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

Abstract

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.

1. 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 C₆₀ 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 metallofullerenes is also being explored in recent times [18]. Guest–host confined systems with a single Ng atom (He-X) [14] and with Ng dimers within the cavity of C₆₀ have been reported both experimentally [19] as well as theoretically, along with that of several other small molecules such as HF, CO₂, H₂, H₂O, etc. [20,21,22,23,24]. Soon after the theoretical prediction of noble gas dimers trapped within fullerene [25], the experimental detection of He₂@C₇₀ and Ne₂@C₇₀ was reported through NMR and mass spectrometry [26,27]. The confined Ng atoms within the fullerene cages exhibit a van der Waals type of interaction between them to attain stability in most cases. There exists no genuine chemical bond between the guest atoms and the cage C atoms. However, the presence of a “chemical bond” between two Xe atoms confined with the C₆₀ cage is claimed by Krapp and Frenking [20]. Other than fullerenes, noble gas confinement within cages with much smaller cavities such as C₁₀H₁₆, C₂₀H₂₀ and Mo₆Cl₈F₆ [28,29,30,31], along with those within smaller BN-fullerenes, such as B₁₂N₁₂ and B₁₆N₁₆, [32] is also performed to study their viability and stability.

Pure boron cages are also explored in this regard. Among them, the borospherene, B₄₀ cage, shows some interesting features regarding its dynamical behaviour. It contains four numbers of heptagonal and two hexagonal holes, with a HOMO-LUMO energy gap comparable to that of C₆₀. A molecular dynamics study on the molecule reveals the dynamical behaviour of the cluster arising from a continuous interconversion between the two types of holes (B₆ and B₇) [33]. The B₄₀ molecule has been previously used to trap Ng atoms, and the dynamical behaviour of the Ng@B₄₀ system is similar to that of the bare cage [34]. The C₆₀, on the other hand, cannot undergo rapid interconversion among its isomers due to its rigid σ-framework. It contains 12 5-C and 20 6-C rings [35]. Previously, it was proposed that any B, N doped structures of fullerene should not have any B-B or N-N bonds. This required the B, N doped structure of C₆₀ to have one C atom in all the five-membered rings, thus producing C₁₂B₂₄N₂₄ [36]. However, studies on the B₃₀N₃₀ ring show that it forms a stable structure, and it was also hypothesized that it can be easily synthesized from borazine [37]. We have encapsulated neon dimers within this cage to study the stability of the resulting complex and the bonding situation therein. The nature of bonding in the noble-gas-encapsulated borospherene is also investigated and compared with those of B₁₂N₁₂, B₃₀N_30, and C₆₀ cages in an attempt to determine whether any possible alterations are brought about by the dynamical nature of the host cage apart from that arising out of confinement. Density functional theory-based computations are performed, and the bonding analysis includes the natural bond order (NBO) [38,39] Bader’s Atoms-in-Molecules electron density analysis (AIM) [40], and energy decomposition analysis (EDA) [41,42,43]. The dynamical picture of the systems is investigated by performing ab initio molecular dynamics simulations using the atom-centred density matrix propagation (ADMP) [44,45,46] technique.

2. Methods and Computational Details

Geometry optimizations and frequency calculations for the B₄₀, B₁₂N₁₂, Ne₂@B_40, and Ne₂@B₁₂N₁₂ systems are performed using the DFT functional, ωB97X-D [47] with 6-311G(d,p) [48,49] basis set, while those of the larger systems, B₃₀N₃₀, C₆₀, Ne₂@B₃₀N_30, and Ne₂@C₆₀, are performed using the 6-31G [50,51] basis set with the same DFT functional. The dissociation energies calculated for the system use the following equation:

$$D_e=E_{N{e}_2}+E_{cage}-E_{N{e}_2@cage}$$

(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:

$$E_{\mathrm{dist}}=E_{expanded cage}-E_{cage}$$

(2)

where E_dist is obtained by removing the Ne₂ 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 Ne₂@B₄₀ and Ne₂@B₁₂N₁₂) and B3LYP-D3(BJ)/TZ2P//ωB97X-D/6-31G (for Ne₂@B₃₀N₃₀ and Ne₂@C₆₀) levels uses the ADF 2014.01 software [54,55] to evaluate different types of interactions between the Ne₂ and the respective cages. The interaction energy between two relevant fragments is evaluated as the total of the Pauli repulsive interaction (ΔE_Pauli) and three attractive interaction energies, viz., electrostatic (ΔE_elstat,), orbital (ΔE_orb), and dispersive (ΔE_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.

3. Results and Discussion

3.1. Structure and Stability

The cages considered for encapsulating neon dimers are the fluxional B₄₀, the non-fluxional B₁₂N₁₂, and the fullerenes C₆₀ and B₃₀N₃₀. The optimized geometries of the encapsulated cage systems are provided in Figure 1. The fluxional borospherene cage has a D_2d symmetry with an ¹A₁ electronic state. The B₄₀ 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 Ne₂, 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 Ne₂ is oriented along the imaginary line connecting the midpoints of the two opposing B₇ rings in the former case and the B₆ rings in the latter case. The formation of both of these Ne₂@B₄₀ systems is thermochemically unfavourable (see

Table 1. Zero-point (ZPE) corrected dissociation energy (D₀), both ZPE and BSSE corrected dissociation energy (D₀^BSSE), entropy change (ΔS) and free energy change (ΔG) for the dissociation processes, Ne₂@cage → Ne₂ + cage, and Ne₂@cage → Ne + Ne@cage along with the distortion energy (E_dist) calculated as Equation (2). The energy values are provided in kcal mol⁻¹ and the entropy change in kcal mol⁻¹ K⁻¹.

Systems	Edist	Processes	D0	D0BSSE	ΔS	ΔG
Ne2@B40 (C2v)	6.395	Ne2@B40 → Ne2 + B40	−70.5	−81.5	0.048	−83.5
		Ne2@B40 → Ne + Ne@B40	−69.3	−80.3	0.030	−77.4
Ne2@B40 (D2d)	8.106	Ne2@B40 → Ne2 + B40	−79.4	−90.6	0.049	−92.7
		Ne2@B40 → Ne + Ne@B40	−78.1	−89.4	0.031	−86.5
Ne2@B12N12	84.687	Ne2@B12N12 → Ne2 + B12N12	−449.8	−462.2	0.047	−462.7
		Ne2@B12N12 → Ne + Ne@B12N12	−352.1	−364.5	0.026	−359.7
Ne2@B30N30	0.604	Ne2@B30N30 → Ne2 + B30N30	5.3	−11.4	0.035	−5.1
		Ne2@B30N30 → Ne + Ne@B30N30	−6.7	−23.5	0.030	−15.1
Ne2@C60	0.983	Ne2@C60 → Ne2 + C60	−6.9	−19.4	0.040	−31.0
		Ne2@C60 → Ne + Ne@C60	−13.5	−26.0	0.033	−35.1

); 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 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, D₀), both ZPE and BSSE corrected dissociation energy (D₀^BSSE), entropy change (ΔS) and free energy change (ΔG) for the dissociation processes, Ne₂@cage → Ne₂ + cage, and Ne₂@cage → Ne + Ne@cage. The D₀^BSSE has higher negative values than the corresponding D₀ values. From the distortion and dissociation energy values, it is clear that the amount of destabilization to the cage caused by the Ne₂ encapsulation decreases with their increasing cavity size. Although the Ne₂@B₃₀N₃₀ 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 Ne₂ within the B₃₀N₃₀ cage seems to be the most likely to be made favourable at lower temperatures. To determine the reaction energy of the process, 2Ne + cage → [Ne₂@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 → [Ne₂], where [Ne₂] indicates the frozen geometry of Ne₂ as obtained within the respective cages; second, the energy [E_dist] involved in the distortion of the cages in their equilibrium state to the geometry they attain after encapsulating the Ne₂, i.e., cage → [cage]; finally, the interaction energy between the frozen dimer and cage, i.e., [Ne₂] + [cage] → [Ne₂@cage]. The aforementioned energies are tabulated in

Table 2. Reaction energies calculated for all Ne₂@cage systems.

Processes	Reactions	ER (kcal mol−1)
2Ne → [Ne2]	$2\mathrm{Ne} \to {[{\mathrm{Ne}}_2]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{40}\right] \left(C_{2v}\right)}$	12.3
	$2\mathrm{Ne} \to {[{\mathrm{Ne}}_2]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{40}\right] \left(D_{2d}\right)}$	14.7
	$2\mathrm{Ne} \to {[{\mathrm{Ne}}_2]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{12}{\mathrm{N}}_{12}\right] }$	54.8
	$2\mathrm{Ne} \to {[{\mathrm{Ne}}_2]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{30}{\mathrm{N}}_{30}\right] }$	3.7
	$2\mathrm{Ne} \to {[{\mathrm{Ne}}_2]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{C}}_{60}\right]}$	5.8
cage → [cage]	${\mathrm{B}}_{40} \to {[{\mathrm{B}}_{40}]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{40}\right] \left(C_{2v}\right)}$	6.4
	${\mathrm{B}}_{40} \to {[{\mathrm{B}}_{40}]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{40}\right] \left(D_{2d}\right)}$	8.1
	${\mathrm{B}}_{12}{\mathrm{N}}_{12} \to {[{\mathrm{B}}_{12}{\mathrm{N}}_{12}]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{12}{\mathrm{N}}_{12}\right] }$	84.7
	${\mathrm{B}}_{30}{\mathrm{N}}_{30} \to {[{\mathrm{B}}_{30}{\mathrm{N}}_{30}]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{B}}_{30}{\mathrm{N}}_{30}\right] }$	0.6
	${\mathrm{C}}_{60} \to {[{\mathrm{C}}_{60}]}_{\mathrm{in} \left[{\mathrm{Ne}}_2@{\mathrm{C}}_{60}\right]}$	1.0
[Ne2] + [cage] → [Ne2@cage]	[Ne2] + [B40] → [Ne2@B40] (C2v)	50.3
	[Ne2] + [B40] → [Ne2@B40] (D2d)	55.1
	[Ne2] + [B12N12] → [Ne2@B12N12]	312.8
	[Ne2] + [B30N30] → [Ne2@B30N30]	−10.9
	[Ne2] + [C60] → [Ne2@C60]	11.5
2Ne + cage → [Ne2@cage]	2Ne + B40 → [Ne2@B40] (C2v)	69.0
	2Ne + B40 → [Ne2@B40] (D2d)	77.8
	2Ne + B12N12 → [Ne2@B12N12]	452.2
	2Ne + B30N30 → [Ne2@B30N30]	−6.6
	2Ne + C60 → [Ne2@C60]	18.3

.

3.2. 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 Ne₂@B₄₀, and 1.602, 2.130, and 2.050 Å in Ne₂@B₁₂N₁₂, Ne₂@B₃₀N₃₀, and Ne₂@C₆₀ 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 B₁₂N₁₂ (q_Ne = 0.137 |e|) than in the larger cages (q_Ne = 0.047 |e|, 0.042 |e|, and 0.010 |e| in B₄₀, B₃₀N₃₀, and C₆₀, respectively) (

Table 3. Natural charges on the encapsulated Ne atoms, the WBI values between two Ne atoms, and the total WBI value of Ne atoms.

Systems	qNe(1)	qNe(2)	WBINe-Ne	Total WBINe
Ne2@B12N12	0.137	0.137	0.0027	0.2736
Ne2@B40 (C2v)	0.047	0.047	0.0003	0.1015
Ne2@B40 (D2d)	0.046	0.046	0.0004	0.1001
Ne2@C60	0.010	0.010	0.0251	0.0231
Ne2@B30N30	0.041	0.042	0.0003	0.0837

). The charge on the cage atoms of the bare host in B₁₂N₁₂ is 1.164 |e| (positive on the B and negative on the N atoms, respectively), which increases after encapsulating the Ne₂. q_B and q_N ranges within 1.221–1.301 |e| and −1.252–−1.272 |e|, respectively, in the Ne₂@B₁₂N₁₂ system. The B atoms in the fluxional B₄₀ cage have varied charges. In each of the two B₆ 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 B₇ rings range within −0.035–0.121 |e|. In the case of B₃₀N₃₀ and C₆₀, 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 Ne₂@B₁₂N₁₂, C_2v (and D_2d) isomer of Ne₂@B₄₀, Ne₂@B₃₀N₃₀, and Ne₂@C₆₀, 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, ∇²ρ(r_c), calculated for the Ne₂@cage systems. Ne-Ne bond paths are present in all the systems. Six numbers of Ne-N bond paths are detected in the Ne₂@B₁₂N₁₂ system, three Ne-N in Ne₂@B₃₀N₃₀, six and four Ne-B in the C_2v and D_2d isomers of Ne₂@B₄₀, respectively, and ten Ne-C in Ne₂@C₆₀ (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 (∇²ρ(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. Topological descriptors (a.u.) at the bond critical points (BCPs) between Ng atoms in Ne₂@cage systems.

Systems	ρ	∇2ρ(rc)	G(rc)	V(rc)	−G(rc)/V(rc)	H(rc)	G(rc)/ρ(rc)	ELF
Ne2@B12N12	0.130	1.255	0.313	−0.312	1.002	0.001	2.412	0.085
Ne2@B40 (C2v)	0.053	0.446	0.103	−0.095	1.089	0.008	1.933	0.042
Ne2@B40 (D2d)	0.059	0.497	0.116	−0.108	1.074	0.008	1.974	0.046
Ne2@C60	0.036	0.298	0.064	−0.054	1.184	0.010	1.810	0.029
Ne2@B30N30	0.029	0.234	0.049	−0.040	1.237	0.009	1.716	0.024

). 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 ∇²ρ(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 Kr₂ and Ar₂, 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), ∇²ρ(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 B₁₂N₁₂ 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 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. Total interaction energy values and components in kcal mol⁻¹ obtained from EDA results Ne₂@cage systems using Ne₂ and the cage as the fragments.

Systems	ΔEint	ΔEPauli	ΔEelstat	ΔEorb	ΔEdisp
Ne2@B12N12	326.0	666.0	−249.4 (73.4)	−82.9 (24.4)	−7.7 (2.3)
Ne2@B40 (C2v)	59.0	132.9	−51.0 (69.0)	−11.7 (15.8)	−11.2 (15.2)
Ne2@B40 (D2d)	64.0	142.4	−54.5 (69.4)	−12.7 (16.2)	−11.3 (14.3)
Ne2@C60	7.6	35.0	−14.2 (51.6)	−2.2 (8.0)	−11.1 (40.4)
Ne2@B30N30	4.2	28.1	−11.6 (48.4)	−2.0 (8.2)	−10.4 (43.4)

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 ΔE_elstat, ΔE_orb, and ΔE_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 Ne₂@B₁₂N₁₂, which decreases in the larger cages, whereas that of the dispersion interaction increases with cage size.

3.3. 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 B₄₀ 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 B₆ and B₇ rings are also observed (see simulation Video S1). Due to the smaller cavity size in B₁₂N₁₂, such movement of the guest molecule is restricted. Despite the larger sizes of the B₃₀N₃₀ and C₆₀ cages, the Ne₂ 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 C₆₀ 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].

4. 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 B₄₀ and some non-fluxional cages such as B₁₂N₁₂, B₃₀N₃₀, and C₆₀ using a density functional theory approach. The formations of all the Ne₂@cage systems are 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 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.