Formation of dimers of light noble atoms under encapsulation within fullerene’s voids
 Tymofii Yu Nikolaienko^{1} and
 Eugene S Kryachko^{2}Email author
https://doi.org/10.1186/s116710150871x
© Nikolaienko and Kryachko; licensee Springer. 2015
Received: 20 October 2014
Accepted: 20 March 2015
Published: 17 April 2015
Abstract
Van der Waals (vdW) He_{2} diatomic trapped inside buckminsterfullerene’s void and preserving its diatomic bonding is itself a controversial phenomenon due to the smallness of the void diameter comparing to the HeHe equilibrium distance. We propound a computational approach, including smaller fullerenes, C_{20} and C_{28}, to demonstrate that encapsulation of He_{2} inside the studied fullerenes exhibits an interesting quantum behavior resulting in a binding at shorter, nonvdW internuclear distances, and we develop a computational model to interpret these HeHe bonding patterns in terms of Bader’s atominmolecule theory. We also conjecture a computational existence of He_{2}@C_{60} on a solid basis of its theoretical UV absorption spectrum and a comparison with that of C_{60}.
Keywords
Review
Background introduction
At the recent lecture of Prof. Ihor R. Yukhnovskii ‘Phase Transition of the First Order Below the LiquidGas Critical Point’, partly published elsewhere [1], one of the authors of the titled work, E.S.K., has been actually impressed by a great vitality of the ingenious idea that lay behind a very simple equation of state, which Johannes Diderik van der Waals derived in his PhD thesis in 1837 and which won him the 1910 Nobel Prize in Physics [2], and that spread over centuries, the idea of the attractive dispersion force referred, after him, to as the van der Waals force. This force is responsible for the correction to the pressure in this equation of state and governs myriads of interactions appearing between atoms and molecules in a variety of chemical and biochemical processes [37]. Nevertheless, this vitality of van der Waals (vdW) interactions lies actually in that they continue  even after 177 years  to wonder: three recent events will serve as good examples.
The spectroscopic detection of the weakly bound van der Waals diatomic LiHe has been reported [8] in 2013. Actually, this system was predicted 14 years earlier, as existing with a single bound rovibrational state in the X^{2}Σ ground electronic state characterized by the average bond length of approximately 28 Å and the binding energy of 0.0039 cm^{−1} (approximately 0.56 mK) [9]. Another surprise came in 2000 when the diffraction experiments [10] of molecular beam consisting of small clusters of He finally resolved the longstanding paradox with the van der Waals ^{4}He_{2} dimer. The paradox  not yet then thought as that  started in 1931 when Slater and Krkwood performed the first calculation of the HeHe potential [11], later corrected by Hirschfelder, Curtiss, and Bird in their famous book [12], and thoroughly reviewed by Margenau and Kestner [13], Hobza and Zahradnik [5], Kaplan [3], and Barash [14] (and the references therein, on the works of L. D. Landau school in particular). The importance of this interaction is hardly to overestimate since helium is the second most abundant element after hydrogen and the second simplest atom in the universe (see, e.g., [15]). The interaction between two helium atoms arises electrostatically, when an electric multipole on one atom creates a surrounding electric field that induces an electric mutipole moment on the other. In contrast to other molecular interactions, the van der Waals one is not related to a charge transfer  according to the Mulliken rule, the charge transfer is completely absent in the HeHe interaction (see e.g., [16], p. 877) due to the enormous ionization potential of He equal to 24.5 eV, its small (in fact, negative) electron affinity, and very small polarizability α = 0.21 Å^{3}. The latter make the HeHe interaction extremely weak: the mean HeHe internuclear distance (the bond length, in a sense) reaches 52 ± 4 Å and its binding energy 1.1 mK [10] (≈0.0022 kcal/mol, compared to quantum chemical accuracy of approximately 1 kcal/mol [17]), thus providing, first, an unusual inertness of He: Toennies [15] mentioned that only ‘a few compounds containing helium have been predicted [1], but none have been found,’ and, second, the breakdown of the BornOppenheimer approximation. In this lies the idea of the aforementioned paradox.
Another surprise came out from an unexpected side, from fullerenes [18]: in 2009, Peng and Wang et al. [19] developed the explosionbased method and prepared the endohedral fullerene He_{2}@C_{60}, which existence was confirmed in their mass spectrum experiments. It is not, however, absolutely clear how C_{60} enables to accommodate He_{2} dimer since its void diameter 0.7 nm = 7 Å is smaller than the aforementioned mean HeHe internuclear distance, and thus, rules out that He_{2} dimer is still bonded therein. These authors claimed that the HeHe bonding in He_{2}@C_{60} arises due to the following mechanism: the repulsive interaction between two helium atoms keeps them away from the center, thus approaching each to C_{60} surface and establishing a charge transfer between He and C_{60}. Altogether, this slightly distorts the C_{60} architecture that was detected in the peak recycling high performance liquid chromatography (HPLC) retention time. Only fewer computational works that have been done in parallel to this experiment mostly fell to agree with the latter and to explain it.
Our aim is to recover the agreement between experiment and theory by conducting a series of computations which include van der Waals effects and to offer the computational model behind the mechanism of bonding in He_{2}@C_{60}. The layout of the present work is the following. The ‘Methodological strategy’ section opens the methodological content of this work. The next section ‘Results and discussion’ focuses on discussing theoretical HeHe and NeNe bonding patterns and offers, and in the ‘Notes: computational experiment’ section, we give a definite computational evidence for the very existence of He_{2}@C_{60} in terms of its theoretical UV absorption spectrum which is experimentally measurable. The work completes with thorough discussions and future perspectives.
Methodological strategy
All systems studied in the present work are divided into three categories: fullerenes, He and Ne@fullerenes, and He_{2} and Ne_{2}@fullerenes where the intermediate one is chosen as the reference origin to examine the HeHe and NeNe bondings in the last category.
Fullerenes
Fullerenes belong to the class of materials with a high ratio of surface to volume. According to the mathematical definition [20], a fullerene is the surface of a simple closed convex 3Dpolyhedron with only 5 and 6gonal faces (pentagons and hexagons). We assert that this surface/volume high ratio definitely predetermines a hollow cage architecture or void within a fullerene whose propensity is to accommodate (incapsulation or embedding) therein guest atom(s) or molecule(s) [21]. The latter system with the fullerene doped by atom is nowadays dubbed, after Cioslowski [22] and Schwarz and Krätschmer [23], as an ‘endohedral fullerene’ that originates from Greek words ενδον (‘endon’  within) and εδρα (‘hedra’  face of geometrical figure) [21].
He@fullerenes and He_{2}@fullerenes
Endofullerenes with encapsulated noble gas (Ng) atoms have been scarcely studied in the past, both experimentally and theoretically, compared to the first endohedral metallofullerene (EMF) La@C_{82}, isolated in 1991 [30]. The reason is in that the Ngencapsulation has very low yields and features a rather tedious separation from the host fullerene [21]. Though, the experiments on hightemperature decomposition of Ng@C_{60} ⇒ Ng + C_{60} have revealed the activation barrier of ca. 90 kcal∙mol^{−1} high [31,32] (see also [33]). On the theoretical side, it is worth mentioning the secondorder MøllerPlesset perturbation and density functional computational approaches [29,33,34] to study Ngendohedral complexes with C_{60}buckminsterfullerene [29,33,34] (see [21,35] for current reviews and references therein).
In the present work, Ng@fullerenes (Ng = He, Ne) and Ng_{2}@fullerenes were studied at the same computational level as their cage fullerenes (see Figure 1). To reveal the bonding patterns in the He_{2}@fullerenes, Bader’s ‘atomsinmolecules’ (AIM) theory [3638] was invoked since it provides a mathematically elegant approach [37] to describe a bonding. It should be noted beforehand that the theme on a chemical bond is rather fragile and subtle (see, e.g., Introduction in [39] and references therein). Once Bader [36] conjectured that oneelectron density ρ(r), r ϵ ℝ^{3} of a given molecule should contain the essence of this molecule’s structure. Precisely, the topology of density is characterized by introducing the corresponding gradient vector field ∇_{ r }ρ(r) given by a bundle of trajectories as curves r = r(s) parametrized by some parameter s and satisfying the equation dr(s)/ds = ∇_{ r } ρ(r( s )). The trajectories with zero gradient defines the zeroflux surface ∂Ω: = {r ϵ ℝ^{3} n(r) ∙∇_{ r } ρ(r) = 0, where n = r/r}. This surface bounds a region in ℝ^{3} that defines a (topological) ‘atom’, or atomic basin. Two atoms are defined as bonded if they share a common interatomic surface. If follows from topology that each zeroflux surface contains a (3,–1)type critical point r _{c} ϵ ℝ^{3} (CP) where ∇_{ r }ρ(r _{c}) = 0 and where the Hessian matrix of ρ has two negative and one positive eigenvalues. The eigenvector of this Hessian matrix corresponding to its positive eigenvalue define two directions in which two trajectories of the ∇_{ r }ρfield originate from the critical point forming a socalled ‘bond path’ connecting two attractors (which typically are atomic nuclei).
It is worth noticing that although AIM itself provides only a definition of a topological atom and does not provide a formal proof for its relevance to ‘chemical’ atoms, numerous examples demonstrate the fruitfulness of equating these concepts. In particular, bond paths have been shown to be a universal indicator of bonded interactions [40,41]. In the latter context, AIM is used in this work. Using Gaussian package, we obtained the electron density distributions at the corresponding computational level using the keyword ‘output = wfn’ and analyzed it with AIMAll package [42] to reveal all (3,–1)type critical points and bond paths of the electron charge density distribution. A bond ellipticity that defines a measure of the extent to which a charge is preferentially accumulated in a given plane [30] was calculated as ε = λ_{1}/λ_{2} − 1, where λ _{1} and λ _{2} are negative eigenvalues (λ _{1} ≥ λ _{2}) of the Hessian matrix \( {H}_{ij}=\frac{\partial^2\rho }{\partial {x}_i\partial {x}_j} \) (i,j = 1,2,3) evaluated at the bond critical point. When possible transformations of a given molecular graph represented by a set of molecular bond paths are considered, it can be shown that ‘the ellipticity of the bond which is to be broken increases dramatically and becomes infinite at the geometry of the bifurcation point’ so that ‘a structure possessing a bond with an unusually large ellipticity is potentially unstable’ [37]. Therefore, the value of bond ellipticity can be considered as the measure of bond stability.
In conclusion, naturally anticipating the contribution of a vdW force into the bonding of the titled endofullerenes (see e.g., [43]), we also employed the ORCA package [44,45] using the densitydependent, nonlocal dispersion functional of Vydrov and Van Voorhis [46] in conjunction with the Ahlrichs’ TZV(2d,2p) polarization functions. Timedependent DFT [47] was invoked within this package to calculate UV absorption spectra of C_{60} and He_{2}@C_{60}.
Results and discussions
Data of the AIM analysis of the electron density distribution in Ng _{ 2 } @C _{ 60 } (Ng = He, Ne)
Bond (A · · · B)  R _{ AB } , Å  ρ ^{ cp } · 10 ^{ 2 } , e / a _{ B } ^{ 3 }  Bond ellipticity 

He_{2}@C_{60}  
He_{1} · · · He_{2}  1.979^{a,b}  1.26  3⋅10^{−6} 
C_{38} · · · He_{1}  2.588  1.07  1.21 
C_{44} · · · He_{2}  2.588  1.07  1.21 
He_{2} ^{ + }(^{2}Σ_{u} ^{+}):  
Present work  1.1881^{c}  
MRCI [69]  1.0816  
B3LYP [56]  1.1454^{d,e}  
1.0806^{f}  
Ne_{2}@C_{60}  
C38 · · · Ne1  2.649  1.57  8.31 
C58 · · · Ne1  2.631  1.63  10.46 
C29 · · · Ne1  2.638  1.60  5.53 
C41 · · · Ne2  2.638  1.60  5.53 
C37 · · · Ne2  2.631  1.63  10.46 
Ne1 · · · Ne2  2.096  3.40  6⋅10^{−5} 
C44 · · · Ne2  2.649  1.57  8.31 
C38 · · · Ne1  2.649  1.57  8.31 
Finally, one can conclude from Table 1 and Figure 5 that in all studied complexes the He · · · He bond is much stronger than that between He and carbon atoms of the fullerene. In this regard, let us compare the HeHe stretching mode ν_{HeHe} in the studied endofullerene with ν_{HeHe} ^{expt} = 1,698.5 cm^{−1} of the dihelium cation [5759]: (a) in [HeHe]^{+0.02}@C_{60} ^{−0.02} ν_{HeHe} contributes to the collective modes centered at 495.7, 504.8, and 531 cm^{−1}. Note, for a purpose of comparison, that (b) in He_{2}@C_{20} ν_{HeHe} peaks at 2,380.6 cm^{−1}, whereas in He_{2}@C_{28} ν_{HeHe} peaks at 1,682.1 cm^{−1}; and (c) in [NeNe]^{+0.08}@C_{60} ^{−0.08} contributes to the collective modes centered at 440.6, 458.2, and 519.4 cm^{−1}. A peculiar feature of He · · · C bond, especially in He_{2}@C_{20} complex, is a rather large magnitude of ellipticity which, along with symmetry considerations, probably indicates that He_{1} · · · C_{14} and He_{2} · · · C_{9} bonds could also exist in He_{2}@C_{20} complex (see Figure 2).

Overlap of the asymptotic tails of the electron densities of carbon atoms with that of He that may lead to negative Mulliken charges q _{He} since the corresponding internuclear distances of 1.98 and 2.37 Å are smaller than the sum of the vdW radii which are correspondingly equal to 1.4 for He and 1.7 Å for carbon [52]. To shed a light on this mechanism, we performed some additional calculations for the same complex geometries with the He atom replaced by the ‘ghost’ (defined as the ‘atom’ with the same set of basic functions and the zero nuclear charge). They show that (i) in He@C_{20}, the ‘ghost’ He charge is −0.358 (vs. −0.240 for real atom) so that insertion of a single He into a void of C_{20} results in overlapping of electronic clouds and thus to negative Mulliken charge on helium; (ii) in He_{2}@C_{20}, the ‘ghost’ He atom acquires a Mulliken charge of −0.101 (cf. −0.340 for real atoms). It should be noted however that each of He_{2} atoms lies apart further from the center of C_{20} void as compared to the situation with the single He atom [28], so that the charge repelled by He_{1} can, in principle, induce an increase of population on He_{2} and vice versa.
Notes: computational experiment
Glancing over Figure 7, our first impression is that the UV absorption spectra of He_{2}@C_{60} system and of C_{60} are quite similar: two peaks, one is narrow, the other is quite broad  and such similarity we have already observed in the UV spectra of Kr@C_{60}, also isolated by HPLC, and C_{60} [21,61]. On the other hand, this similarity emphasizes a distinguished difference of the studied UV spectra and therefore, the way to experimentally discriminate between the corresponding systems, He_{2}@C_{60} and C_{60}.
Conclusions
After 30 years since the serendipitous discovery of fullerenes by Sir Kroto and coworkers [62], let us recall the statement by Ashcroft [63] that ‘The issue for C60 seems to go deeper.’ This is precisely what has been done in the present work which provides a solid computational basis for the existence of the buckminsterfullerene with the vanderWaalsbonded He dimer which has been recently isolated in the HPLC experiments. A variety of its computational properties, from spectroscopic to the bonding ones, calculated by invoking Bader’s ‘atomsinmolecules’ quantum theory, have been discussed and presented to identify its experimental ‘fingerprints’ and to reveal the mechanism of its bonding after trapping of He_{2} inside C_{60}.
Declarations
Acknowledgements
We thank the GRID computational facilities of the Bogolyubov Institute for Theoretical Physics for the excellent computational service. One of the authors, ESK, would like to acknowledge stimulating discussions with Sigrid Peyerimhoff and Stefan Grimme. This work was supported by the Research Grant of the Alexander von Humboldt Foundation and conducted within the Programme ‘Microscopic and Phenomenological Models of Fundamental Physical Processes in Micro and MacroWorlds’ of the Division of Physics and Astronomy (PR No. 0112U000056), Natl. Acad. Sci. Ukraine.
Note: After submission of this work, one of the present authors, ESK, received a copy of the 2015th paper [54] which systematically studied the encapsulation of raregas atoms into a series of fullerenes, from C_{20} to C_{60}, using the dispersioncorrected DFT.
Authors’ Affiliations
References
 Yukhnovskii IR, Kolomiets VO, Idzyk IM. Liquidgas phase transition at and below the critical point. Condens Matter Phys. 2013;16:1–6.View ArticleGoogle Scholar
 van der Waals JD, The Nobel Prize in Physics 1910. Nobelprize.org. Nobel Media AB. http://www.nobelprize.org/nobel_prizes/physics/laureates/1910/. Accessed 12 April 2015
 Kaplan IG. Introduction to theory of intermolecular interactions. Nauka, Moscow: John Wiley & Sons; 1982. p. 20. in Russian.Google Scholar
 Kaplan IG. Intermolecular interactions: physical picture, computational methods and model potentials. Chichester: Wiley; 2006.View ArticleGoogle Scholar
 Hobza P, Zahradnik R. Intermolecular complexes. Prague: Academia; 1988.Google Scholar
 Israelachvili JN. Intermolecular and surface forces. London: Academic; 1985. Section 2.4.Google Scholar
 Kipnis AY, Yavelov BE, Rowlinson JS. van der Waals and molecular sciences. New York: Oxford; 1996.Google Scholar
 Tariq N, Taisan NA, Singh V. Weinstein J D. Phys Rev Lett. 2013;110:153201.View ArticleGoogle Scholar
 Kleinekathöfer U, Lewerenz M, Mladenović M. Phys Rev Lett. 1999;83:4717.View ArticleGoogle Scholar
 Grisenti RE, Schöllkopf W, Toennies JP, Hegerfeldt GC, Köhler T, Stoll M. Determination of the bond length and binding energy of the helium dimer by diffraction from a transmission grating. Phys Rev Lett. 2000;85:2284–7.View ArticleGoogle Scholar
 Slater JC, Kirkwood JG. The van der Waals forces in gases. Phys Rev. 1931;37:682–97.View ArticleGoogle Scholar
 Hirschfelder JE, Curtiss CF, Bird RB. Molecular theory of gases and liquids. N. Y: Wiley; 1954.Google Scholar
 Margenau H, Kestner NR. Theory of intermolecular forces. N. Y.: Pergamon; 1971.Google Scholar
 Barash Yu S, van der Waals Forces (in Russian), Nauka, Moscow, 1988.Google Scholar
 Toennies JP. Helium clusters and droplets: microscopic superfluidity and other quantum effects. Mol Phys. 2013;111:1879.View ArticleGoogle Scholar
 Collins JR, Gallup GA. Contributions to interatomic and intermolecular forces. II. Interaction energy of two He atoms. Mol Phys. 1983;49:871–9.View ArticleGoogle Scholar
 Morokuma K. New challenges in quantum chemistry  quests for accurate calculations for large molecular systems. Phil Trans Roy Soc Lond A. 2002;360:1149–64.View ArticleGoogle Scholar
 Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE. C_{60}: buckminsterfullerene. Nature. 1985;318:162.View ArticleGoogle Scholar
 Peng RF et al. Preparation of He@C60 and He2@C60 by an explosive method. J Mat Chem. 2009;19:3602–5.View ArticleGoogle Scholar
 Deza M, Sikiric MD, Shtogrin MI. Fullerenes and diskfullerenes. Russian Math Surv. 2013;68:665–720.View ArticleGoogle Scholar
 Popov AA, Yang S, Dunsch L. Endohedral fullerenes. Chem Rev. 2013;113:5989–6113.View ArticleGoogle Scholar
 Cioslowski J, Fleischmann ED. Endohedral complexes: atoms and ions inside the C_{60} cage. J Chem Phys. 1991;94:3730–4.View ArticleGoogle Scholar
 Weiske T, Böhme DK, Hrusak J, Krätschmer W, Schwarz H, Angew. Endohedral cluster compounds: inclusion of helium within C_{60} ^{·+} and C_{70} ^{·+} through collision experiments. Chem Int Ed Engl. 1991;30:884–7.View ArticleGoogle Scholar
 Roland C, Larade B, Taylor J, Guo H. Ab initio IV characteristics of short C20 chains. Phys Rev B. 2001;65:041401(R).View ArticleGoogle Scholar
 Fowler PW, Manolopoulos DE. An atlas of fullerenes. Oxford: Clarendon; 1995.Google Scholar
 Makurin Yu N, Sofronov AA, Gusev AI, Ivanovsky AL. Electronic structure and chemical stabilization of C28 fullerence. Chem Phys. 2001;270:293–308.View ArticleGoogle Scholar
 Dunk PW, Kaiser NK, MuletGas M, RodriguezFortea A, Poblet JM, Shinohara H, et al. The smallest stable fullerene M@C28 (M = Ti, Zr, U). J Am Chem Soc. 2012;134:9380–9.View ArticleGoogle Scholar
 Kryachko ES, Ludeña EV. Density functional theory: foundations reviewed. Phys Rep. 2014;544(2):123–239.View ArticleGoogle Scholar
 Darzynkiewicz R, Scuseria GE. Noble gas endohedral complexes of C_{60} buckminsterfullerene. J Phys Chem A. 1997;101:7141–4.View ArticleGoogle Scholar
 Chai Y, Guo T, Jin CM, Haufler RE, Chibante LPF, et al. Fullerenes with metals inside. J Phys Chem. 1991;95:7564–8.View ArticleGoogle Scholar
 Shimshi R, Khong A, JiménezVázquez HA, Cross RJ, Saunders M. Release of noble gases from inside fullerenes. Terahedron. 1996;52:5143.View ArticleGoogle Scholar
 Saunders M, JiménezVázquez HA, Cross RJ, Poreda RJ. Stable compounds of helium and neon: He@C_{60} and Ne@C_{60}. Science. 1993;259:1428–30.View ArticleGoogle Scholar
 Ramachandran CN, Roy D, Sathyamurthy N. Hostguest interaction in endohedral fullerenes. Chem Phys Lett. 2008;461:87–92.View ArticleGoogle Scholar
 Patchkovskii S, Thiel W. Equilibrium yield for helium incorporation into buckminsterfullerene: quantumchemical evaluation. J Chem Phys. 1997;106:1796–9.View ArticleGoogle Scholar
 RodriguezFortea A, Irle S, Poblet JM. Fullerenes: formation, stability, and reactivity. WIREs Comput Mol Sci. 2011;1:350–66.View ArticleGoogle Scholar
 Bader RFW. Atoms in molecules: a quantum theory. Oxford: Oxford University Press; 1994.Google Scholar
 Bader RF. A quantum theory of molecular structure and its applications. Chem Rev. 1991;1991(91):893–928.View ArticleGoogle Scholar
 Matta CF, Boyd RJ. Chapter 1. In: Matta CF, Boyd RJ, editors. The quantum theory of atoms in molecules. Weinheim: Wiley; 2007. p. 1–34.View ArticleGoogle Scholar
 Krapp A, Frenking G. Is this a chemical bond? A theoretical study of Ng2@C60 (Ng = He, Ne, Ar, Kr, Xe). Chem Eur J. 2007;13:8256–70.View ArticleGoogle Scholar
 Guo J, Ellis DE, Bader RFW, MacDougall PJ. Topological analysis of the charge density response of a Ni_{4} cluster to a probe H_{2} molecule. J Cluster Sci. 1990;1:201–21.View ArticleGoogle Scholar
 Bader RF. A bond path: a universal indicator of bonded interactions. J Phys Chem A. 1998;102:7314–23.View ArticleGoogle Scholar
 Keith T A, AIMAll (Version 14.04.17), TK Gristmill Software, Overland Park KS, USA. http://aim.tkgristmill.com. Accessed 12 April 2015
 Pyykko P, Wang C, Straka M, Vaara J. A Londontype formula for the dispersion interactions of endohedral A@B systems. Phys Chem Chem Phys. 2007;9:2954–8.View ArticleGoogle Scholar
 Neese, F, ORCA  an ab initio, DFT and semiemprical SCFMO package (Version 2.9.1), University of Bonn, Bonn, Germany. https://orcaforum.cec.mpg.de/. Accessed 1 November 2012
 Grimme S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J Comput Chem. 2004;25:1463–76.View ArticleGoogle Scholar
 Vydrov OA, Van Voorhis TJ. Nonlocal van der Waals density functional: the simpler the better. Chem Phys. 2010;133:244103 (originally termed VV10).Google Scholar
 DeBeerGeorge S, Petrenko T, Neese F. Timedependent density functional calculations of ligand Kedge Xray absorption spectra. Inorg Chim Acta. 2008;361:965–72.View ArticleGoogle Scholar
 Deleuze MS, Francois JP, Kryachko ES. The fate of dicationic states of molecular clusters of benzene and related compounds. J Am Chem Soc. 2005;127:16824–34.View ArticleGoogle Scholar
 Sure R, Tonner R, Schwerdtfeger P. A systematic study of rare gas atoms encapsulated in small fullerenes using dispersion corrected density functional theory. J Comput Chem. 2015;36:88–96.View ArticleGoogle Scholar
 Cheng C, Sheng L. Abinitio study of heliumsmall carbon cage systems. Int J Quantum Chem. 2013;113:35–8.View ArticleGoogle Scholar
 Kobayashi K, Nagase S. Structures and electronic states of M@C_{82} (M = Sc, Y and La). Chem Phys Lett. 1998;282:325–9.View ArticleGoogle Scholar
 Popov AA, Dunsch L. Bonding in endohedral metallofullerenes as studied by quantum theory of atoms in molecules. Chem Eur J. 2009;15:9707–29.View ArticleGoogle Scholar
 Pavanello M, Jalbout AF, Trzaskowski B, Adamowicz L. Fullerene as an electron buffer: charge transfer in Li@C_{60}. Chem Phys Lett. 2007;442:339–43.View ArticleGoogle Scholar
 Bader RFW. An introduction to the electronic structure of atoms and molecules. Toronto: Clarke Irwin & Co Ltd; 1970. Ch. 6.Google Scholar
 Pauling L. The nature of the chemical bond. 3rd ed. Ithaca, NY: Cornell University Press; 1960.Google Scholar
 Bally T, Sastry GN. Incorrect dissociation behavior of radical ions in density functional calculations. J Phys Chem A. 1997;101:7923–5.View ArticleGoogle Scholar
 Huber KP, Herzberg G. Constants of diatomic molecules. N. Y: Van Nostrand Reinhold; 1979.View ArticleGoogle Scholar
 Frenking G, Cremer D. The chemistry of noble gas elements helium, neon, and argon  the experimental facts and theoretical predictions. Struct Bond. 1973;73:17–93.View ArticleGoogle Scholar
 Balanarayan P, Moiseyev N. Strong chemical bond of stable He_{2} in strong linearly polarized laser fields. Phys Rev A. 2012;85:032516.View ArticleGoogle Scholar
 Lieber CM, Chen CC. Preparation of fullerenes and fullerenebased materials. Solid State Physics. 1994;48:109–48.Google Scholar
 Yamamoto K et al. Isolation and spectral properties of Kr@C60, a stable van der Waals molecule. J Am Chem Soc. 1999;121:1591–6.View ArticleGoogle Scholar
 Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE. C_{60}: buckminsterfullerene. Nature. 1985;318:162–3.View ArticleGoogle Scholar
 Ashcroft NW. Elusive diffusive liquids. Nature. 1993;365:387–8.View ArticleGoogle Scholar
 Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, et al. Gaussian 09, Revision A.02, Gaussian, Inc., Wallingford CT, 2009.Google Scholar
 Zhao Y, Truhlar DG. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: Two new functionals and systematic testing og our M06class functionals and 12 other functionals. Theor Chem Acc. 2008;120:215–41.View ArticleGoogle Scholar
 Zhao Y, Truhlar DG. Density functionals with broad applicability in chemistry. Acc Chem Res. 2008;41:157–67.View ArticleGoogle Scholar
 Tishchenko O, Truhlar DG. Atomcage charge transfer in endohedral metallofullerenes: trapping atoms within a spherelike ridge of avoided crossings. J Phys Chem Lett. 2013;4:422–5.View ArticleGoogle Scholar
 Häser M, Almlöf J, Scuseria GE. The equilibrium geometry of C_{60} as predicted by secondorder (MP2) perturbation theory. Chem Phys Lett. 1991;181:497–500.View ArticleGoogle Scholar
 Cencek W, Rychlewski J. Manyelectron explicitly correlated Gaussian functions. II. Ground state of the helium molecular ion He_{2} ^{+}. J Chem Phys. 1995;102:2533–8.View ArticleGoogle Scholar
 Khatua M, Pan S, Chattaraj PK. Movement of Ng2 molecules confined in a C60 cage: an ab initio molecular dynamics study. Chem Phys Lett. 2014;610611:351–6.View ArticleGoogle Scholar
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