Computed En apsulation Energeti s for MetallofullerenesZden�ek Slanina,�;�� Filip Uhl��k��� and Shigeru Nagase����Department of Theoreti al and Computational Mole ular S ien e, Institute for Mole ular S ien e, Myodaiji, Okazaki444-8585, Ai hi, Japan���S hool of S ien e, Charles University, 128 43 Prague 2, Cze h Republi Abstra t: Some alkali and alkaline-earth metals an be now en apsulated in fullerenes. For example, Li�C60 andLi�C70 an be produ ed by the low-energy bombardment method while Ca�C74, Sr�C74, and Ba�C74 an beprepared by high-temperature syntheses. Hen e, their omputations at higher levels of theory are also of interest.In the report, the omputations are arried out on Li�C60, Li2�C60 and Li3�C60 with the B3LYP and MPWB1Kdensity-fun tional theory (DFT) treatment in the standard 3-21G and 6-31G* basis sets. The omputed energeti ssuggests that Lix�C60 spe ies ould be produ ed for several small x values if the Li pressure is enhan ed suÆ iently.The B3LYP DFT approa h is also applied to Mg�C74, Ca�C74, Sr�C74, and Ba�C74 and produ tion populationsare thus rationalized.Keywords: Endohedral fullerenes; arbon-based nanote hnology; mole ular modeling; mole ular ele troni stru ture;metallofullerene stabilities.INTRODUCTIONThere has been a renewed interest [1-22℄ in systems ontaining alkali metals and fullerenes, in parti ular Li�C60 andLi�C70 produ ed by low energy ion implantation [11,13,14℄ in bulk amounts. Re ently, su h systems have also beensubje ted to more advan ed omputations [19-22℄, espe ially in the studies by Gurin [19,20℄. The vibrational spe -tra were obtained [13,14℄ for Li�C60 and Li�C70. Li2�C60 was also eviden ed in observations [11℄ though in a smallamount ompared to Li�C60. In addition to alkali metals, even alkaline-earth metals an be en apsulated into fullerene ages and a tually a whole rea tion series Ca�C74, Sr�C74, and Ba�C74 is now available from high-temperature te h-niques [23-25℄. The observations moreover suggest a qualitative information on their relative populations. The re entexperimental progress makes omputations of the spe ies even more interesting and, in parti ular, some theoreti alrationalization of the observed relative stabilities would be useful. In the report, the omputations are arried out onLi�C60, Li2�C60, and Li3�C60, and also on Mg�C74, Ca�C74, Sr�C74, and Ba�C74, using the density-fun tionaltheory (DFT) treatments. The report supplies illustrative examples what kind of information an be obtained from al ulations for su h metallofullerene systems. Both potential energy and Gibbs free energy terms are evaluated. Thestudy for the �rst time allows for omputational rationalization of the observed relative populations for metallofullerenerea tion series.COMPUTATIONSThe geometry optimizations were arried out with Be ke's three parameter fun tional [27℄ with the non-lo alLee-Yang-Parr orrelation fun tional [28℄ (B3LYP) in the standard 3-21G basis set (B3LYP/3-21G). The geometryoptimizations were arried out with the analyti ally onstru ted energy gradient as implemented in the Gaussianprogram pa kage [29℄. Although the 3-21G basis set is a small basis, its appli ation has been ustomary for fullerenegeometries owing to the omputational demands (though a he k with larger basis sets would in future be useful).In the optimized B3LYP/3-21G geometries, the harmoni vibrational analysis was arried out with the ana-lyti al for e- onstant matrix. In the same optimized geometries, higher-level single-point energy al ulations werealso performed, using the standard 6-31G* basis set, i.e., the B3LYP/6-31G* level (or, more pre isely, B3LYP/6-31G*//B3LYP/3-21G). As Li�C60 and Li3�C60 are radi als, their omputations were arried out using the unre-stri ted B3LYP treatment for open shell systems (UB3LYP). The ultra�ne integration grid was used for the DFTnumeri al integrations throughout.Re ently, Zhao and Truhlar [30-35℄ performed a series of test DFT al ulations with a on lusion [35℄ that theMPWB1K fun tional (the modi�ed Perdew and Wang ex hange fun tional MPW [36℄ and Be ke's meta orrelationfun tional [37℄ optimized against a kineti s database) is the best ombination for evaluations of nonbonded intera tionswith a relative averaged mean unsigned error of only 11%. The MPWB1K fun tional is also applied in this report.RESULTS AND DISCUSSIONOne atom-type stepwise en apsulations: Lix�C60The UB3LYP approa h is preferred here over the restri ted open-shell treatment (ROB3LYP) as the latter a tually�The orresponding author - e-mail: zdenek�ims.a .jp

94 The Open Chemical Physics Journal, 2008, 1, 94-99

1874-4125/08 2008 Bentham Open

Fig. 1. B3LYP/3-21G optimized stru tures of Lix�C60 (the Li atoms are darkened).exhibits a slow SCF onvergen y for the present systems (and ROB3LYP analyti al frequen ies are not implementedin the Gaussian program pa kage [29℄). Although the unrestri ted Hartree-Fo k (UHF) approa h an be faster, it analso be in uen ed by the so alled spin ontamination [38℄ and indeed, this fa tor was an issue in our previous [15℄UHF SCF al ulations as the UHF/3-21G spin ontamination turned out to be higher than re ommended threshold[38℄ in the expe tation value for the < S2 > term where S stands for the total spin. As long as the deviations fromthe theoreti al value are smaller than 10%, the unrestri ted results have traditionally been onsidered [38℄ appli able.This requirement is well satis�ed for the Li�C60 and Li3�C60 spe ies - for example, at the B3LYP/3-21G level theexpe tation value is 0.7552 and 0.7546 for Li�C60 and Li3�C60, respe tively, i.e., loser less than 1% to the theoreti alvalue. Fig. 1 shows the omputed stru tures of Li�C60, Li2�C60, and Li3�C60. In all the three ases the Li atomsin the optimized stru tures are shifted from the age enter towards its wall. In parti ular, in the Li�C60 spe iesthe shortest omputed Li-C distan e is 2.26 �A while in a entral lo ation (whi h is a saddle point) the shortest Li-Cdistan e at the UB3LYP/3-21G level is 3.49 �A. As for the energeti s of the entri and o�- entri lo ation, the saddlepoint is pla ed by some 9.9 k al/mol higher at the UB3LYP/3-21G level. However, the energy separation is furtherin reased in the UB3LYP/6-31G*//UB3LYP/3-21G treatment, namely to 15.0 k al/mol.In the Li2�C60 ase, the shortest Li-C distan e is even bit shorter, 2.14 �A. On the other hand, the Li-Li separationis omputed as 3.29 �A, i.e., substantially longer than the observed value in the free Li2 mole ule (2.67 �A, f. refs.[39-41℄). In the third spe ies, Li3�C60, the shortest omputed Li-C onta t is even further redu ed to 2.05 �A. TheLi-Li distan es in the en apsulated Li3 luster are not equal - they are omputed as 2.70, 2.76 and 2.84 �A. In identally,while the observed Li-Li distan e for free Li2 is [39-41℄ 2.67 �A, the B3LYP/3-21G omputed value is 2.725 �A (it hangesto 2.723 �A at the B3LYP/6-31G* level). Similarly, also the observed values for the free Li3 luster are available [42,43℄,a tually for two triangular forms - opened (2.73, 2.73, 3.21 �A) and losed (3.05, 3.05, 2.58 �A). The UB3LYP/3-21G omputed distan es in the free Li3 opened luster are 2.78, 2.78, and 3.30 �A. Hen e, there is a good theory-experimentagreement. The formal Mulliken harge (the largest value) found on the Li atoms is somewhat de reasing in theLi�C60, Li2�C60, and Li3�C60 series with the UB3LYP/3-21G values of 1.16, 1.10, and 0.86, respe tively. Still, thetotal harge transferred to the age is in reasing in the series: 1.16, 2.21, and 2.46 (the harges are redu ed in the6-31G* basis).The vibrational analysis enables to test if a true lo al energy minimum was found. All the omputed frequen iesfor the stru tures in Fig. 1 are indeed real and none imaginary (though we ould also lo ate some saddle pointsnot dis ussed here). Moreover, the vibrational frequen ies are primarily used here (together with other omputedmole ular parameters) for the entropy and thus Gibbs free-energy evaluations. Hen e, the spe tros opi aspe ts arenot of parti ular interest in our onne tions, however, let us mention relationships to some observed values. The lowest omputed vibrational frequen ies are mostly represented by motions of the Li atoms. Obviously, owing to symmetryredu tions upon en apsulation, the symmetry sele tion rules do not operate any more in the way they simplify theC60 vibrational spe tra [44℄. Hen e, the vibrational spe tra of Lix�C60 must be onsiderably more omplex than forthe i osahedral (empty) C60 age with just four bands in its IR spe trum [44℄. This in reased spe tral omplexityhas indeed been observed [13,14℄. In identally, the observed harmoni frequen y [39-41℄ for free Li2 is 351 m�1 whilethe omputed B3LYP/3-21G term is 349 m�1 (and the B3LYP/6-31G* value 342 m�1). For the endohedrals, alarger-basis frequen y al ulations are not yet ommon, at least not throuhout a larger rea tion series, though a he kat a more advan ed level would be interesting.Di�erent atom-type en apsulations: X�C74There is a general stability problem related to fullerenes and metallofullerenes - either the absolute stability of thespe ies or the relative stabilities of lusters with di�erent stoi hiometries. One an onsider an overall stoi hiometryof a metallofullerene formation: xY(g) + Cn(g) = Yx�Cn(g): (1)

Computed Encapsulation Energetics for metallofullerenes The Open Chemical Physics Journal, 2008, Volume 1 95

Table 1. Computed en apsulation potential-energy hanges �EYx�Cn (k al/mol) for Lix�C60Spe ies �EYx�Cn �EYx�Cnx aB3LYPbLi�C60 -28.4 -28.4Li2�C60 -51.1 -25.6Li3�C60 -71.0 -23.7MPWB1K Li�C60 -34.9 -34.9Li2�C60 -62.8 -31.4Li3�C60 -101.0 -33.7aThe relative term related to one Li atom. bComputed at the B3LYP/6-31G*//B3LYP/3-21G level. Computed atthe MPWB1K/6-31G*//MPWB1K/3-21G level.The en apsulation pro ess is thermodynami ally hara terized by the standard hanges of, for example, enthalpy�HoYx�Cn or the Gibbs energy �GoYx�Cn . In a �rst approximation, we an just onsider the en apsulation potential-energy hanges �EYx�Cn (i.e., the di�eren es in the total ele troni energy between rea tants and produ ts). Table1 presents the terms for Lix�C60. Their absolute values in rease with the in reasing number of the en apsulated Liatoms. In order to have some dire tly omparable relative terms, it is onvenient to onsider the redu ed �EYx�Cnxterms related to one Li atom. Although the absolute values of the redu ed term de rease with in reasing Li ontent,the de rease is not parti ularly fast (so that, a further in rease of the en apsulated Li atoms ould still be possible).The MPWB1K terms are somewhat more pronoun ed and there is even a di�erent trend for Li3�C60 omparedto B3LYP (as shown by Zhao and Truhlar [30-35℄, B3LYP is however not parti ularly reliable for su h situations).The omputational �ndings help to rationalize why also the Li2�C60 spe ies ould be observed [11℄. The basis setsuperposition error (BSSE) is not estimated for the presented values (a straightforward appli ation of the Boys-Bernardi ounterpoise method would be rather questionable in this situation, partly owing to a substantial agedeformation upon the en apsulation). However, the BSSE terms ould be to some extent additive and thus, theyshould somewhat an el out in a rea tion series. Interestingly enough, the stabilization of metallofullerenes is mostlyele trostati as do umented [45,46℄ using the topologi al on ept of 'atoms in mole ules' (AIM) [47,48℄ whi h showsthat the metal- age intera tions form ioni (and not ovalent) bonds.The problem of the relative stabilities of lusters with di�erent stoi hiometries an also be onsidered in a serieswith variable metal, like Mg�C74, Ca�C74, Sr�C74, and Ba�C74. Let us onsider an overall stoi hiometry of ametallofullerene formation: X(g) + Cn(g) = X�Cn(g): (2)The en apsulation pro ess is thermodynami ally hara terized by the standard hanges of, for example, enthalpy�HoX�Cn or the Gibbs energy �GoX�Cn . The equilibrium omposition of the rea tion mixture is ontrolled by theen apsulation equilibrium onstants KX�Cn;p: KX�Cn;p = pX�CnpXpCn ; (3)expressed in the terms of partial pressures of the omponents. The en apsulation equilibrium onstant is interrelatedwith the the standard en apsulation Gibbs energy hange:�GoX�Cn = �RTlnKX�Cn;p: (4)Temperature dependen y of the en apsulation equilibrium onstant KX�Cn;p is then des ribed by the van't Ho�equation: dlnKX�Cn;pdT = �HoX�CnRT 2 (5)where the �HoX�Cn term is typi ally negative (as shown by available omputations and also as expe ted in order toget a signi� ant stabilization) so that the en apsulation equilibrium onstants de rease with in reasing temperature.Let us further suppose that the metal pressure pX is a tually lose to the respe tive saturated pressure pX;sat.While the saturated pressures pX;sat for various metals are known from observations [49℄, the partial pressure of Cn is

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Table 2. The produ ts of the en apsulation equilibrium onstanta KX�Cn;p with the metal saturated-vapor pressurebpX;sat for Mg�C74, Ca�C74, Sr�C74, and Ba�C74 omputed for illustrative temperatures T = 1500 and 2000 KEndohedral KX�C74;p pX;sat pX;satKX�C74;p pX;satKX�C74;ppBa;satKBa�C74;p(atm�1) (atm)T = 1500 KMg�C74 5.78x10�8 2.53 1.46x10�7 4.2x10�9Ca�C74 0.00919 0.162 0.00149 4.3�10�5Sr�C74 0.518 0.355 0.184 5.3�10�3Ba�C74 1332.6 0.0261 34.82 1.00T = 2000 KMg�C74 2.01x10�7 16.6 3.34x10�6 6.1x10�7Ca�C74 0.00144 3.773 0.00542 9.9�10�4Sr�C74 0.02694 6.124 0.1650 0.030Ba�C74 10.399 0.528 5.489 1.00a The potential-energy hange evaluated at the B3LYP/6-31G��dz level and the entropy part at the B3LYP/3-21G�dzlevel; the standard state - ideal gas phase at 101325 Pa pressure.b Extra ted from available observed data [49℄.less lear as it is obviously in uen ed by a larger set of pro esses (though, pCn should exhibit a temperature maximumand then vanish). However, at present there is no observed information available on the Cn values. Therefore, weavoid the latter pressures in our onsiderations - in fa t, they an el out anyhow within a rea tion series whi h is our ase here. As already mentioned, the omputed equilibrium onstants KX�Cn;p have to show a temperature de reasewith respe t to the van't Ho� equation (5). However, if we onsider the ombined pX;satKX�Cn;p term:pX�Cn � pX;satKX�Cn;p; (6)that dire tly ontrols the partial pressures of various X�Cn en apsulates in an endohedral series (based on one ommonCn fullerene), we get a di�erent pi ture. The onsidered pX;satKX�Cn;p term an frequently (though not ne essarily)be in reasing with temperature so that a temperature enhan ement of metallofullerene formation in the ele tri -ar te hnique is still possible. An optimal produ tion temperature ould be evaluated in a more omplex model that alsoin ludes temperature development of the empty-fullerene partial pressure.If we however want to evaluate produ tion abundan es in a series of metallofullerenes like Mg�C74, Ca�C74,Sr�C74 and Ba�C74, just the produ t pX;satKX�C74;p terms an straightforwardly be used. The metal atoms are omputed here in a dz basis set [50℄ with the e�e tive ore potential (ECP) so that the geometry optimizations are arried out at the B3LYP/3-21G�dz level and the energeti s then evaluated at the B3LYP/6-31G*�dz level. It is a ommon pra ti e to ompute entropy at a lower level and the related energeti s at a higher level of theory as entropydoes not hange onsiderably with a basis-set extension. The results in Table 2 show several interesting features.While for Mg�C74 and Ca�C74 the pX;satKX�C74;p quotient in reases with temperature, it is about onstant forSr�C74 for the onsidered temperatures, and it de reases with temperature for Ba�C74. This behavior results from a ompetition between the de reasing en apsulation equilibrium onstants and in reasing saturated metal pressures. Asthe en apsulation enthalpy �HoX�Cn has the most negative value for Ba�C74, its en apsulation equilibrium onstanthas to exhibit the fastest temperature de rease that already annot be over ompensated by the temperature in reaseof the saturated metal pressure so that the pX;satKX�C74;p quotient de reases with temperature in this ase. In orderto allow for an ellation of various fa tors introdu ed by the omputational approximations involved, it is better todeal with the relative quotient pX;satKX�C74;ppBa;satKBa�C74;p . Table 2 shows that the produ tion yield of Sr�C74 should be bytwo or three orders of magnitude smaller than that for Ba�C74. For Ca�C74 the produ tion yield for the onsideredtemperatures is omputed to be between three and �ve orders of magnitude lower than for Ba�C74, and for Mg�C74between seven and nine orders. In prin iple, an endohedral with lower value of the en apsulation equilibrium onstant an still be produ ed in larger yields if a onvenient over- ompensation by higher saturated metal pressure an takepla e.Although the energy terms are likely still not pre ise enough, their errors are frequently omparable in rea tion se-ries. If this is also the ase in our en apsulation series, the errors should an el out in the relative term pX;satKX�C74;ppBa;satKBa�C74;p .This should be the ase of, for example, the basis set superposition error important for evaluation of the en apsulationpotential-energy hanges. A similar an ellation should also operate for the higher orre tions to the rigid-rotor andharmoni -os illator partition fun tions, in luding motions of the en apsulate. The motion of the endohedral atom is

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highly anharmoni , however, its des ription is yet possible only with simple potential fun tions. Another option forevaluation of the entropi term ould be lassi al mole ular dynami s. As long as we are interested in the relativeprodu tion yields, the anharmoni e�e ts should at least to some extent be an elled out in the relative quotientpX;satKX�C74;ppBa;satKBa�C74;p . In identally, the omputed stability proportions do orrelate with qualitative abundan es knownfrom observations. For Ba�C74 even mi ro rystals ould be prepared [26℄ so that a di�ra tion study was possible,while for Sr�C74 at least various spe tra ould be re orded [25℄ in solution, Ca�C74 was studied [24℄ only by NMRspe tros opy, and Mg�C74 is yet unknown from experiment.ACKNOWLEDGMENTSThe reported resear h has been supported by a Grant-in-aid for NAREGI Nanos ien e Proje t, for S ienti� Resear h on Priority Area (A), and for the Next Generation Super Computing Proje t, Nanos ien e Program, MEXT,Japan, and by the Cze h National Resear h Program 'Information So iety' (Cze h A ad. S i. 1ET401110505). Lastbut not least, the onstru tive suggestions from referees are highly appre iated, too.REFERENCES[1℄ Haddon RC, Hebard AF, Rosseinsky MJ, Murphy DW, Du los SJ, Lyons KB, Miller B, Rosamilia JM, FlemingRM, Kortan AR, Glarum SH, Makhija AV, Muller AJ, Ei k RH, Zahurak SM, Ty ko R, Dabbagh G, Thiel FA.Condu ting �lms of C60 and C70 by alkali-metal doping. 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Computed Encapsulation Energetics for metallofullerenes The Open Chemical Physics Journal, 2008, Volume 1 99