Calculations of Metallofullerene Yields

Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 8, 1–7, 2011

Calculations of Metallofullerene Yields Zdenˇek Slanina1 ∗ , Filip Uhlík2 , Shyi-Long Lee3 , Ludwik Adamowicz4 , Takeshi Akasaka1 , and Shigeru Nagase5 1

Center for Tsukuba Advanced Research Alliance, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan 2 Department of Physical and Macromolecular Chemistry, Charles University in Prague, Faculty of Science, Albertov 6, 128 43 Praha 2, Czech Republic 3 Department of Chemistry and Biochemistry, National Chung-Cheng University, Chia-Yi 62117, Taiwan 4 Department of Chemistry, University of Arizona Tucson, AZ 85721-0041, USA 5 Department of Theoretical and Computational Molecular Science Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Aichi, Japan The paper reports computations for Mg@C74 , Ca@C74 , Sr@C74 , Ba@C74 , and all lanthanoids, based on encapsulation into the only C74 IPR (isolated pentagon rule) cage, and for Al@C82 , Sc@C82 , Y@C82 and La@C82 based on encapsulation into the IPR C2v C82 cage. Their structural and energetic characteristics are used for evaluations of the relative production yields, employing the encapsulation Gibbs-energy terms and saturated metal pressures. It is moreover shown that the encapsulation potential-energy changes in such series can be well related to the mere ionization potentials of the free metal atoms.

Keywords: Metallofullerene Stability and Production, Metallofullerene Electronic Properties, Molecular Memories.

In 1985, fullerenes appeared1 as a surprise, however, after the 1990 synthesis,2 just the opposite took place—so many new forms3 of nanocarbon have simply been seen as very obvious. Still, the fullerene/metallofullerene/nanotube stabilities are far from being well understood. On the theoretical/computational side, prediction tools are built with numerous simplifications and approximations so that interpretation of the observed facts is possible. For example, the computations could confirm4–6 particular stability of the all isolated isomers of empty fullerenes, whether they are selected from numerous cages by thermodynamics or by kinetics. Then, interest has been shifted to metallofullerenes and nanotubes. A topical example is presented here with stability evaluations in a series of metallofullerene formations X@Cn with one common cage Cn and variable (though somehow similar or linked) metals X. There are several well-established families of metallofullerenes based on one common carbon cage, for example X@C74 or Z@C82 . Although the empty C74 fullerene7 is not yet available in solid form, several related endohedral species X@C74 have been known like Ca@C74 ,8 9 ∗

Author to whom correspondence should be addressed.

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Sr@C74 ,10 Ba@C74 11 (while Mg@C74 was never isolated), and also for some lanthanoids, especially La@C74 ,12–14 all based on the isolated pentagon rule (IPR) D3h C74 cage. Another common metallofullerene family, Z@C82 , is based on the IPR C2v C82 cage—for example Sc@C82 ,15 Y@C82 16 and La@C82 12 17 (while Al@C82 was never isolated). The present paper deals with computational evaluations of the structural, bonding and stability features in the homologous series X@C74 (X = Mg, Ca, Sr, Ba, and all lanthanoids) and also Z@C82 (Z = Al, Sc, Y, La). Special interest is paid to the Gibbs-energy evaluations for estimations of the relative populations. Fullerenes and metallofullerenes have represented objects of very vigorous research activities in connection with their expected promising nanoscience and nanotechnology applications, see e.g., Refs. [18–23]. In particular, various endohedral cage compounds have been suggested as possible candidate species for molecular memories and other future molecular-electronic devices. One approach is built on endohedral species with two possible location sites of the encapsulated atom19 while another concept of quantum computing aims at a usage of spin states of N@C60 20 or fullerene-based molecular transistors.21 Although there can be three-dimensional

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doi:10.1166/jctn.2011.1950

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1. INTRODUCTION

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rotational motions of encapsulates in the cages, the internal motions can be restricted by a cage derivatization22 thus in principle allowing for a versatile control of the endohedral positions needed for the molecular-memory applications. However, a still deeper knowledge of various molecular aspects of the endohedral compounds is needed before their tailoring to nanotechnology applications is possible and computations do play4–6 24–37 a significant role in the process.

rotator and harmonic oscillator (RRHO) approximation, also applied in our previous studies.4–6 42 The present study is mostly based on calculations, only a small portion of observed data43–45 is employed in some steps. Although the temperature region where fullerene or metallofullerene electric-arc synthesis takes place is not yet known, there are some arguments to expect it around or above 1500 K. Thus, the calculations here are presented for two illustrative temperatures of 1500 and 2000 K.

2. COMPUTATIONS

3. RESULTS AND DISCUSSION

The full geometry optimizations were carried out using density-functional theory (DFT), namely employing Becke’s three parameter functional with the non-local LeeYang-Parr correlation functional (B3LYP) in the combined basis set of the 3-21G basis for C atoms and the LanL2DZ basis set with the LANL2 effective core potential for the metal atoms (3-21G∼dz) as implemented in the Gaussian 03 program package.38 In the optimized B3LYP/3-21G∼dz geometries, the harmonic vibrational analysis was then performed. Moreover, in the optimized geometries, higherlevel single-point energy calculations were also carried out with the standard 6-311G∗ and 6-31 + G∗ (6-311G∗ ∼dz and 6-31 + G∗ ∼dz) basis set for C atoms, and finally also with the standard 6-311 + G∗ basis for carbon atoms and the SDD basis with the SDD effective core potential for the metals (6-311 + G∗ ∼sdd). The B3LYP/3-21G∼dz geometries have been known18 to be comparable with the B3LYP/6-31G∗ ∼dz structures for fullerenes and metallofullerenes. Moreover, at the B3LYP/3-21G∼dz level the vibrational analysis is relatively feasible. In the case of the lanthanoid atoms (where the LanL2DZ basis set is mostly not available), the geometry optimization was carried out at the B3LYP/3-21G∼cep level. In addition to the traditional B3LYP functional, a newer MPWB1K functional suggested recently by Zhao and Truhlar39 as the best combination for evaluations of long-range interactions has also partly been employed in this study. The basis set superposition error (BSSE) was estimated by the Boys-Bernardi counterpoise method.40 41 The original Boys-Bernardi counterpoise method was suggested40 for dimers with a fixed geometry. Although a BSSE-respecting geometry optimization would be possible,41 it is rather practical only for simpler systems. Still, in order to reflect the cage distortion, a steric-corrected BSSE treatment is also applied here (labelled e.g., B3LYP/6-31 + G∗ ∼dz and steric) which includes the difference between the energy of the carbon-cage geometry simply taken from, e.g., Z@C82 and the energy of the related fully-optimized empty IPR C2v C82 cage. The Gibbs energies were evaluated using the rotationalvibrational partition functions constructed from the calculated structural and vibrational data using the rigid

The X@C74 series with alkaline earth metals is rather homogeneous as for example documented18 by the B3LYP/3-21G∼dz formal Mulliken atomic charge on X in X@C74 —it reads 2.15, 2.04, and 2.32 for Ca@C74 , Sr@C74 , and Ba@C74 , respectively. Similarly, the Z@C82 metallofullerenes formed18 via metal encapsulations into the IPR C2v C82 exhibit strong charge transfer from the metal to the cage leaving the metal between the Z2+ and Z3+ states. For example, the Mulliken atomic charge in La@C82 , Y@C82 , and Sc@C82 is at the B3LYP/3-21G∼dz level calculated as 2.67, 2.38, and 2.44, respectively. However, the natural population analysis, for example at the B3LYP/6-311 + G∗ ∼sdd level, produces for La@C82 , Y@C82 , and Sc@C82 charges of 2.32, 2.05, and 1.77, respectively. In such homologous series of metallofullerene formations with one common cage X@Cn , a rather straightforward stability relationship can be suggested. Let us consider three formal reaction steps for the illustrative series like Mg@C74 , Ca@C74 , Sr@C74 , Ba@C74 (or Al@C82 , Sc@C82 , Y@C82 , La@C82 ): (i) double- (or triple-) ionization of the free metal, (ii) double (or triple-) charging of the empty cage, and (iii) placing the metal di- (or tri-) cation into the di(or tri-) anionic cage. Let us stress that these three steps are purely formal (as allowed in thermodynamic considerations) and they are not suggested as important steps in the real (unknown) formation mechanism. It should moreover be pointed out that throughout this paper we deal only with the observed ionization potentials and only for isolated (free) metal atoms. The (ii) energy is identical for all members of the series, and the (iii) terms should be similar as they are controlled by electrostatics. For example, the bonding situation in Al@C82 , Sc@C82 , Y@C82 and La@C82 can be surveyed by the highest C-Z Wiberg bond index. The very low values of the C-Z Wiberg index (at the B3LYP/6-311 + G∗ ∼sdd level: 004 ∼ 021) in Z@C82 indicate that instead of a covalent bond, an ionic bond is formed between the metal and cage. Moreover, the feature that the stabilization of metallofullerenes is mostly electrostatic was already documented46 using the topological concept47 48 of ’atoms in molecules’ (AIM) which indeed shows that the metal-cage interactions form

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ionic (and not covalent) bonds. The Wiberg-index analysis can be considered as an additional support for the previous finding. Hence, the free-metal ionization potentials should actually represent a critical yield-controlling factor—the computed relative potential-energy changes upon encapsulation rel E and the relative observed ionization potentials of the free atoms rel IP should— according to the above three-step analysis—be correlated: rel E ∼ rel IP . This interesting conclusion is documented in Figures 1–3. In the three figures, the observed45 second and first or third ionization potentials (IP) for the X atoms in the X@C74 , the Z atoms of the Z@C82 , and for the lanthanoid atoms in the L@C74 series are used (as the Mulliken atomic charge on the metals are computed between 2 and 3). Figure 1 presents the correlation for the MPWB1K/6-31G∗ ∼dz relative potential-energy changes upon encapsulation rel E in the series Mg@C74 , Ca@C74 , Sr@C74 , and Ba@C74 . Although in this case both the observed second and first ionization potentials are presented, the second IP ionization potentials should be more relevant according to the computed Mulliken

charges. In fact, step (ii) in the above decomposition scheme (charging the cage with the same charge) just requires the metal charges to be about the same in the series, not to be equal to some preselected number like 2 or 3. The B3LYP/6-31+G∗ ∼dz potential-energy changes for the Z@C82 series are presented in Figure 2. The suggested correlation works well in both series. Finally, the computed B3LYP/sdd potential-energy changes in the lanthanoid encapsulation L@C74 (L: La−Lu) are plotted against the observed free-atom relative ionization potentials in Figure 3. The correlation is not particularly good in this third series—the uniformity of the metal charges is not obeyed in the L@C74 series. For example, the B3LYP/3-21G∼cep Mulliken charge on L in L@C74 equals to 3.03, 1.67, and 1.35 for La@C74 , Yb@C74 , and Lu@C74 , respectively. Nevertheless, all the three Figures basically support the suggested relationship: rel E ∼ rel IP . In fact, such a correlation should operate for any homologous reaction series of metal encapsulations, i.e., into any type of a common carbon nanostructure. Moreover, this sort of reasoning should step by step explain the fullerene-encapsulation stability islands known throughout the periodic

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δrelIP (kcal/mol)

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δrel∆E (kcal/mol) Fig. 1. The computed MPWB1K/6-31G∗ ∼dz relative potential-energy changes upon encapsulation rel E and the observed45 relative ionization potentials (IP) of the free atoms rel IP for the series Mg@C74 , Ca@C74 , Sr@C74 , and Ba@C74 (solid line—the second IP, dashed line—the first IP).


Sc

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δrel∆E (kcal/mol) Fig. 2. The computed B3LYP/6-31+G∗ ∼dz & BSSE relative potentialenergy changes upon encapsulation rel E and the observed45 relative ionization potentials (IP) of the free atoms rel IP for Z@C82 (solid line— 3-rd IP, dashed line—2-nd IP).

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interrelated with the the standard encapsulation Gibbs energy change GoX@Cn :

2nd IP: 3rd IP:

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GoX@Cn = −RT ln KX@Cn p

Temperature dependency of the encapsulation equilibrium constant KX@Cn p is then described by the van’t Hoff equation:

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δrel∆E (kcal/mol) Fig. 3. The computed B3LYP/sdd//B3LYP/3-21G∼cep & BSSE/steric relative potential-energy changes upon encapsulation rel E and the observed45 relative ionization potentials (IP) of the free atoms rel IP for lanthanoid encapsulation L@C74 (L: La–Lu; black —3-rd IP, — 2-nd IP).

system (though the underlying calculations are rather demanding).49 Let us consider now an overall stoichiometry of a metallofullerene formation: Xg + Cn g = X@Cn g

(1)

although it is not really relevant what kind of reactants is on the left side as they will in the end cancel out in our considerations. The encapsulation process is thermodynamically characterized by the standard changes o of, for example, enthalpy HX@C or the Gibbs energy n o GX@Cn . An illustration will be given here on two reaction series: Mg@C74 , Ca@C74 , Sr@C74 , Ba@C74 and La@C74 , Yb@C74 , Lu@C74 . The equilibrium composition of the reaction mixture is controlled by the encapsulation equilibrium constants KX@Cn p : KX@Cn p =

pX@Cn pX pCn

(2)

expressed in the terms of partial pressures of the components. The encapsulation equilibrium constant is 4

(3)

=

o HX@C n

RT 2

(4)

o where the HX@C term is typically negative so that the n encapsulation equilibrium constants decrease with increasing temperature. Let us further suppose that the metal pressure pX is actually close to the respective saturated pressure pXsat . While the saturated pressures pXsat for various metals are known from observations43 44 (and belong to the essential input set of experimental information43–45 still necessary for our otherwise purely computational treatment), the partial pressure of Cn is less clear as it is obviously influenced by a larger set of processes (though, pCn should exhibit a temperature maximum and then vanish). Therefore, we avoid the latter pressure pCn in our considerations at this stage, and this particular step can actually be done in a rigorous form. In order to observe the relative populations in a metallofullerene series, one can think on an experiment where all the considered metals are simultaneously placed in the electric-arc chamber. This experiment would obviously ensure the same conditions for every member of the series. Moreover, the term pCn in Eq. (2) will be in this arrangement just common for all the members of the series and thus, it can be canceled out. Hence, we can just consider the combined pXsat KX@Cn p terms:

pX@Cn ∼ pXsat KX@Cn p

(5)

that actually control the relative partial pressures of various X@Cn encapsulates in the endohedral series (based on one common Cn fullerene). In this way we get a simpler, applicable scheme. As already mentioned, the computed equilibrium constants KX@Cn p themselves have to show a temperature decrease with respect to the van’t Hoff equation (Eq. 4) which however does not necessarily mean a yield decrease with increasing temperature. Actually, the considered pXsat KX@Cn p product term can frequently (though not necessarily) be increasing with temperature. An optimal production temperature could be evaluated in a more complex model that also includes temperature development of the empty-fullerene partial pressure. Hence, if we want to evaluate production abundances in a series of metallofullerenes like Mg@C74 , Ca@C74 , Sr@C74 and Ba@C74 , just the product pXsat KX@C74 p terms can straightforwardly be used—some representative examples are shown in Table I. While for Mg@C74 and J. Comput. Theor. Nanosci. 8, 1–7, 2011

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Table I. The products of the encapsulation equilibrium constanta KX@Cn p with the metal saturated-vapor pressureb pXsat for Mg@C74 , Ca@C74 , Sr@C74 , and Ba@C74 computed for illustrative temperatures T = 1500 and 2000 K. Endohedral

pXsat KX@C74 p

pXsat KX@C74 p pBasat KBa@C74 p

T = 1500 K Mg@C74 Ca@C74 Sr@C74 Ba@C74

190 × 10−7 233 × 10−3 0.397 89.0

21 × 10−9 26 × 10−5 45 × 10−3 1.0

T = 2000 K Mg@C74 Ca@C74 Sr@C74 Ba@C74

406 × 10−6 760 × 10−3 0.293 11.1

37 × 10−7 68 × 10−4 0.026 1.00

a The standard state—ideal gas phase at 101325 Pa pressure; the potential-energy change evaluated at the B3LYP/6-311G∗ ∼dz level and the entropy part at the B3LYP/3-21G∼dz level. b Extracted from available observed data.43 44


KL@C74 p (atm−1 )

pLsat KL@C74 p

pLsat (atm)

pL sat KL@C74 p

pLa sat KLa@C74 p

T = 1500 K La@C74 831 × 10+10 Yb@C74 334 × 10+0 Lu@C74 169 × 10+1

223 × 10−9 130 × 10+0 121 × 10−8

185 × 10+2 436 × 10+0 205 × 10−7

1.0 235 × 10−2 111 × 10−9

T = 2000 K La@C74 679 × 10+6 Yb@C74 785 × 10−2 Lu@C74 337 × 10−1

927 × 10−6 770 × 10+1 314 × 10−5

629 × 10+1 604 × 10+0 106 × 10−5

1.0 960 × 10−2 168 × 10−7

Endohedral

a The potential-energy change evaluated at the B3LYP/6-311G∗ ∼sdd//B3LYP/321G∼cep & BSSE level, the entropy part at the B3LYP/3-21G∼cep level. b Extrapolated from available observed data.43 44

The motion of the endohedral atom is highly anharmonic, however, such its description is yet possible only with simple potential functions. As long as we are interested in the relative production yields, the anharmonic effects should at least to some extent be canceled out in the relative quotient pXsat KX@C74 p /pBasat KBa@C74 p . Table II presents similar comparison for three members of the lanthanoid encapsulation series L@C74 (L: La, Yb, Lu). La@C74 comes as the most abundant member of the series, in agreement with the fact that even its X-ray analysis is available.14 One to two orders of magnitude less populated is Yb@C74 (where no structural data could be obtained but in fact two isomers were isolated).50 Finally, the least populated Lu@C74 species has never been isolated. In fact, we are dealing with a special case of clustering under saturation conditions.51–54 The saturation regime is a useful simplification—it is well defined, however, it is not necessarily always achieved. Under some experimental arrangements, under-saturated or perhaps super-saturated metal vapors are also possible. This reservation is applicable not only to the electric-arc treatment but even more likely to newly introduced ion-bombardment production technique.55 56 Still, Eqs. (2) and (5) remain valid, however, the metal pressure has to be described by the values actually relevant. A generalized treatment of this type can be designed for multi-atom encapsulations like57 58 Lix @C60 . For some volatile metals their critical temperature could be overcome and the saturation region thus abandoned (though practically speaking, this could come into consideration with mercury and cesium). Anyhow, the saturation regime can give a kind of upper-limit estimates of the production yields. Acknowledgments: The reported research has been supported by a Grant-in-aid for Scientific Research on Innovative Areas (No. 20108001, “-Space”), Scientific Research (A) (No. 20245006), Nanotechnology Support 5

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Ca@C74 the pXsat KX@C74 p factor increases with temperature, it is roughly constant for Sr@C74 at the considered temperatures, and it decreases with temperature for Ba@C74 . The behavior results from competition between the decreasing encapsulation equilibrium constants and increasing saturated-metal pressures. As the encapsulation o has the most negative value for Ba@C74 , enthalpy HX@C n its encapsulation equilibrium constant has to exhibit the fastest temperature decrease that already cannot be overcompensated by the temperature increase of the saturated metal pressure so that the pXsat KX@C74 p term decreases relatively so fast with temperature in this case. In principle, an endohedral with lower value of the encapsulation equilibrium constant can still be produced in larger yields if a convenient over-compensation by higher saturated metal pressure can take place. In order to allow for cancellation of various factors introduced by the computational approximations involved, it is however better to deal with the related relative term pXsat KX@C74 p /pBasat KBa@C74 p . The computed production yield of the (never observed) Mg@C74 species should be by three to four orders of magnitude smaller than that for Ca@C74 . On the other hand, Ba@C74 should be the most abundant endohedral in the series (Table I). This stability picture qualitatively agrees with the observed facts: for Ba@C74 even microsrystals could be prepared11 so that a diffraction study was possible, while for Sr@C74 at least various spectra could be recorded10 in solution, and Ca@C74 was studied9 only by NMR spectroscopy. Although the energy terms are likely still not precise enough, their errors could be comparable in the series and thus, they should cancel out approximately in the relative term pXsat KX@C74 p /pBasat KBa@C74 p . This should be the case of, for example, the higher corrections to the RRHO partition functions, including motions of the encapsulate.

Table II. The products of the encapsulation equilibrium constanta KL@C74 p with the metal saturated-vapor pressureb pLsat for three lanthanoids computed for illustrative temperatures T = 1500 and 2000 K.


Project, the Next Generation Super Computing Project (Nanoscience Project), Nanotechnology Support Project, and Scientific Research on Priority Area (Nos. 20036008, 20038007) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan; by the National Science Council, Taiwan-ROC; and by the Ministry of Education of the Czech Republic (MSM0021620857) and the Czech Science Foundation/GACR (P208/10/0179).

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Received: 25 December 2010. Accepted: 23 January 2011.

RESEARCH ARTICLE


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