3.2.1. Energy and Structural Arguments
It is understandable that encapsulating an atom into the interior of the superphane molecule will affect its physicochemical properties and its structure. It should be expected that this influence will be more pronounced the bulkier the encapsulated atom is. The relatively high flexibility of the superphane molecule structure allows for an elegant visualization of these changes. The values (
B97X-D/6-311++G(2d,p)) of the binding energy, the strain energy, and the selected geometric parameters for the superphane and Ng@superphane complexes are presented in
Table 4. For a better visualization of changes, the considered distances and energies are also shown in
Figure 2.
What is most important is that the binding energies are clearly positive, and therefore, according to the definition given by Equation (
1), they indicate the fact that the Ng⋯superphane interactions inside the cage are
destabilizing. It should be noted that positive binding energies, indicating an internal
repulsive effect, were obtained for the vast majority of endohedral complexes [
10,
11,
12,
36,
37]. This result reflects the fact that both entities, i.e., Ng and superphane, prefer not to form an endohedral complex with each other, but rather to be separated from each other. This conclusion is consistent with the statement of Moran et al. that “... exohedral coordination is preferred to endohedral complexes without exceptions.” [
36].
As expected, the destabilization effect of the endohedral Ng@superphane complex increases monotonically as the size of the noble gas atom increases (
Table 4 and
Figure 2). The binding energy is as high as 551.6 kcal/mol for the Kr atom. It is also true that, in general, the binding energy depends on the size of the cage, e.g., for the smaller adamantane and even yet smaller cubane the binding energy with the encapsulated helium atom is as high as 154.7 kcal/mol (B3LYP/6-311++G(2d,2p)) [
10,
11] and 322.4 kcal/mol (B3LYP/6-311++G(d,p)) [
12], respectively, while for He@superphane it is only 81.3 kcal/mol (
Table 4).
The encapsulated noble gas atom has a significant impact on the structure of the Ng@superphane complex. It is especially visible during the analysis of the distance between benzene rings, i.e., . This distance in a superphane is 2.650 Å. Encapsulation of a helium atom inside it leads to an increase in this distance to 2.817 Å, i.e., by 0.167 Å, which gives ca. 6% of the initial distance. The exchange of He to Ne leads to a further increase to 3.126 Å, i.e., a further 0.309 Å, or 11%, an increase of 18% over the superphane value. The effect of the exchange of Ne on Ar is very significant. The distance increases by a further 0.436 Å (to 3.562 Å), or 14%, which is more than a third (34%) of the value in the superphane molecule. Replacing the Ar atom with the Kr atom, however, does not cause such a significant change (0.136 Å, i.e., ca. 4%). Nevertheless, in relation to superphane, the distance increases by 1.048 Å, which is 40%. This result best confirms the considerable flexibility of the superphane cavity, which makes the superphane molecule a convenient species for research on the effect of encapsulation.
It should be remembered that both benzene rings of the superphane molecule are bound together by six ethylene bridges (see
Figure 1). It should be expected that the distancing of the benzene rings from each other, forced by encapsulation of the noble gas atom, has an unfavorable effect on the structure of these bridges. First of all, as can be seen from
Table 4 (and
Figure 2), the C-C bond undergoes a significant and monotonic elongation, from 1.591 Å in superphane to 1.753 Å in Kr@superphane, i.e., by 10%. It is worth mentioning that it is one of the longest C–C bonds ever reported [
63,
64,
65], not including record breaking carboranes [
66,
67,
68] and especially tetracyanoethylene [
69] and tetracyanopyrazinide [
70] dimer dianions.
Encapsulation changes the value of the C-C-C angle in the bridge () from the favorable characteristic for the tetrahedron (about 109.5) to the unfavorable one, amounting to as many as 128 in the complex containing the largest Kr atom. It is also worth emphasizing that encapsulation with a larger and larger Ng atom leads to the elongation of C-C bonds in the benzene rings, from 1.406 Å in superphane to 1.456 Å (on average) in Kr@superphane. An interesting fact is that the full geometry optimization of the Kr@superphane complex leads to a clear distinction between C-C bonds in benzene rings, so that they have two different lengths (1.429 and 1.482 Å) and alternate. This change shows that benzene rings are no longer aromatic.
It is worth noting that, as indicated by the values of the dihedral angle
shown in the last column of
Table 4, the ethylene bridges are slightly twisted and the C
C
C
C
carbons are not in one plane. Such a defect is most likely aimed at minimizing the unfavorable angular stresses in the ethylene connectors. This angle in the case of superphane amounts to ca. 6
and increases slightly (by ca. 1–3
) in complexes with He, Ne or Ar. On the other hand, in the case of the Kr@superphane complex, significantly lower values (2.0° and 4.0
) and their alternation (similar to the C-C ring bond) have been obtained.
It is clear that the structural changes caused by encapsulation are adverse to the host superphane molecule. Apart from the already discussed structural changes, a strong energetic destabilization of the superphane should be expected. This destabilization can be easily described by the strain energy defined by Equation (
2). As can be seen from
Table 4 and
Figure 2, this energy actually increases rapidly from 8.7 kcal/mol in He@superphane to as much as 241.6 kcal/mol in Kr@superphane, which is ca. 11% and as much as 44% of the binding energy. It is clear that the strain energy should depend to a large extent on the volume of the cavity inside the encapsulating molecule. For example, this energy is as high as 15.3 kcal/mol for the He@adamantane complex [
10,
11].
The values of the structural parameters presented in
Table 4 are sufficient to conclude that the superphane molecule ‘swells’ as soon as an atom of noble gas is introduced into it. This effect is greater the larger this atom is. Of course, this effect is energetically unfavorable for the superphane molecule, as evidenced by the rapid increase in the strain energy. It is possible that increasing the atom’s size to Xe and Rn will disrupt the endohedral complex while releasing the encapsulated noble gas atom. Of course, it is to be expected that this process will be associated with the release of energy.
This will be the subject of future research. Nevertheless, an additional result clearly indicating an unfavorable energetics of the Kr@superphane endohedral complex is the result of its full optimization which has been obtained utilizing the 6-31G(d) basis set together with some exchange-correlation functionals (see
Table 1 and
Table 2). Although this result may be due only to a fairly small basis set 6-31G(d) (although it did not occur in the case of the even smaller 6-31G and using the Hartree–Fock method or either the M06-2X or
B97X-D exchange-correction functional), it is worth discussing, which is presented in the next subsection.
3.2.2. About How Krypton Escapes from the Cage
In the footnotes to
Table 1 and
Table 2, it has already been mentioned that the full geometry optimization of the Kr@superphane endohedral complex leads in some cases to the spontaneous expulsion of the Kr atom from the superphane cage, which of course leads to a decrease in the total energy of the Kr–superphane system. This interesting result has been obtained in the case of geometry optimizations utilizing B3LYP, B3LYP-D3, M06, or M06-HF exchange-correlation functionals, but only together with the basis set 6-31G(d).
However, this result has not occurred in the case of the M06-2X and
B97X-D functionals or the methodologically simpler Hartree–Fock method. Moreover, all of these methods have not given this result as long as either the 6-31G or the 6-311G basis set was used (see
Table 1 and
Table 2). Therefore, on the one hand, this result could be considered as an artifact (which, however, has happened), but on the other hand, it is possible that some other basis sets and some other functionals (and there are hundreds of both [
49,
62]) would lead to this the same result, i.e., to a spontaneous ejection of the trapped krypton atom out of the superphane cage.
Of course, although the search for other method/basis set combinations may be a student project (albeit rather arduous), it was not the main goal of the research described here. Nevertheless, it is worth taking a look at the total energy change that occurs during the optimization procedure. The B3LYP-D3/6-31G(d) case is shown in
Figure 3. It represents the other three cases mentioned earlier.
Figure 3 clearly shows that when optimizing the geometry of the initially endohedral Kr@superphane complex, spontaneous (i.e., without any energy barrier) expulsion of the encapsulated krypton atom out of the superphane cage with simultaneous formation of the exohedral superphane⋯Kr complex takes place. It is worth noting that, after this process, the superphane molecule is closed again and significantly flattened. The entire process of releasing the krypton atom is strongly exoenergetic, as much as 540 kcal/mol is released. It is also worth noting that even a slight opening of the cage of ‘swollen’ superphane is very energetically beneficial. The moment when the initially trapped atom Kr is in the carbon window is already an energy gain of about 150–250 kcal/mol.
In our opinion, this process is a powerful illustration of the intra-cage repulsion of the Kr⋯superphane type (or more generally Ng@host). The fact that this process has not been observed at many other levels of theory is not at all evidence of the attractive interaction of the Ng⋯host type, but simply because, at a given level of theory, disrupting the host cage is too energetically expensive. As is well known, the purpose of geometry optimization is to find the equilibrium structure, i.e., the most favorable energetically, which is related to the minimization of the total energy during its course.
For this reason alone, in most cases, we do not observe the removal of the encapsulated entity outside the host’s cage, and this is by no means a symptom of a beneficial attractive interaction inside the cage. In the case of the previously described encapsulating hydrocarbons [
36], such as, for example, adamantane [
10,
11], cubane [
12,
13], dodecahedrane [
12,
13], or fullerenes (e.g., the most famous C
[
12,
13,
37]) CC bonds are too strong to break, and therefore the encapsulated entity cannot spontaneously escape beyond the host’s cage while optimizing the geometry of the E@host complex. It seems that superphane is also unique in this respect and is therefore suitable for research on the influence of an encapsulated entity on the properties of a host molecule.
3.2.3. On Whether a Bond Path Really Is Evidence of a Stabilizing Interatomic Interaction
As already mentioned in the Introduction, Cioslowski suggested for the first time that in some cases, especially in highly steric congested systems, a bond path may be associated with a repulsive rather than an attractive interatomic interaction [
6,
7,
8,
9]. Since then, there has been a wealth of literature on the subject [
10,
11,
12,
13,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35]. Merino et al. [
12] pointed out that a large number of bond paths terminating at an atom may be the result of a high symmetry of a system. For example, in the case of the He@cubane complex, they have found eight He⋯C bond paths, while Ar@C
experiences as many as 60 Ar⋯C bond paths [
12].
In the case of the Ng@superphane (Ng = He, Ne, Ar) complexes, in the vast majority of cases (depending on the Ng atom itself as well as the adopted level of theory), we have obtained 12 bond paths between the Ng atom and all the carbon atoms forming the benzene rings, but the picture becomes much more complicated in the case of the Kr@superphane complex. In the context of the ongoing discussion, this fact deserves more attention. Only four molecular graphs obtained using basis sets 6-31G, 6-311+G(d), 6-311++G(d,p), and 6-311++G(2d,p) together with the
B97X-D functional are presented in
Figure 4.
Figure 4 shows that, depending on the basis set used to determine the molecular graph of Kr@superphane, 12 (6-31G and 6-311++G(2d,p)), 7 (6-311+G(d)) or only 6 (6-311++G(d,p)) Kr⋯C bond paths have been obtained. Since the number of chemical bonds cannot be so easily varied, it is clear that a bond path cannot mean a chemical bond. It is also worth noting that although basis sets 6-31G and 6-311++G(2d,p) have led to the same number (12) of Kr⋯C bond paths, these paths differ significantly from each other. Namely, they are straight (and on the vertical planes of the rings) in the former case, whereas clearly curved in the latter. Also, the change of the number of bond paths when changing the basis set 6-311++G(d,p) to 6-311++G(2d,p), and thus only after doubling the number of d-type functions on C and Kr atoms, should be of concern; does the number of bonds Kr⋯C change?
Other examples of the dependence of a molecular graph on the level of theory have recently been shown [
28,
29,
30,
71]. Two such cases will be mentioned here. The first one concerns the O⋯O type bond paths in the fully optimized C(NO
)
anion [
30] (
Figure 5). Data concerning the presence of the O⋯O bond paths are presented in
Table 5.
Indeed, the results presented in
Table 5 show that the presence of a given bond path (O⋯O in this case) in the molecular graph can be highly dependent on the methodology used. More specifically, within the set of considered basis sets, the presence of the O⋯O bond path is not dependent on the basis set when either the Hartree–Fock method or the M06-2X functional is used, while this dependence is manifested in the case of B3LYP,
B97X-D and the second-order Møller–Plesset perturbation theory (MP2) [
72].
In these cases, larger and better balanced basis sets (6-311++G(2df,2pd), cc-pVTZ, cc-pVQZ, aug-cc-pVTZ, aug-cc-pVQZ) did not give the O⋯O bond path, although it does not seem to be the rule, because the smallest basis set 6-31G in combination with MP2 also did not give such a bond path. This result also showed that a molecular graph obtained with a large and more reliable basis set can sometimes also be obtained with a much smaller and less reliable basis set [
30]. The case of Kr@superphane molecular graphs shown in
Figure 4 also proves this very well. Namely, the smallest and generally unreliable basis set 6-31G reproduces (neglecting the clear differences in the curvature of the bond paths) the molecular graph obtained using the largest and most reliable 6-311++G(2d,p), whereas the slightly simpler form of the latter, i.e., 6-311++G(d,p), gives different number of bond paths. Therefore, the dependence of a molecular graph on the computational methodology makes the reliability of the molecular graph a somewhat problematic issue.
The second case is a bit more complex and concerns the presence of different types of bond paths on molecular graphs while using different methodologies. The molecular graphs obtained [
71] with the four levels of theory for the H
SiH⋯CCl
dimer are shown in
Figure 5. The use of the
B97X-D/6-311++G(2df,2pd) level of theory (
b) yielded only one intermolecular bond path, namely of the H⋯C type. The use of the aug-cc-pVTZ basis set (i.e., the
B97X-D/aug-cc-pVTZ level of theory;
b ) provided an additional H⋯Cl bond path. In contrast, the MP2 method led to three bond paths for intermolecular interactions. Two of them are of the H⋯Cl type, while one is of H⋯C. However, both of these basis sets differentiate hydrogen atoms due to the number of such paths. Namely, in the case of the smaller 6-311++G(2df,2pd) basis set (
b), one of the hydrogen atoms of SiH
is linked with two chlorine atoms of CCl
. However, in the case of the larger aug-cc-pVTZ basis set (
b), the same hydrogen atom creates a bond path to only one chlorine.
Returning to the main issue, however, whether a bond path must necessarily prove a stabilizing effect (especially in the case of endohedral complexes), it should be remembered that the encapsulation of the noble gas atom inside the superphane molecule has led to its high destabilization. As mentioned earlier, in the vast majority of cases of endohedral complexes, the trapped entity does not escape from the cage during the geometry optimization procedure just because the barrier associated with opening of the cage is too large to be overcome (nevertheless, such spontaneous ejection of the encapsulated entity is, indeed, possible as shown in the example of Kr@superphane in
Figure 3).
The combination of the presence of counterintuitive Ng⋯C bond paths with a simultaneous strong destabilization of the endohedral complex and therefore the urgent desire to eject the encapsulated atom out of the cage is, in our opinion, sufficient evidence to conclude that the presence of a bond path does not necessarily indicate interatomic stabilization (attraction).
3.2.4. Negative Mayer Bond Orders
The strength of a chemical bond is often described by the bond order. Still a popular variety of this property is the so-called Wiberg Bond Index (WBI) [
55], which, however, is already somewhat outdated. Bridgeman et al. [
59] have shown that the newer Mayer Bond Order (MBO) [
56,
57,
58] is a very valuable parameter. Importantly, its definition (see Equation (
4)) also allows for negative values. The calculated MBO values for the Ng⋯C contact and for the C-C bond of the ethylene spacer and the benzene ring are shown in
Table 6. Additionally, their WBI counterparts are also shown for comparison.
As would be expected, the WBI value for the C-C spacer bond is close to 1 as it should be for a single bond. The WBI value for the C-C bond within the benzene ring is 1.37 and is slightly lower than the corresponding value for benzene (1.44). In the case of the Kr@superphane complex, a variation in value has been obtained which corresponds to a variation in length (see
Table 4). More importantly, however, in the case of the Ng⋯C contacts, very small values, close to zero, have been obtained, which suggests negligibly weak Ng⋯C interactions.
However, it is instructive now to look at the Mayer Bond Orders (MBO). The corresponding values for the C-C spacer bond are again close to 1, and for the ring C-C slightly lower than or very similar to the benzene value (1.57). For Kr, two values have been obtained again. The most important result, however, is that the MBO values for the Ng⋯C contacts are negative! This result indicates the antibonding nature of the interaction between the noble gas atom and the ring carbon atoms of the superphane molecule. The antibonding nature of this interaction increases with increasing size of the noble atom, i.e., He→Ne→Ar→Kr, and in the case of Kr it is significant (−0.902). It can be added that we have also obtained negative MBO values, e.g., for the He dimer with an interatomic distance of 2 Å (−0.003), or the FF dimer (−0.035 for 3 Å and as much as −0.290 for 2 Å).
The obtained values of the bond order for C-C bonds slightly increase with the increase in the size of the Ng atom, while in the case of WBI these changes are negligible. This result may seem very astonishing, as a larger bond order should, according to Pauling’s formula (BO =
) [
54], mean a shorter bond, but the C-C bonds actually become longer (the superphane ‘swelling’ effect) as the noble gas atom becomes larger (see
Table 4). The Mayer Bond Order, however, depends on the charge delocalization [
59] and the increase in value is most likely due to the strong charge transfer from the noble gas atom to the carbon atoms of the caging superphane molecule. As a result of this electron density transfer, the Ng atoms have partial positive charges.