 Nano Review
 Open Access
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
 Fullerene confinement
 Noble atoms dimers
 Bonding patterns
 QTAIM
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
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