Can the Fluxionality in Borospherene Influence the Confinement-Induced Bonding between Two Noble Gas Atoms?

A density functional theory study is performed to determine the stability and bonding in the neon dimer inside the B30N30 fullerene cage, the fluxional B40 cage, and within non-fluxional cages such as B12N12 and C60. The nature of bonding in the Ne2 encapsulated B40 is compared with the that in other cages in an attempt to determine whether any possible alterations are brought about by the dynamical nature of the host cage apart from the associated confinement effects. The bonding analysis includes the natural bond order (NBO), Bader’s Atoms-in-Molecules electron density analysis (AIM), and energy decomposition analysis (EDA), revealing the non-covalent nature of the interactions between the Ne atoms and that between the Ne and the cage atoms. The formation of all the Ne2@cage systems is thermochemically unfavourable, the least being that for the B30N30 cage, which can easily be made favourable at lower temperatures. The Ne-Ne distance is lowest in the smallest cage and increases as the cage size increase due to steric relaxation experienced by the dimer. The dynamical picture of the systems is investigated by performing ab initio molecular dynamics simulations using the atom-centred density matrix propagation (ADMP) technique, which shows the nature of the movement of the dimer inside the cages, and by the fact that since it moves as a single entity, a weak bonding force holds them together, apart from their proven kinetic stability.


Introduction
Encapsulation of noble gas (Ng) atoms and their dimers is widely studied in many caged systems in an attempt to understand the possible bonding between the so-called "inert" elements. Owing to their high and low IP and EA, respectively, noble gases tend to be chemically "inert: towards other elements unless they are subjected to a strong polarizing source that would facilitate a deformation in their otherwise rigid electron density to induce a donor-acceptor type of interaction. Noble gas dimers only have weak dispersive interactions within them. Confining them within molecular cages allows them to interact with the cage atoms as well as with each other. Although in most cases the complex formation is not thermochemically favourable, they have kinetic stability and hence do not disintegrate easily once formed. The most common caged compounds to form stable endohedral complexes with varied types of atoms and molecules are the carbon fullerene compounds. Confining noble gas atoms within neutral as well as cationic fullerene systems was experimentally performed using high-energy bimolecular collision reactions for the charged ones. For the neutral systems, generating C 60 from graphite in a He atmosphere, using high temperatures to achieve the "window mechanism" where the Ng atoms are inserted via the formation of a temporary window by breaking one or more of the cage C-C bonds, was adopted. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Later on, the insertion process involved the use of cationic Ng beams on the fullerene cage [15], "molecular surgery" [16,17], etc. The study of endohedral D e = E Ne 2 + E cage − E Ne 2 @cage (1) The energy corresponding to the distortion that the host undergoes due to the encapsulation of the guest molecules (E dist ) is calculated as follows: where E dist is obtained by removing the Ne 2 molecule and by evaluating single-point energy of the expanded cage. Basis set superposition error (BSSE) is corrected by using the counterpoise method [52].
The natural charge on the atoms and Wiberg bond index between two atoms are calculated using NBO 3.1 [38] at the ωB97X-D/6-311G (d,p) level of theory. Electron density analysis is performed using the Multiwfn software [53] to calculate the relevant topological descriptors. The energy decomposition analysis performed at B3LYP-D3(BJ)/TZ2P//ωB97X-D/6-311G(d,p) (for Ne 2 @B 40 and Ne 2 @B 12 N 12 ) and B3LYP-D3(BJ)/TZ2P//ωB97X-D/6-31G (for Ne 2 @B 30 N 30 and Ne 2 @C 60 ) levels uses the ADF 2014.01 software [54,55] to evaluate different types of interactions between the Ne 2 and the respective cages. The interaction energy between two relevant fragments is evaluated as the total of the Pauli repulsive interaction (∆E Pauli ) and three attractive interaction energies, viz., electrostatic (∆E elstat ,), orbital (∆E orb ), and dispersive (∆E disp ) interactions. Lastly, atom-centered density matrix propagation (ADMP) is utilized to carry out the ab initio molecular dynamics simulation. All the above computations are executed using the Gaussian 16 [56] program package in the supercomputing facility, PARAM Shakti, at IIT Kharagpur.

Structure and Stability
The cages considered for encapsulating neon dimers are the fluxional B 40 , the nonfluxional B 12 N 12 , and the fullerenes C 60 and B 30 N 30 . The optimized geometries of the encapsulated cage systems are provided in Figure 1. The fluxional borospherene cage has a D 2d symmetry with an 1 A 1 electronic state. The B 40 cage has two six-membered and four seven-membered ring structures, the continuous interconversion among which causes its fluxional behaviour akin to that of a nanobubble. On encapsulation of Ne 2 , two possible geometries can be obtained, one with a C 2v symmetry and the other with a D 2d symmetry, where the former is marginally more stable. The bond axis of Ne 2 is oriented along the imaginary line connecting the midpoints of the two opposing B 7 rings in the former case and the B 6 rings in the latter case. The formation of both of these Ne 2 @B 40 systems is thermochemically unfavourable (see Table 1); however, once formed, they have kinetic activation barriers high enough to protect from their spontaneous dissociation (~67.5 kcal mol −1 at ωB97X-D/def2-TZVP//ωB97X-D/def2-SVP level) [34]. The distance between the Ne atoms within the cavity reduces from 3.061 to 1.913 Å and to 1.877 Å in the C 2v and D 2d isomers, respectively. The distortion undergone by the host cage as a result of confining the Ne dimers within it is calculated as 6.395 and 8.106 kcal mol −1 for the C 2v and D 2d isomers, respectively.  The B30N30 cage is a BN analogue of the C60 fullerene. Out of the three possible geometries, as reported by Yin et al. [57], we have selected the one with the minimum energy where the cage contains six N-N bonds. It primarily differs from carbon fullerene in the fact that, unlike C60, it has non-uniform diameters of the rings throughout the cage and is slightly larger than C60. The encapsulation of Ne2 within this cage is marginally unfavourable at room temperature (ΔG for the dissociation process is −5.073 kcal mol −1 ), which indicates that at slightly lower temperatures, its formation will be thermochemically feasible. The interatomic distance between the Ne atoms is found to be 2.130 Å , which is relatively larger than that in the B40 cage owing to the availability of ample space within the B30N30 cage. For this reason, the encapsulation of dimers of higher noble gas atoms is also viable without distorting the cage, much like the case in its carbon analogue. The B12N12 cage, on the other hand, has a much smaller cavity with four-and six-membered rings. On Ne2 encapsulation, it becomes distorted to a much higher extent than all the other cages in consideration in this study (see Table 1). The Ne-Ne distance is also much lower (1.602 Å ) due to the higher confinement effect of the small host cage. All attempts of inserting Ar2 within the cage failed due to insufficient space within the cage. Insertion of Ng dimers within C60 has been reported earlier. It can accommodate dimers of higher noble gas elements as well with distortions increasing with increasing size of the Ng atoms. In the case of Ne2@C60, the internuclear distance between the Ne atoms is 2.050 Å . Several geometries with different point group symmetries are possible for this system, which are The B 30 N 30 cage is a BN analogue of the C 60 fullerene. Out of the three possible geometries, as reported by Yin et al. [57], we have selected the one with the minimum energy where the cage contains six N-N bonds. It primarily differs from carbon fullerene in the fact that, unlike C 60 , it has non-uniform diameters of the rings throughout the cage and is slightly larger than C 60 . The encapsulation of Ne 2 within this cage is marginally unfavourable at room temperature (∆G for the dissociation process is −5.073 kcal mol −1 ), which indicates that at slightly lower temperatures, its formation will be thermochemically feasible. The interatomic distance between the Ne atoms is found to be 2.130 Å, which is relatively larger than that in the B 40 cage owing to the availability of ample space within the B 30 N 30 cage. For this reason, the encapsulation of dimers of higher noble gas atoms is also viable without distorting the cage, much like the case in its carbon analogue. The B 12 N 12 cage, on the other hand, has a much smaller cavity with four-and six-membered rings. On Ne 2 encapsulation, it becomes distorted to a much higher extent than all the other cages in consideration in this study (see Table 1). The Ne-Ne distance is also much lower (1.602 Å) due to the higher confinement effect of the small host cage. All attempts of inserting Ar 2 within the cage failed due to insufficient space within the cage. Insertion of Ng dimers within C 60 has been reported earlier. It can accommodate dimers of higher noble gas elements as well with distortions increasing with increasing size of the Ng atoms. In the case of Ne 2 @C 60 , the internuclear distance between the Ne atoms is 2.050 Å. Several geometries with different point group symmetries are possible for this system, which are energetically close to each other. This occurs due to the precessional motion of the dimer inside the cage [58]. Table 1 presents the zero-point corrected dissociation energy (ZPE, D 0 ), both ZPE and BSSE corrected dissociation energy (D 0 BSSE ), entropy change (∆S) and free energy change (∆G) for the dissociation processes, Ne 2 @cage → Ne 2 + cage, and Ne 2 @cage → Ne + Ne@cage. The D 0 BSSE has higher negative values than the corresponding D 0 values. From the distortion and dissociation energy values, it is clear that the amount of destabilization to the cage caused by the Ne 2 encapsulation decreases with their increasing cavity size. Although the Ne 2 @B 30 N 30 cage showed a small positive dissociation energy, the introduction of the BSSE correction produced a small negative dissociation energy. Among all the cages considered, the encapsulation of Ne 2 within the B 30 N 30 cage seems to be the most likely to be made favourable at lower temperatures. To determine the reaction energy of the process, 2Ne + cage → [Ne 2 @cage] (where the third brackets indicate that the system is frozen in that geometry), we need to calculate the energies step by step: first, the energy required to bring the two Ne atoms to their equilibrium distance within the cages, i.e., 2Ne →

Processes
Reactions

Bonding
The interactions between the two Ne atoms inside the cages are investigated in terms of NBO, AIM, and EDA. The internuclear bond distances of the free Ne dimer is reported to be 3.091 Å [59], and the atoms are bound to each other with weak van der Waals forces of attraction. In the studied complexes, the Ne-Ne distances are 1.913 and 1.877 Å in the C 2v and D 2d isomers of Ne 2 @B 40 , and 1.602, 2.130, and 2.050 Å in Ne 2 @B 12 N 12 , Ne 2 @B 30 N 30 , and Ne 2 @C 60 systems, respectively. There is clearly an increase in the interatomic interaction, the extent of which is discussed to determine whether the resulting situation can be considered a genuine chemical bond.
The natural charges on the Ne atoms calculated within the cage systems are much higher in the smaller B 12 N 12 (q Ne = 0.137 |e|) than in the larger cages (q Ne = 0.047 |e|, 0.042 |e|, and 0.010 |e| in B 40 , B 30 N 30 , and C 60 , respectively) ( Table 3). The charge on the cage atoms of the bare host in B 12 N 12 is 1.164 |e| (positive on the B and negative on the N atoms, respectively), which increases after encapsulating the Ne 2 . q B and q N ranges within 1.221-1.301 |e| and −1.252-−1.272 |e|, respectively, in the Ne 2 @B 12 N 12 system. The B atoms in the fluxional B 40 cage have varied charges. In each of the two B 6 rings, two opposing B atoms have q B = −0.210 |e| while the others have q B = 0.118 |e|, whereas the charges on the rest of the B atoms constituting the B 7 rings range within −0.035-0.121 |e|. In the case of B 30 N 30 and C 60 , very little to no change is observed in the natural charges on the cage atoms. Positive charges on the Ne atoms on encapsulation indicate that a partial charge cloud transfer occurs from them to the caged atoms. Wiberg bond indices calculated for the interaction between the two Ne atoms reveal very low values (ranging within 0.0003−0.0004), which are just marginally higher than that in the free Ne dimer (0.0002). WBI values in such low ranges reveal the presence of a van der Waals type of interaction and the absence of any covalent interactions. The total WBI values for the Ne atoms are 0.274, 0.102 (and 0.100), 0.0837, and 0.0231 in Ne 2 @B 12 N 12 , C 2v (and D 2d ) isomer of Ne 2 @B 40 , Ne 2 @B 30 N 30 , and Ne 2 @C 60 , respectively. The higher the total WBI values, the higher the extent of interactions between the Ne and the respective cage atoms. Bader once claimed that the presence of a bond path between two atoms in a molecule in its equilibrium state indicates that there exists a bond between the two [60]. This statement is quite controversial [61][62][63][64][65][66] and is proven not to be true in many cases. We, therefore, rely on the AIM topological analysis to determine the nature of the interaction, rather than claim the presence of a bond just based on the presence of a bond path. Figure 2 depicts the contour diagrams for the Laplacian of the electron density, ∇ 2 ρ(r c ), calculated for the Ne 2 @cage systems. Ne-Ne bond paths are present in all the systems. Six numbers of Ne-N bond paths are detected in the Ne 2 @B 12 N 12 system, three Ne-N in Ne 2 @B 30 N 30 , six and four Ne-B in the C 2v and D 2d isomers of Ne 2 @B 40 , respectively, and ten Ne-C in Ne 2 @C 60 (see Figure 3). We are interested in the nature of interactions that exists between the two Ne atoms under confinement; thus, we have calculated the relevant topological descriptors, viz., electron density (ρ(r c )), its Laplacian (∇ 2 ρ(r c )), local potential, kinetic, and total energy densities (V(r c ), G(r c ), and H(r c )), at the bond critical point (BCP) between the said atoms (Table 4). A positive value of the Laplacian of electron density indicates a depletion of the same, suggesting a noncovalent type of interaction. Although it is a necessary criterion, it is not a sufficient one. A positive value of the summation of the local energy densities (i.e., H(r c ) = G(r c ) + V(r c )) along with positive ∇ 2 ρ(r c ) again points towards the nature of interaction being a non-covalent one [67]. In our cases, the H(r c ) values are positive, but their values are so small (pretty close to zero) that they may change their sign in case we try different levels of theory and stroke all basis sets. It may be noted that for the dimers of larger Ng atoms such as Kr 2 and Ar 2 , negative H(r c ) values are obtained [34]. This means that a partial covalent bond exists in those two dimers. It follows the prognosis made by Linus Pauling that it is possible for heavier Ng atoms to form bonds owing to the fact that they contain loosely bound electrons [68]. The magnitude of ρ(r c ), ∇ 2 ρ(r c ), G(r c ) and V(r c ) gradually decreases with the increase in the size of the cage, whereas H(r c ) increases. The ratio −G(r c )/V(r c ) > 1 in all the systems suggests a purely non-covalent interaction between the Ne atoms [69]. Again, the ratio of G(r c ) to ρ(r c ) provides an insight into the type of interaction: greater than 1 suggests the absence of any covalent interactions.
It has earlier been reported that in the B 12 N 12 cage, some degree of covalent character can be induced between two He atoms [32,70]. Thus, this interesting behaviour of developing partial covalent bonding interaction can be investigated for heavier noble gas dimers as well by subjecting them to a similar degree of confinement. Of course, that would require increasing size of cages for increasing size of the noble gas elements.    The energy decomposition analysis is carried out considering Ne 2 and the cage as two separate fragments. The total interaction energies between the Ne dimer and the different cages are provided in Table 5 along with its components. The Pauli interaction energy is always positive, being repulsive in nature, and it decreases with increasing size of the host cage, highest and lowest corresponding to B 12 N 12 and C 60 , respectively. Among the attractive type of interaction energies, the electrostatic interaction dominates the other two. The percentage contribution to the net attractive energy of ∆E elstat , ∆E orb , and ∆E disp are provided in parentheses, which clearly supports the above statement. In all the cases, the repulsive interaction is so high that it overcompensates all the attractive energies to result in an overall repulsive (positive) total interaction energy, the extent of which decreases with increasing cage size. Substantial orbital contribution is present in the case of Ne 2 @B 12 N 12 , which decreases in the larger cages, whereas that of the dispersion interaction increases with cage size. The energy decomposition analysis is carried out considering Ne2 and the cage as two separate fragments. The total interaction energies between the Ne dimer and the different cages are provided in Table 5 along with its components. The Pauli interaction energy is always positive, being repulsive in nature, and it decreases with increasing size of the host cage, highest and lowest corresponding to B12N12 and C60, respectively. Among the attractive type of interaction energies, the electrostatic interaction dominates the other two. The percentage contribution to the net attractive energy of ΔEelstat, ΔEorb, and ΔEdisp are provided in parentheses, which clearly supports the above statement. In all the cases, the repulsive interaction is so high that it overcompensates all the attractive energies to result in an overall repulsive (positive) total interaction energy, the extent of which decreases with increasing cage size. Substantial orbital contribution is present in the case of Ne2@B12N12, which decreases in the larger cages, whereas that of the dispersion interaction increases with cage size.

ADMP
The interconversion between the six-and seven-membered ring structures in the borospherene cage causes its fluxional behaviour, which has an activation free energy barrier of 14.3 kcal mol −1 [33]. Such transformation is possible owing to the appropriate σ-and π-delocalization among the two possible isomers along with the corresponding transition state. The above transformation is observed in BOMD simulations performed at high temperatures of 1200 and 1500 K, which occur by the movement of one B atom during cage distortion from one of the heptagonal to the neighbouring hexagonal ring. Note that the higher temperatures are not indicative of the fact that the transformation occurs at those temperatures; this means that the higher temperatures help the system overcome the energy barrier within the very small simulation time window (in the order of ps). In realistic time scales, this can be observed at far lower temperatures. Upon encapsulating the noble gas monomer, no significant change is observed in the dynamical behaviour of the cage. ADMP simulations performed for the C 2v isomer of the dimer-encapsulated B 40 at 400 K for a runtime of 500 fs show that the dimer undergoes slight precessional motion along the axis joining the centres of the two opposing heptagonal rings. The interconversions between the B 6 and B 7 rings are also observed (see simulation Video S1). Due to the smaller cavity size in B 12 N 12 , such movement of the guest molecule is restricted. Despite the larger sizes of the B 30 N 30 and C 60 cages, the Ne 2 inside their cavities does not move separately, but as a single entity, proving the existence of some bonding interaction, albeit weak, that holds them together. The dynamical study of the empty C 60 cage investigated earlier at higher temperatures reported no fluxional behaviour because of the fact that the associated energy barrier for the transformation to its other isomeric forms is very high and is thermally forbidden by Woodward-Hoffmann rules [71].

Conclusions
The fluxional behaviour in any system arises due to the presence of low-lying energetically accessible isomers on the potential energy surface. If the energy barrier is low enough, the interconversion is easily observed. In the case of caged systems, this phenomenon can be monitored and utilized to influence the bonding, stability, and reactivity of any confined system or to catalyse a chemical reaction. The stability and bonding in the neon dimer are studied inside the fluxional B 40 and some non-fluxional cages such as B 12 N 12 , B 30 N 30 , and C 60 using a density functional theory approach. The formations of all the Ne 2 @cage systems are thermochemically unfavourable, the least being that for the B 30 N 30 cage, which can easily be made favourable at lower temperatures. The Ne-Ne distance is lowest in the smallest cage, and it increases as the cage size increases due to steric relaxation experienced by the dimer. NBO, AIM, and EDA reveal the non-covalent nature of the interaction between the Ne atoms and that between the Ne and the cage atoms. The dynamical study revealed the nature of movement of the dimer inside the cages and the fact that since it moves as a single entity, a weak bonding force holds them together. Since the fluxionality of the borospherene cage exists at a relatively higher temperature range, its effect on the bonding aspects is not very pronounced at lower temperatures. Additionally, the cage is too small to effectively host a chemical reaction without rupturing. Larger-sized fluxional cages may be inspected with different common reactions within to determine if a dynamic coupling exists between the confined reaction and the fluxional conversion (if so, whether both of them happen simultaneously or sequentially). The combined effect of confinement and fluxionality on certain chemical reactions may lead to some interesting and unusual results.