Cage‐size effects on the encapsulation of P2 by fullerenes

The classic pnictogen dichotomy stands for the great contrast between triply bonding very stable N2 molecules and its heavier congeners, which appear as dimers or oligomers. A banner example involves phosphorus as it occurs in nature as P4 instead of P2, given its weak π‐bonds or strong σ‐bonds. The P2 synthetic value has brought Lewis bases and metal coordination stabilization strategies. Herein, we discuss the unrealized encapsulation alternative using the well‐known fullerenes' capability to form endohedral and stabilize otherwise unstable molecules. We chose the most stable fullerene structures from Cn (n = 50, 60, 70, 80) and experimentally relevant from Cn (n = 90 and 100) to computationally study the thermodynamics and the geometrical consequences of encapsulating P2 inside the fullerene cages. Given the size differences between P2 and P4, we show that the fullerenes C70–C100 are suitable cages to side exclude P4 and host only one molecule of P2 with an intact triple bond. The thermodynamic analysis indicates that the process is favorable, overcoming the dimerization energy. Additionally, we have evaluated the host‐guest interaction to explain the origins of their stability using energy decomposition analysis.

bond rule, suggesting a weak π-bond between the phosphorus atoms. 11 In contrast, Kutzelnigg discussed that taking into account only the overlap populations, it is expected that the triple bond in N 2 and P 2 are approximately equal in strength. 12 However, he pointed out that the bond strength also depends on how strongly the valence electrons are attached to the elements, and hence, in qualitative terms, the bond energy on N 2 is significantly stronger than P 2 . 12 Many years later, Jerabek and Frenking brought a quantitative assessment using energy decomposition analysis (EDA). 13 Notably, they found that the contribution of the π-bonding in P 2 (40.5%) is higher than in N 2 (34.4%). Thus, the tendency of P 2 to dimerize is related to the enhanced stability due to the σ-bonds formed rather than π-bonding lost.
The constant quest for environmentally friendly processes which are of preparative value has triggered many investigations on the introduction of phosphorus atoms under mild conditions. 14 In this vein, P 2 is an interesting reagent for producing heterocycles via cycloadditions, for instance. 15 However, its use in synthesis requires the stabilization preserving the triply bonded P P moiety.
Strategies based on transition-metal mediated degradation of the P 4 into P 2 units have been extensively explored in the past (Scheme 1A). 16 The outcome of these reactions is a side-on bridging M 2 P 2 coordination mode, yielding a significant decrease in the multiple bond character relative to free P 2 . 17 Nonetheless, a landmark study by Cummins and co-workers demonstrated that niobium-based coordination can be used for the thermal transfer of P 2 to 1,3-dienes. 18 End-on M=PÀP=M coordination modes are rare cases, and the formation is ascribed to the steric hindrance between the metal moieties. 19 The center moiety has been described as a PÀP single bond given the redox activity of the metals. 20 Recently, Schneider, Holthausen, and co-workers reported the use of redox inactive platinum ligands as an unprecedented platform for the stabilization of P 2 as a neutral, triply bonded unit. 21 Furthermore, the stabilization by strong σ-donor Lewis bases has been also evaluated, counting N-heterocyclic carbene (NHC), 22 cyclic(alkyl)amino carbene (CAAC), 23 and the boryl, 24 and silylene analogues (Scheme 1B). 25 Similarly, the loss of the multiple bond character is observed as a result of a strong donor-acceptor interaction between the Lewis base lone pair and the π* orbitals of the P 2 species. 26 Strategies based on transition-metal and Lewis bases coordination have been long-standing within the P 2 feedstock methods.
There is, however, an unrealized alternative based on the size change from P 2 to P 4 species. In this sense, one could envisage an approach using the ability of fullerenes to encapsulate and form stable endohedral complexes with atoms and small molecules. 27 The chemistry of endohedral started directly after the seminal discovery of fullerenes with the characterization of La@C 60 . 28  study aims to give a comprehensive overview of the structural and electronic features of endohedral fullerenes C n (n = 50, 60, 70, 80, 90, and 100) upon encapsulation of P 2 , and also the evaluation of the dominating physicochemical factors.

| METHODS
All structures were optimized with a combination of Turbomole 7.3.1 software 46 and Gaussian 16 C.01 software. 47 Initial fullerene geometries of a given isomer were extracted from the Fullerene software. 48 This program uses the face-spiral algorithm of Manolopoulos and Fowler with a force field optimization to generate the fullerene coordinates. 49 53 The formal partial charges were obtained using the topological fuzzy Voronoi cells (TFVC) atomic definition 54 as implemented in APOST3D code. 55 The nature of the host-guest interaction was investigated by means of the EDA, developed by Morokuma 56 and by Ziegler and Rauk, 57 at the BP86-D3(BJ)/TZ2P 58 level of theory using ADF2019.101. Core electrons were treated by the frozen-core approximation and scalar relativistic effects have been incorporated by the zeroth-order regular approximation (ZORA). 59 The bonding analysis focuses on the instantaneous interaction energy ΔE int of a bond A-B between two fragments A and B in the particular electronic reference state and in the frozen geometry AB.
This energy is divided into four main components (Equation 1).
The term ΔE elst corresponds to the quasiclassical electrostatic interaction between the unperturbed charge distributions of the pre- 3 | RESULTS AND DISCUSSION

| Geometries and energetics
The most stable isomer of C 60 is the well-known C 60 ÀI h (#1812), the only C 60 isomer obeying the isolated pentagon rule (IPR). 63 Similarly, for C 70 the experimentally characterized isomer is the C 70 ÀD 5h (#8149) one. 64 There is no IPR structure for C 50 and previous computational studies point to the C 50 ÀD 5h (#271) and C 50 ÀD 3 (#270) isomers as the most stable ones, 65 depending on the particular level of theory applied. For C 80 , the experimentally characterized isomer is the C 80 ÀD 2 (#31919). 66 Sure et al. computational studied all 31,924 isomers of C 80 and found several additional isomers close in energy. 67 In particular, isomer C 80 ÀD 5d (#31918) was found the most stable one at PBE-D3/def2-TZVP level of theory, while C 80 ÀD 2 (#31919) was found the lowest energy isomer for DLPNO-CCSD(T)/CBS energies.
Finally, Koenig et al. have recently experimentally characterized tubular C 90 ÀD 5h and C 100 ÀD 5d isomers. 68 It is worth noting that for C 80 , the cage of I h symmetry is the most unstable among the seven isomers of C 80 that satisfy the IPR. 69 However, this cage leads to the most favored EMFs when two La atoms or a Sc 3 N unit are present inside C 80 ÀI h . This result shows that the relative stability of the different cages can change when atoms or metallic clusters are encapsulated inside the cage. However, as we will show later, interaction of P 2 with the cage is relatively weak, and, therefore we do not expect major changes in the stability of the cages due to P 2 encapsulation. Moreover, the determination of the global minima of all P 2 @C n , n = 50, 60, 70, 80, 90, and 100 is out of the scope of this work. In addition to the particular C n isomer, one has to take into account that the P 2 moiety can exhibit different orientations inside the cage. For this reason, we have performed an exploratory study of the P 2 @C 60 species. Thus, we have considered first the C 60 ÀI h isomer and up to four different well-defined orientations of the P 2 unit inside the cage (see Table S3). Note that the encapsulation of P 2 lowers the symmetry of the pristine cage, depending on its specific position inside. In the D 5d geometry, the internuclear P-P bond axis is collinear with the center of opposing pentagon poles. Similarly, a C 3v symmetry is achieved by placing the P-P bond collinear with the center of opposing six-membered rings (6-MRs). The third and the fourth structures were considered where the P-P bond axis is collinear with the midpoints of two opposing 6,6-and 5,6-type C-C bonds, with symmetries D 2h and C 2h , respectively (see Figures S1-2). Both the D 5d and C 2h structures correspond to local minima, while D 2h and C 3v correspond to first-and second-order saddle points at the current level of theory. All stationary points are almost degenerated (i.e., within 0.3 kcal/mol), which would indicate that the host-guest interaction is not directional and essentially the P 2 moiety exhibits free rotation inside the cage.
Next, we have considered the P 2 encapsulation into two additional C 60 cage isomers, namely the C 60 ÀC 2v (#1809) and C 60 ÀD 3 (#1804). As shown in Table 1 67 For small fullerenes, the effect of P 2 encapsulation on the relative energies of the C 60 isomers is not negligible. This effect can be seen in the case of C 50 , where the (#271) isomer is lower in energy than the (#270), which is found to be lower in energy for the pristine cage. However, when considering larger fullerenes such as C 80 , the relative energies of the isomers are barely affected by the encapsulation of P 2 . Still, pristine C 80 ÀD 2 (#31919) and C 80 ÀD 5d (#31918) isomers are found to be within <1 kcal/mol at the current level of theory, and upon P 2 encapsulation the lowest energy structure is P 2 @C 80 ÀD 2 (#31919) by merely 0.5 kcal/mol (see Table 1). Figure 1 depicts the final optimized geometries of all at P 2 @C n and P 4 @C n , n = 50-100, systems, obtained the BP86-D3(BJ)/def2-SVP level of theory together with their symmetry and the P-P bond length. On the one hand, in P 2 @C n the P-P bond axis is collinear with the center of pentagon poles when possible by the geometry of the cage, as described for the C 60 cage. This leads to P 2 @C 70 -D 5h , P 2 @C 90 -C 5v and P 2 @C 100 -C 5v structures. In addition, only in the larger cages (C 90 and C 100 ) the center of mass of the P 2 unit is slightly shifted from the geometrical center of the cage. When the cage is C 50 , the P 2 unit is almost collinear with two opposing (5,6) C-C bonds, leading to a P 2 @C 50 -C 2v structure. Finally, in P 2 @C 80 -D 2 , the P 2 bond axis is collinear with the center of two opposing (6,6) C-C bonds of the cage. On the other hand, P 4 @C n endohedrals achieve lower symmetry levels than P 2 @C n . In this case, the P 4 tetrahedron edges can point towards the center of the pentagon (P 4 @C 50 -C s ) or the hexagon (P 4 @C 60 -C 3v ). Alternatively, one of the P-P bonds matches opposing (6,6) C-C bonds furnishing P 4 @C 70 -C 2 , P 4 @C 80 -D 2 , and P 4 @C 90 -C 2 .
In C 100 , the P 4 molecule is shifted from the cage center, raising the P 4 @C 100 -C s .
At the current level of theory, the bond length of free P 2 and P 4 are 1.917 and 2.233 Å, respectively, in rather good agreement with the experimentally measured for P 2 (1.893 Å) 45a  The encapsulation of P 2 into the smaller size cages induces a shortening of P-P bond length, down to 1.823 Å in the case of C 50 .
From C 80 and larger cages the P 2 distance remains essentially unaffected, already pointing to the absence of electronic effects (e.g. charge-transfer) from the cage.
Encapsulation energies, given by the following equation provide a hint about the feasibility of the formation of the endohedral species. We have also considered the encapsulation of P 4 by the fullerenes, to yield the corresponding P 4 @C n species. The electronic and Gibbs energies can be also found in the Table 2. Our calculations T A B L E 1 Relative energies (in kcal/mol) of selected C n isomers for pristine cages and upon P 2 encapsulation. suggest a highly endergonic process in all cases with values well over +50 kcal/mol, except for the case of C 80 where the spherical shape helps a better fit of the P 4 inside the cage. Still, the overall formation of P 4 @C 80 is not favored with respect to that of P 2 @C 80 .
We explored the possible relationship between the energetics of the encapsulation and geometrical parameters of the cages for P 2 @C n . Two parameters have been introduced to quantify the deformation of the C n cages upon encapsulation. On the one hand, d max is defined as the difference (in Å) between the maximum CÀC distance of the endohedral species and that of the pristine cage. On the other hand, one can also consider, for each C atom of the cage, which is the furthest one. Averaging over all C atoms gives an average maximum distance (the corresponding standard deviation would measure its    Table S2 of the Supporting Information. The encapsulation energies exhibit good correlation with the d aver parameter, but not quite if one focuses on the larger cages where the d aver values are much smaller than for C 50 or C 60 . Even worst correlation is found between the encapsulation energy and the P 2 deformation. However, the encapsulation energies do correlate very well with the total host-guest geometry deformation (defined as d aver + ΔP 2 ), as shown in Figure 2. The smaller the deformation, the better.

| Frontier molecular orbitals
It is not easy to trace the origin of the deformation of each cage.
However, that of the P 2 moiety should be related to the shape of the molecular orbitals in which the P 2 unit is primarily involved and the corresponding PÀP bond order. In the free P 2 species, the σ and two π bonding orbitals are occupied, consistent with a formal triple bond.
Charge transfer from the cage to the P 2 host would populate its antibonding orbitals, leading to a decrease of the bond order and a concomitant PÀP stretch. Nevertheless, similarly, any charge transfer from the P 2 moiety to the cage would depopulate PÀP bonding orbitals, causing the same effect. This means that the PÀP bond order can only decrease upon encapsulation, disregarding the PÀP distance.
A similar analysis involves P 4 unit, where the frontier orbitals consist of σ (e, t 2 ) and σ* (t 2 ) PÀP orbitals. 70 Thus, the observed compression of the PÀP bond upon encapsulation on the smaller cages is due to the steric pressure of the cage (this will be more evident from the EDA analysis below). Table 3 gathers the P 2 and P 4 bond orders and partial charges obtained with a Hilbert-space (NAO) and a real-space (TFVC) atomic definitions. The large disagreement between different atomic population analysis in endohedral fullerenes has been pinpointed. For instance, in the endohedral borospherene complex Cl@B 39 , the charge on Cl changes from À0.62 to 0.76 e depending on the method used. 71 In that work, the authors found that realspace QTAIM charges are reliable. We use here real-space TFVC charges because they provide similar results to QTAIM charges at much lower cost.
The TFVC method predicts a charge transfer from the P 2 moiety to the cage up to C 70 , and the opposite effect from C 80 . The charge transfer is very modest (ca. 0.3e) except for the smaller cage. On the other hand, NPA charges are usually negative for P 2 , and much smaller.
Such a charge flow is equally distributed over the entire cage (see Figure S3 in the ESI). As mentioned above, any charge transfer (positive or negative) should induce a decrease of the PÀP bond order. This is exactly what is observed with both schemes. The predicted effect on the bond order is much more pronounced for the TFVC method, going down to 2.09 for the P 2 @C 50 species, where the charge transfer is maximal. However, for the most interesting larger cages the bond order of the PÀP bond remains similar to that of the free P 2 unit, indicating that upon encapsulation, the triple bond character of the P 2 host is maintained. Similarly, P 4 shows a significant charge transfer for the small cages, which is related to the reduction of the PÀP bond order. With the size increase, charge transfer becomes smaller, and the bond order approaches the one observed for free P 4 molecule.
Molecular orbital analysis has been carried out focusing on the σ and π orbitals of the P 2 fragment at the BP86-D3(BJ)/def2-TZVP level of theory. As shown on Figure 3, the encapsulation induces an inversion of the relative energies of the σ and π orbital of P 2 . Thus, while for free P 2 the σ is lower in energy, the contrary is found upon encapsulation for the smaller cages C 50 and C 60 . This is likely to be due to the increased Pauli repulsion suffered by the σ electrons that are closer to the cage than the π ones. Also, in the smaller cages (C 50 , C 60 ), the σ and π orbital of P 2 orbitals are energetically destabilized for the same reason. In fact, in both cases the σ orbital becomes the HOMOÀ1. Then, the larger the cage, the more stabilized the σ and π orbitals become. The contrary occurs for the σ* and π* ones (not shown). It is also worth to note that the degeneracy of the π and π* orbitals is lost in the case of P 2 @C 50 and P 2 @C 80 .

| Energy decomposition analysis
More detailed information about the nature of the interaction between P 2 and C n (n = 50, 60, 70, 80, 90, and 100) fullerene cages T A B L E 3 P P bond orders (BO, in a.u.) and partial charges (Q, in a.u.) of the P 2 and P 4 unit from Hilbert-space (NAO) and real-space (TFVC) analyses are provided by the results of the EDA method. 72 EDA has proven to be a useful tool to assess the nature of the chemical bond in main group compounds and transition metal compounds, 73 as well as the interaction in endohedrals. 74 Nonetheless, a recent discussion has been placed about the path function nature of the energy components. 75 Within EDA scheme, the interaction formation between two (or more) fragments is divided into Pauli repulsion, electrostatic interaction, and orbital interaction (for further details, see the computational section).  Table 4, revealing a small charge transfer from P 2 to the fullerenes. Table 4 also gathers the EDA results for the encapsulation of P 4 with C n (n = 50, 60, 70, 80, 90, and 100) in the singlet reference state.
As discussed above, this process is thermodynamically unfavorable for all cases, resulting in negative dissociation energy values. The only exception is C 80 , where the internal space is enough to host P 4 with a slightly positive D e value of 6.5 kcal/mol. Nonetheless, bigger cages C 90 and C 100 yield negative D e values as a consequence of the spheroidal shape. The preparation energy values (ΔE prep ) suggest high energy penalties upon complexation, going from 114.5 kcal/mol (C 50 ) to 21.1 kcal/mol (C 80 ). In addition, the interaction energy values (ΔE int ) reveal a destabilizing effect by the encapsulation caused by strong Pauli repulsion between the host and the guest.

| CONCLUSIONS
The feasibility of the encapsulation of P 2 in C n fullerenes has been computationally assessed for cages from n = 50 to n = 100. We show fullerenes C 70 to C 100 are suitable cages to incorporate P 2 instead of P 4 , which is the most stable form of phosphorous. Upon inclusion of thermal and entropic effects, only the formation of endohedral C 80 to C 100 overcome the energetic penalty for the required P 4 dissociation into two P 2 dimers. Orbital analysis indicates that the triple bond in P 2 remains intact within the endohedral system, with very small hostguest charge-transfer. EDA shows that Pauli repulsion is roughly twice the amount of the (favorable) electrostatic interaction along the series.
The dispersion energy contribution amounts to ca. À55 to À60 kcal/ mol for all cages except the smallest one. From n = 70 on, the dispersion becomes dominant and accounts for the favorable encapsulation energies of P 2 in C 70 to C 100 cages.

ACKNOWLEDGMENTS
The work at University of Saarland has been supported by the ERC