K)

Fig. 32.4. Relative concentrations of the Ca@C72 isomers based on the B3LYP/6-31Gp energetics and the B3LYP/3-21G , dz entropy. (Reproduced from Ref. [150] with permission of Elsevier BV.)

[160] measured the 13C NMR spectra of all the four isomers and assigned the symmetry for the isomers (I), (II), (III) and (IV) as Cs ; C3v ; C2 and C2v ; respectively. The Ca@C82 structure – energetics relationships were also computed [156] and a qualitative agreement with the experiment found [41]. The computations were performed at the HF and DFT levels and in both cases the C2v structure was the lowest isomer in the potential energy. There were still three other low energy species—Cs ; C2 ; and C3v : The combined stability computations are also available [161] for the full set of nine isomers [156] of Ca@C82. The nine C82 IPR cages [97] produce nine Ca@C82 endohedrals. Fig. 32.5 presents their structures optimized [161] at the B3LYP/3-21G , dz level. It has turned out that in five cases the original HF structures after the DFT reoptimizations within the same symmetry lead to saddle points with imaginary vibrational frequencies, not to the required local energy minima. When the five saddle points are relaxed and reoptimized, the following local minima are obtained: C3v ðbÞ ! Cs ; C2v ! Cs ; C2 ðaÞ ! C1 ; C2 ðbÞ ! C1 ; Cs ðbÞ ! C1 : Fig. 32.6 presents the temperature development [161] of the relative concentrations of the nine Ca@C82 isomers in a wide temperature region. The enthalpy part of the Gibbs energy is taken from the B3LYP/6-31Gp//B3LYP/3-21G , dz calculations, the entropy References pp. 908 – 917

904

Chapter 32

C3v(a)

C3v(b)

C2v

C2(a)

C2(b)

C2(c)

Cs(a)

Cs(b)

Cs(c)

Fig. 32.5. B3LYP/3-21G , dz optimized structures of Ca@C82 isomers. (Reproduced from Ref. [161] with permission of American Institute of Physics.)

part is evaluated at the B3LYP/3-21G , dz level. At very low temperatures the structure o lowest in the DH0;i scale must be prevailing. However, at a temperature of 1700 K the relative concentrations of the C2v ! Cs ; and Cs ; (c) structures are interchanged and beyond the point the Cs (c) structure is always somewhat more populated. The Cs (c) isomer and also C2 (c) exhibit a temperature maximum. Then, there are still two other structures with significant populations at high temperatures: C3v ðbÞ ! Cs and Cs ðbÞ ! C1 : Although the former species is a bit more populated, their concentrations are rather close. In contrast to the five species with significant populations at least in some temperature regions, the remaining four isomers are computed to be negligible at any temperature. The C3v (a) structure is remarkable as it is lower in potential energy than C3v ðbÞ ! Cs ; however, it is actually suppressed by entropy. The observed yields [160] of the isomers were nearly equal except for the considerably less-produced C3v species (though the HPLC chromatograms [155] could indicate somewhat larger production differences). Fig. 32.6 is in reasonable agreement with the qualitative population

Computational chemistry of isomeric fullerenes and endofullerenes

905

Fig. 32.6. Relative concentrations of the Ca@C82 isomers based on the B3LYP/6-31Gp energetics and the B3LYP/3-21G , dz entropy. (Reproduced from Ref. [161] with permission of American Institute of Physics.)

information in a relatively wide temperature interval. However, the fifth isomer also pointed out by the computations, Cs ðbÞ ! C1 ; has not been observed. The fifth species could be hidden in a chromatographic fraction as a minor component. The last illustrative example deals with La@C82, i.e. an electronic open-shell system. The La@C82 metallofullerene is one of the very first endohedrals that was macroscopically produced [371]. Recently, structures of two of its isomers were clarified [163,164] using 13 C NMR spectra of their monoanions generated electrochemically. The major isomer [163] was thus assigned C2v symmetry and the minor species [164] Cs : The C2v structure was moreover confirmed by an X-ray powder diffraction study [372]. The findings stand in contrast to Ca@C82 with four known isomers. Computations at ab initio HF and DFT levels pointed out [121,162,373] just three IPR cages with a sufficiently low energy after La atom encapsulation: C2v ; C3v (b) and Cs (c). The fourth lowest La endohedral species, C2 (a), is actually already rather high in energy to be significant in experiment. A partial agreement with experiment can be reached [374] for temperatures roughly from 1000 to 1300 K. However, the agreement can be further improved by References pp. 908 – 917

906

Chapter 32

100

80

xi (%)

60

Cs(c) 40

C2v

20

C3v(b)

C2(a) 0

500

1000 1500 2000 2500 3000 3500 4000 4500 T (K)

Fig. 32.7. Relative concentrations [364] of the La@C82 isomers derived within the FEM approach.

an alternative approach to description of the encapsulate motions. One can expect that if the encapsulate is relatively free to move within the cage then, at sufficiently high temperatures, its behavior in different cages will bring about the same contribution to the partition functions. The contributions would then cancel out in Eq. (1). This simplification can be called free, fluctuating or floating encapsulate model (FEM). In the FEM model, in addition to the removal of the three lowest vibrational frequencies, the symmetries of the cages should be treated as the highest possible, considering the averaging effect of the large amplitude motions of the encapsulate. Fig. 32.7 shows the results for the La@C82 isomeric system evaluated within the FEM approach. A good agreement with the observed facts [163,164] is reached in the treatment. There is still another aspect to be considered. The fullerene and metallofullerene production is not always close to the inter-isomeric equilibrium. This factor may be pertinent to the La@C82 case. Lian et al. [375] reported a Ni-catalyzed production of La@C82 with a considerably variable isomeric ratio, indicating a possible kinetic control.

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32.9 CONCLUDING REMARKS This survey shows the ongoing productive interaction between theory and experiment in fullerene research that allows for a deeper insight into the observed facts. For fullerenes and metallofullerenes the mutual interaction is indeed essential, given the complexity of the systems. The considerable thermal effects on the relative isomeric populations revealed by the quantum-chemical and statistical-mechanical computations result from a complex interplay between rotational, vibrational, electronic, potential energy terms, chirality factors, etc. Such effects would never be seen if only energetics is considered and entropy neglected. The treatment is, however, built upon the presumption of the inter-isomeric thermodynamic equilibrium. We do not know yet to which degree this presumption is satisfied in reality though there a few indications that the equilibrium is not necessarily always established. We can only acknowledge that the thermodynamic equilibrium treatment has already produced a reasonable computation – observation agreement for the isomeric systems from C76 till C96, with no serious failure, and also for a few first metallofullerenes. This relatively large tested set supports the belief in a still wider applicability of the equilibrium treatment. There are also other aspects of the combined treatment that should further be studied—especially anharmonicity of vibrations, motions of encapsulates in cages and reliability of inter-isomeric energetics. The non-IPR structures have not been sufficiently explored yet. Heterofullerenes, like cages based on boron nitride, should also attract more attention. Once the thermodynamic treatment is completely exposed, the interest will gradually shift towards complex kinetic schemes and to fullerene sets with variable stoichiometry. The steadily growing family of fullerene and metallofullerene cages will further attract interest of more applied branches of nanoscience and nanotechnology [376], including quantum computing [377] and superconductivity [378]. Further developments are ahead for the cages, the first public exposure [379] of which (namely in Konstanz just 15 years ago) W. Kra¨tschmer1 commented with a modest: “Fullerene molecules must belong to a very frequently overlooked molecular species”. 32.10 ACKNOWLEDGEMENTS The reported research has been supported by a Grant-in-aid for NAREGI Nanoscience Project, Scientific Research on Priority Area (A), and Scientific Research (B) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and also by the Japan Society for the Promotion of Science. The authors also wish to thank the following organizations for kindly permitting the reprinting of copyrighted material: American Institute of Physics; Elsevier Scientific Publishing Company. 1

Interestingly enough, the first association [23,24,380] which came to W. Kra¨tschmer’s mind, when asked by R. Smalley to come to the ISSPIC-5 conference in Konstanz [26], was the fact that, in 1415, the Czech reformer and president of Charles University of Prague, Jan Hus, was also invited to a conference in Konstanz, and then burnt there at the stake as a heretic for the revolutionary theses he presented. Times are changing—575 years later, the Kra¨tschmer–Huffman cross-disciplinary breakthrough was in Konstanz loudly applauded [26,381,382].

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