The Acid-Base Through-the-Cage Interaction as an Example of an Inversion in a Cage Isomerism

We define a new inversion in a cage isomerism (ic): X@C· · ·Y Symmetry 2020, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/symmetry Article The Acid-Base Through-the-Cage Interaction as an Example of an Inversion in a Cage Isomerism Jan Cz. Dobrowolski * and Sławomir Ostrowski Institute of Nuclear Chemistry and Technology, 16 Dorodna Street, 03-195 Warsaw, Poland; s.ostrowski@ichtj.waw.pl * Correspondence: j.dobrowolski@nil.gov.pl Received: 19 June 2020; Accepted: 29 July 2020; Published: date Abstract: We define a new i version in a cage isomerism (ic): X@C∙∙∙Y ₪ic Y ∙∙∙X, (₪ is the isomerism relation) as an isomerism in the three-component system of molecules X, Y, and a cage C, in which one of the molecules is located inside and the other outside the cage. The ic isomerism is similar to the endo-exo one, which occurs only if either the interior or exterior of C is empty. By contrast, ic occurs only if neither the interior nor the exterior of C is empty. We also discuss the other closely related types of isomerisms are also discussed. Calculations of the XH∙∙∙NH3@C60 and NH3∙∙∙HX@C60 ic isomers were performed at the ωB97XD/Def2TZVP level. The calculated energies demonstrated that the systems with the HX acid outside (X = F, Cl) and the NH3 base inside the cage, XH∙∙∙NH3@C60, are more stable than their ic isomers, NH3∙∙∙HX@C60, by about 4–8 kcal/mol. This is because NH3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the studied systems and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always a Lewis base. The C60 molecule with HX inside or outside the cage is also an acid for the NH3 base positioned outside or inside the cage. On the other hand, the C60 cage is truly amphoteric because it is simultaneously an acid and a base. We define a new i version in a cage isomerism (ic): X@C∙∙∙Y ₪ic Y ∙∙∙X, (₪ is the isomerism relation) as an isomerism in the three-component system of molecules X, Y, and a cage C, in which one of the molecules is located inside and the other outside the cage. The ic isomerism is similar to the endo-exo one, which occurs only if either the interior or exterior of C is empty. By contrast, ic occurs only if neither the interior nor the exterior of C is empty. We also discuss the other closely related types of isomerisms are also discussed. Calculations of the XH∙∙∙NH3@C60 and NH3∙∙∙HX@C60 ic isomers were performed at the ωB97XD/Def2TZVP level. The calculated energies demonstrated that the systems with the HX acid outside (X = F, Cl) and the NH3 base inside the cage, XH∙∙∙NH3@C60, are more stable than their ic isomers, NH3∙∙∙HX@C60, by about 4–8 kcal/mol. This is because NH3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the studied systems and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always a Lewis base. The C60 molecule with HX inside or outside the cage is also an acid for the NH3 base positioned outside or inside the cage. On the other hand, the C60 cage is truly amphoteric because it is simultaneously an acid and a base.


Introduction
Searching for new compounds, new materials, and their new properties are among the primary tasks of chemistry. Geometry, topology and set theory can provide a flash of inspiration for the discovery of new molecules. This is because without a name a thing does not exist in our minds, and the name for a plethora of geometric, topological and set theory relations between the objects is usually already known in mathematics. One can find new molecules through modifications of known structures, discovery of new syntheses, finding new natural products, but also through a search for new types of isomerism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical summary formula but demonstrating different properties during the time characteristic for a given measurement.
Fullerenes form cages and exhibit quite specific types of isomerism connected to both connectivity between the atoms constituting the cage and the presence of a well-defined interior, boundary, and exterior. A cage isomerism occurs at the boundary and is due to the arrangement of the C-atoms composing the cage in polygons (mostly pentagons and hexagons) [4]. For example, there are possibly as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only pentagons and hexagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, 19, 3, and 1 belong to the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus 1720 of C60 fullerenes occur in two enantiomeric forms, which makes the existence of 3532 C60 fullerenes composed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with

Introduction
Searching for new compounds, new materials, and their new properties are among the primary tasks of chemistry. Geometry, topology and set theory can provide a flash of inspiration for the discovery of new molecules. This is because without a name a thing does not exist in our minds, and the name for a plethora of geometric, topological and set theory relations between the objects is usually already known in mathematics. One can find new molecules through modifications of known structures, discovery of new syntheses, finding new natural products, but also through a search for new types of isomerism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical summary formula but demonstrating different properties during the time characteristic for a given measurement.
Fullerenes form cages and exhibit quite specific types of isomerism connected to both connectivity between the atoms constituting the cage and the presence of a well-defined interior, boundary, and exterior. A cage isomerism occurs at the boundary and is due to the arrangement of the C-atoms composing the cage in polygons (mostly pentagons and hexagons) [4]. For example, there are possibly as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only pentagons and hexagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, 19, 3, and 1 belong to the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus 1720 of C60 fullerenes occur in two enantiomeric forms, which makes the existence of 3532 C60 fullerenes composed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with

Introduction
Searching for new compounds, new materials, and their new properties are among the primary tasks of chemistry. Geometry, topology and set theory can provide a flash of inspiration for the discovery of new molecules. This is because without a name a thing does not exist in our minds, and the name for a plethora of geometric, topological and set theory relations between the objects is usually already known in mathematics. One can find new molecules through modifications of known structures, discovery of new syntheses, finding new natural products, but also through a search for new types of isomerism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical summary formula but demonstrating different properties during the time characteristic for a given measurement.
Fullerenes form cages and exhibit quite specific types of isomerism connected to both connectivity between the atoms constituting the cage and the presence of a well-defined interior, boundary, and exterior. A cage isomerism occurs at the boundary and is due to the arrangement of the C-atoms composing the cage in polygons (mostly pentagons and hexagons) [4]. For example, there are possibly as many as 1812 C 60 (non-isomorphic graphs of) fullerenes composed of only pentagons and hexagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, 19, 3, and 1 belong to the chiral C 1 , C 2 , D 2 , D 3 , and D 5 point symmetry groups, respectively [5]. Thus 1720 of C 60 fullerenes occur in two enantiomeric forms, which makes the existence of 3532 C 60 fullerenes composed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with at least one C-atom replaced by a heteroatom [6]. The cage isomerism of heterofullerenes is even richer because of the combinatorial possibility of the heteroatom location at the cage. Moreover, a heterofullerene can also be chiral if it does not exhibit an orientation-reversing self-isometry. For example, there are 14 chiral out of 23 (or 28 enantiomers out of 37) different C 58 X 2 heterofullerenes based on the buckminsterfullerene cage [7,8].
In 1991, Weiske et al. [9] and Cioslowski and Fleischmann [10] independently introduced the term endohedral fullerenes for structures containing species inside the fullerene cage. Now, the scientific community well recognizes the chemistry and physics of the endohedral fullerenes, yet they are still intensively studied e.g., [11][12][13]. The exohedral fullerenes [14] are those that interact with species outside the cage. The possibility for the existence of the same species inside and outside a fullerene gives rise to the endohedral-exohedral isomerism. However, the interaction inside and outside the cage have to be similar; otherwise, for instance, if a bond forms only on one side of the cage, the isomerism will be classified as a kind of constitutional one. The endohedral-exohedral isomerism can be, e.g., related to internal and external interaction with noble gases, alkali cations or halogen ions, or noble metal atoms with a fullerene cage [15][16][17][18][19][20].
Simmons and Park introduced the idea of in-out isomerism for macrobicyclic amines [21,22]. Dietrich, Lehn, Sauvage, and Blanzat also studied it as exo-exo, exo-endo, and endo-endo isomerisms abbreviated to xx, xn, and nn [23]. However, they pointed out the ambiguity and inconsistency of the in-out term [24] because they referred the in and out descriptors to the non-rigid cycles. In such a case, the CIP priority rules [25], generalized if needed [26], are sufficient to correctly distinguish the non-rigid macrobicyclic isomers. Nevertheless, for rigid fullerenes, the in-out terminology is intuitive and useful [27,28], even though the configuration and conformation of these isomers is again sufficient for an unequivocal distinction.
Here, we turn our attention to the possibility of the existence of yet another isomerism connected to the species present in and out of the cage: the inversion in a cage isomerism (ic). We demonstrate an example of the inversion in a cage isomerism supported by quantum-chemical DFT calculations of model through-the-cage acid-base interactions. Then we consider two situations: an acid is in and a base is out of the cage and oppositely. They are compared to the system(s) in which the acid-base interaction occurs without or outside the cage.
We have constructed the manuscript as follows. The technical issues are gathered in the Calculations section. The Results and Discussion section is divided into two independent sections-in the first, the new isomerism is defined, discussed from the point of view of the isomer classification tree, and compared with definitions of the closely related isomerisms. In the second, we give a computational example of the inversion in a cage isomer, an example of the new isomerism, which refers to the theme of a special issue-symmetry in acid-base chemistry. Although the calculations serve for the new isomerism illustration rather than for revealing meticulous details of the illustrative example, we performed a careful interaction energy consideration at the DFT level with the relatively large def2TZVP basis set, the modern ωB97XD functional accounting for dispersion forces, and calculations of the basis set superposition error for two-and three-component systems of which our example is composed. The results are summarized in the Conclusions section.

Calculations
The DFT calculations were done using the ωB97XD functional [29] combined with small 6-31G** [30] and large def2TZVP [31] basis sets using the Gaussian 09 suite of programs (Gaussian Inc.: Wallingford, CT, USA, see also at the Gaussian web page: https://gaussian.com/glossary/g09/) [32]. the ωB97XD functional was applied using the exact Hartree-Fock exchange in both short-and long-range, including correction for the dispersion forces, and is effective in dealing with charge-transfer states [33,34]. In comparison to many other dispersion-corrected functionals, ωB97X-D was shown to perform slightly better for non-covalent sysgtems but much better for covalent systems [35][36][37]. Here, 6-31G** is a valence double-zeta basis set with polarization functions set on both heavy and hydrogen atoms, while def2TZVP is a size-consistent basis set for all atoms and includes triple-ζ and polarization functions. The harmonic frequencies of all optimized structures were real, indicating that only true energy minima on potential energy hypersurfaces were considered. We estimated the Gibbs free energies at 298 K. To better evaluate the interaction energies, the species-cage interaction energies were corrected for basis set superposition error using the seven-point method, ∆E 7 [38][39][40][41], including the Boys-Bernardi counterpoise correction, ∆E CP [38]; and the cage deformation, ∆E def . To calculate ∆E 7 (Equation (1)), we further corrected the interaction energies (Equation (2)) with the Boys and Bernardi counterpoise method (Equation (3)) as well as deformation energy (Equation (4)) [39,40]. Pure BSSE is done as the difference between interaction energy and ∆E 7 (Equation (5)).
where E AB denotes the energy of dimer, E A A , and E B B denote the energies of dimer components optimized in their own basis sets, E AB A,de f and E AB B,de f denote the energies of monomers with geometries taken from dimer in basis set of dimer, and E A A,de f and E A B,de f denote the energies of monomers with geometries taken from dimer in a basis set of appropriate monomers.
In the case of ternary systems, the equations for the counterpoise and deformation energies implemented in Gaussian 09 [32] are the following: where the indices have analogous meanning.in description of energy E and the superscript denotes the basis set in which the system indicated in the subscript is calculated. The energetics are collected in Tables 1, A1 and A2, where A denotes data gathered in Appendix C. The XYZ coordinates and pictures presenting the two-and three-components systems with the most important intermolecular distances shown after atoms in molecules analysis of bond critical points are shown in the Supplementary Information file. Notice that because the calculations are for illustrating a new type of isomerism rather than for studying details of the systems that may be compared to the real systems, we have not considered the role of environment here. Consequently, please take notice that the uncertainty of DFT calculations is ca. 2 kcal/mol.

Inversion in a Cage Isomerism
Consider an acid A and a base B forming a complex A· · · B in an equilibrium : wherein one neglects a difference between an AB salt and an A· · · B complex. Consider also a cage C. Depending on the A, B, and C size, different species can potentially be observed. In the most general case, in which one can include A + B in C, the following can be observed: A@C + B A@C· · · B (10) (A + B)@C (A· · · B)@C (12) In the above, we have assumed that the interaction between A and B is much stronger than either of them is with C. Therefore, in (9), the competing interactions between A and C, between B and C, as well as both interactions (in many possible variants), were neglected for the sake of clarity. Similarly, Formally, one can call all systems at the two sides of Equations (9)-(12) isomeric, even if the "+" symbol can also mean an infinite separation of the components. Let us denote the isomeric relation by the www.mdpi.com/journal/symmetry raction as an merism arsaw, Poland; @C•••Y ₪ic Y@C•••X, (₪ is the of molecules X, Y, and a cage ide the cage. The ic isomerism r or exterior of C is empty. By pty. We also discuss the other ns of the XH•••NH3@C60 and level. The calculated energies ) and the NH3 base inside the X@C60, by about 4-8 kcal/mol. matter of 6.5 kcal/mol). In the nd the NH3 molecule is always s also an acid for the NH3 base is truly amphoteric because it perties are among the primary a flash of inspiration for the oes not exist in our minds, and lations between the objects is rough modifications of known , but also through a search for stence of molecules of identical time characteristic for a given somerism connected to both nce of a well-defined interior, d is due to the arrangement of d hexagons) [4] a new inversion in a cage isomerism (ic): X@C•••Y ₪ic Y@C•••X, (₪ is the as an isomerism in the three-component system of molecules X, Y, and a cage e molecules is located inside and the other outside the cage. The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By ly if neither the interior nor the exterior of C is empty. We also discuss the other s of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and mers were performed at the ωB97XD/Def2TZVP level. The calculated energies he systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the subsystems, the HX molecules are Lewis acids and the NH3 molecule is always 0 molecule with HX inside or outside the cage is also an acid for the NH3 base r inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. ral fullerenes; isomerism; acid-base interaction compounds, new materials, and their new properties are among the primary eometry, topology and set theory can provide a flash of inspiration for the ecules. This is because without a name a thing does not exist in our minds, and ora of geometric, topological and set theory relations between the objects is n in mathematics. One can find new molecules through modifications of known of new syntheses, finding new natural products, but also through a search for m [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical t demonstrating different properties during the time characteristic for a given cages and exhibit quite specific types of isomerism connected to both the atoms constituting the cage and the presence of a well-defined interior, r. A cage isomerism occurs at the boundary and is due to the arrangement of ng the cage in polygons (mostly pentagons and hexagons) [4]. For example, many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only ons [4] as an isomerism in the three-component system of molecules X, Y, and a cage e of the molecules is located inside and the other outside the cage. The ic isomerism e endo-exo one, which occurs only if either the interior or exterior of C is empty. By rs only if neither the interior nor the exterior of C is empty. We also discuss the other types of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and ic isomers were performed at the ωB97XD/Def2TZVP level. The calculated energies that the systems with the HX acid outside (X = F, Cl) and the NH3 base inside the 3@C60, are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. NH3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the s and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always he C60 molecule with HX inside or outside the cage is also an acid for the NH3 base side or inside the cage. On the other hand, the C60 cage is truly amphoteric because it sly an acid and a base. ohedral fullerenes; isomerism; acid-base interaction or new compounds, new materials, and their new properties are among the primary try. Geometry, topology and set theory can provide a flash of inspiration for the molecules. This is because without a name a thing does not exist in our minds, and plethora of geometric, topological and set theory relations between the objects is known in mathematics. One can find new molecules through modifications of known very of new syntheses, finding new natural products, but also through a search for merism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical la but demonstrating different properties during the time characteristic for a given form cages and exhibit quite specific types of isomerism connected to both ween the atoms constituting the cage and the presence of a well-defined interior, xterior. A cage isomerism occurs at the boundary and is due to the arrangement of posing the cage in polygons (mostly pentagons and hexagons) [4]. For example, ly as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only exagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, ng to the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus erenes occur in two enantiomeric forms, which makes the existence of 3532 C60 osed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with symbol, will clarify the isomerism type, e.g., ww.mdpi.com/journal/symmetry ction as an erism aw, Poland; ••Y ₪ic Y@C•••X, (₪ is the molecules X, Y, and a cage the cage. The ic isomerism exterior of C is empty. By . We also discuss the other of the XH•••NH3@C60 and el. The calculated energies d the NH3 base inside the 60, by about 4-8 kcal/mol. ter of 6.5 kcal/mol). In the he NH3 molecule is always o an acid for the NH3 base ruly amphoteric because it ies are among the primary lash of inspiration for the not exist in our minds, and ns between the objects is gh modifications of known t also through a search for ce of molecules of identical e characteristic for a given erism connected to both of a well-defined interior, due to the arrangement of xagons) [4]. For example, lerenes composed of only 1812 structures, 1508, 189, ups, respectively [5]. Thus the existence of 3532 C60 lerenes are fullerenes with con and www.mdpi.com/journal/symmetry nteraction as an Isomerism -195 Warsaw, Poland; (ic): X@C•••Y ₪ic Y@C•••X, (₪ is the ystem of molecules X, Y, and a cage r outside the cage. The ic isomerism interior or exterior of C is empty. By C is empty. We also discuss the other culations of the XH•••NH3@C60 and TZVP level. The calculated energies F, Cl) and the NH3 base inside the H3•••HX@C60, by about 4-8 kcal/mol. X (a matter of 6.5 kcal/mol). In the cids and the NH3 molecule is always cage is also an acid for the NH3 base 60 cage is truly amphoteric because it ction w properties are among the primary ovide a flash of inspiration for the ing does not exist in our minds, and ory relations between the objects is ules through modifications of known ducts, but also through a search for he existence of molecules of identical g the time characteristic for a given of isomerism connected to both presence of a well-defined interior, ry and is due to the arrangement of ns and hexagons) [4]. For example, hs of) fullerenes composed of only ace, out of 1812 structures, 1508, 189, metry groups, respectively [5]. Thus ch makes the existence of 3532 C60 Heterofullerenes are fullerenes with R,S can denote constitutional and R,S stereo-isomerism, respectively. Notice that Symmetry 2020, 12 as an isomerism in the three-component system of molecules X, Y, and a cage C, in which one of the molecules is located inside and the other outside the cage. The ic isomerism is similar to the endo-exo one, which occurs only if either the interior or exterior of C is empty. By contrast, ic occurs only if neither the interior nor the exterior of C is empty. We also discuss the other closely related types of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and NH3•••HX@C60 ic isomers were performed at the ωB97XD/Def2TZVP level. The calculated energies demonstrated that the systems with the HX acid outside (X = F, Cl) and the NH3 base inside the cage, XH•••NH3@C60, are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. This is because NH3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the studied systems and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always a Lewis base. The C60 molecule with HX inside or outside the cage is also an acid for the NH3 base positioned outside or inside the cage. On the other hand, the C60 cage is truly amphoteric because it is simultaneously an acid and a base.

Introduction
Searching for new compounds, new materials, and their new properties are among the primary tasks of chemistry. Geometry, topology and set theory can provide a flash of inspiration for the discovery of new molecules. This is because without a name a thing does not exist in our minds, and the name for a plethora of geometric, topological and set theory relations between the objects is usually already known in mathematics. One can find new molecules through modifications of known structures, discovery of new syntheses, finding new natural products, but also through a search for new types of isomerism [1-3]. Isomerism is the phenomenon of the existence of molecules of identical summary formula but demonstrating different properties during the time characteristic for a given measurement.
Fullerenes form cages and exhibit quite specific types of isomerism connected to both connectivity between the atoms constituting the cage and the presence of a well-defined interior, boundary, and exterior. A cage isomerism occurs at the boundary and is due to the arrangement of the C-atoms composing the cage in polygons (mostly pentagons and hexagons) [4]. For example, there are possibly as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only pentagons and hexagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, 19, 3, and 1 belong to the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus 1720 of C60 fullerenes occur in two enantiomeric forms, which makes the existence of 3532 C60 fullerenes composed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with is the equivalence relation (Appendix B).
Taking into account the assumptions listed in Appendix A, let us focus on the isomerism between the systems at the right sides of Equations (10) and (11): a new inversion in a cage isomerism (ic): X@C•••Y ₪ic Y@C•••X, (₪ is the as an isomerism in the three-component system of molecules X, Y, and a cage e molecules is located inside and the other outside the cage. The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By ly if neither the interior nor the exterior of C is empty. We also discuss the other s of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and mers were performed at the ωB97XD/Def2TZVP level. The calculated energies he systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the subsystems, the HX molecules are Lewis acids and the NH3 molecule is always 0 molecule with HX inside or outside the cage is also an acid for the NH3 base r inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. ral fullerenes; isomerism; acid-base interaction compounds, new materials, and their new properties are among the primary eometry, topology and set theory can provide a flash of inspiration for the ecules. This is because without a name a thing does not exist in our minds, and ora of geometric, topological and set theory relations between the objects is n in mathematics. One can find new molecules through modifications of known of new syntheses, finding new natural products, but also through a search for m [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical t demonstrating different properties during the time characteristic for a given cages and exhibit quite specific types of isomerism connected to both the atoms constituting the cage and the presence of a well-defined interior, r. A cage isomerism occurs at the boundary and is due to the arrangement of ng the cage in polygons (mostly pentagons and hexagons) [4]. For example, many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only ns [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus s occur in two enantiomeric forms, which makes the existence of 3532 C60 f 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with The isomerism, (₪ is the component system of molecules X, Y, and a cage and the other outside the cage. The ic isomerism if either the interior or exterior of C is empty. By e exterior of C is empty. We also discuss the other scussed. Calculations of the XH•••NH3@C60 and B97XD/Def2TZVP level. The calculated energies outside (X = F, Cl) and the NH3 base inside the isomers, NH3•••HX@C60, by about 4-8 kcal/mol. cage than HX (a matter of 6.5 kcal/mol). In the s are Lewis acids and the NH3 molecule is always outside the cage is also an acid for the NH3 base r hand, the C60 cage is truly amphoteric because it -base interaction and their new properties are among the primary heory can provide a flash of inspiration for the t a name a thing does not exist in our minds, and and set theory relations between the objects is d new molecules through modifications of known natural products, but also through a search for omenon of the existence of molecules of identical perties during the time characteristic for a given ecific types of isomerism connected to both age and the presence of a well-defined interior, t the boundary and is due to the arrangement of stly pentagons and hexagons) [4]. For example, orphic graphs of) fullerenes composed of only ensional space, out of 1812 structures, 1508, 189, 5 point symmetry groups, respectively [5]. Thus forms, which makes the existence of 3532 C60 ons possible. Heterofullerenes are fullerenes with , of the through-the-cage C interactions between the acid A and the base B Equation (14), describes a situation when either A is in cage C and B is outside, or the opposite. One can understand it as an interchange of A with B and B with A with fixed C. Such an interchange resembles a transformation called inversion in a sphere or spherical inversion. Inversion in a sphere is a bijection of R3\{0} onto itself in which every internal point of a sphere, except the origin, is transformed on one and only one external point of the sphere, keeping the sphere fixed. As the spherical inversion is not only a reversible, but also continuous transformation, it is a homeomorphism of R 3 \{0} onto itself. An analogy between the isomerism demonstrated in Equation (14) and the inversion in a sphere mapping prompted us to call this kind of isomerism the inversion in a cage (ic) isomerism and use the ction as an erism aw, Poland; ••Y ₪ic Y@C•••X, (₪ is the molecules X, Y, and a cage the cage. The ic isomerism exterior of C is empty. By . We also discuss the other of the XH•••NH3@C60 and el. The calculated energies d the NH3 base inside the 60, by about 4-8 kcal/mol. ter of 6.5 kcal/mol). In the he NH3 molecule is always o an acid for the NH3 base ruly amphoteric because it ies are among the primary lash of inspiration for the not exist in our minds, and ns between the objects is gh modifications of known t also through a search for ce of molecules of identical e characteristic for a given erism connected to both of a well-defined interior, due to the arrangement of xagons) [4]. For example, ic symbol for the relation between the ic isomers. Let us note that, while the ic isomers were already studied [19,20], the phenomenon of the ic isomerism itself, has not been the issue, yet. The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By y if neither the interior nor the exterior of C is empty. We also discuss the other of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and ers were performed at the ωB97XD/Def2TZVP level. The calculated energies e systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the subsystems, the HX molecules are Lewis acids and the NH3 molecule is always molecule with HX inside or outside the cage is also an acid for the NH3 base inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. al fullerenes; isomerism; acid-base interaction compounds, new materials, and their new properties are among the primary ometry, topology and set theory can provide a flash of inspiration for the cules. This is because without a name a thing does not exist in our minds, and ra of geometric, topological and set theory relations between the objects is ic Y@C· · · X (15) can be classified [42]. To simplify the reasoning, assume that X and Y themselves are not isomers of any type. First, the ic isomerism is not a constitutional isomerism because the molecular connectivity does not change with the change of the ic isomer. This, however, requires a widely accepted though not indisputable [41,43] agreement that the intermolecular and in-the-cage interactions are not categorized as ordinary chemical bonds and do not contribute to the molecular connectivity. So X@C· · · Y and Y@C· · · X are the stereoisomers. Second, they are not mirror images of each other, hence, they are not enantiomers, but diastereoisomers. Two diastereoisomers superimposable by a rotation about a single bond, or a finite series of such rotations, are called conformational diastereoisomers; otherwise, they are configurational [42]. A single bond binds neither X with C nor Y with C nor X with Y. Consequently, the X@C· · · Y and Y@C· · · X pair of isomers must fall into the class of configurational diastereoisomers. This conclusion is counterintuitive since the considered ic isomers are placed in one class with the achiral compounds containing multiple chiral centers or with those exhibiting the (E,Z)-diastereoisomerism [44], or with rigid compounds able to be stably twisted, such as cyclododeca-octaenes [45] or some dicyclopentadienes [46].
However, if we agree that the intermolecular and in-the-cage interactions are not ordinary chemical bonds and do not change molecular connectivity, the X@C· · · Y and Y@C· · · X systems are Symmetry 2020, 12, 1291 6 of 15 three-component topological molecules, which were not considered in a classical isomer classification scheme [42]. Some 20 years ago, we studied model topological isomers such as knots, catenanes, Möbius strips, etc., e.g., references [47][48][49][50]. As a reflection on those structures, we modified the isomer classification scheme to include topological compounds which, unlike common molecules, can, inter alia, be as multicomponent as the catenanes and rotaxanes [51]. In the modified classification tree, as soon as two molecules are recognized as true isomers, a question about the number of independent connected components is asked [51]. Until very recently [41], we have not considered endohedral systems. Yet, just as with endohedral fullerenes and nanotubes, the number of independent connected components constituting them is crucial. Indeed, the endohedral structures may contain many species in a very convoluted manner [52][53][54][55][56][57].
The three-component ic isomers Equation (14)  We also discuss the other f isomerisms are also discussed. Calculations of the XH•••NH3@C60 and s were performed at the ωB97XD/Def2TZVP level. The calculated energies ystems with the HX acid outside (X = F, Cl) and the NH3 base inside the more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the systems, the HX molecules are Lewis acids and the NH3 molecule is always olecule with HX inside or outside the cage is also an acid for the NH3 base side the cage. On the other hand, the C60 cage is truly amphoteric because it d and a base. fullerenes; isomerism; acid-base interaction mpounds, new materials, and their new properties are among the primary etry, topology and set theory can provide a flash of inspiration for the les. This is because without a name a thing does not exist in our minds, and of geometric, topological and set theory relations between the objects is mathematics. One can find new molecules through modifications of known ew syntheses, finding new natural products, but also through a search for -3]. Isomerism is the phenomenon of the existence of molecules of identical monstrating different properties during the time characteristic for a given ges and exhibit quite specific types of isomerism connected to both atoms constituting the cage and the presence of a well-defined interior, cage isomerism occurs at the boundary and is due to the arrangement of the cage in polygons (mostly pentagons and hexagons) [4]. For example, ny as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus cur in two enantiomeric forms, which makes the existence of 3532 C60 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with ndo-exo C· · · X (16) where X, C, @, and · · · denote the same as in Equations (8)- (12) and in the exo form X is interacting with the cage surface e.g., [19,20,[58][59][60]. Indeed, if we agree that a ghost molecule Ø could be included in the set of molecules, then Equation (16)  The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By y if neither the interior nor the exterior of C is empty. We also discuss the other of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and ers were performed at the ωB97XD/Def2TZVP level. The calculated energies e systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the subsystems, the HX molecules are Lewis acids and the NH3 molecule is always molecule with HX inside or outside the cage is also an acid for the NH3 base inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. al fullerenes; isomerism; acid-base interaction compounds, new materials, and their new properties are among the primary ometry, topology and set theory can provide a flash of inspiration for the cules. This is because without a name a thing does not exist in our minds, and ra of geometric, topological and set theory relations between the objects is in mathematics. One can find new molecules through modifications of known f new syntheses, finding new natural products, but also through a search for [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical demonstrating different properties during the time characteristic for a given cages and exhibit quite specific types of isomerism connected to both the atoms constituting the cage and the presence of a well-defined interior, r. A cage isomerism occurs at the boundary and is due to the arrangement of g the cage in polygons (mostly pentagons and hexagons) [4]. For example, many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only ns [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, he chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus occur in two enantiomeric forms, which makes the existence of 3532 C60 f 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with Introducing the Ø ghost molecule reveals both similarity and dissimilarity between the two isomerisms. Removing the ghost molecule from (17) makes the system two-component and thus disqualifies it from the class of the ic isomers. Then, the end-exo isomerism may occur for two-or more-, while the ic isomerism for three-or more-component systems. Thus, A more general class of isomerism, which could include both the endo-exo and ic types of isomerisms in which the same number of components is involved with respect to the same cage C, can be called the cage-host in-out isomerism. This isomerism is similar to the cage-configuration in-out isomerism, which describes, for example, the C m R 2 molecules, where m is large enough and one can direct some R-groups in and the other out of the cage [28]. Notice that the designation, cage, is inevitable because the term in-out isomerism was first and foremost used to describe equilibria in the macrobicyclic molecules [21][22][23][24][25][26]. A pair of isomers composed by X, Y, and C, wherein one isomer belongs to the endo-exo isomers (e.g., (X + Y)@C) while the other, to the ic isomers (e.g., X@C· · · Y), is an example the configurational cage-in-out isomerism: is the s an isomerism in the three-component system of molecules X, Y, and a cage e molecules is located inside and the other outside the cage. The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By y if neither the interior nor the exterior of C is empty. We also discuss the other of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and ers were performed at the ωB97XD/Def2TZVP level. The calculated energies e systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the ubsystems, the HX molecules are Lewis acids and the NH3 molecule is always molecule with HX inside or outside the cage is also an acid for the NH3 base inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. Let us also remember that recently, we introduced [41] Definition 6. The cage isomerism of the endohedral molecules is the phenomenon whereby the same internal individuum is captured in two or more cages of differing cage connectivity.

Definition 7.
The endohedral isomerism is the phenomenon whereby an internal individuum captured in a cage can occupy more than one stable position without changing the cage connectivity.
At the end of this section, one should remark that the onion-like fullerene and nanotube structures [52][53][54][55][56][57] offer a chance for the existence of a huge number of the isomerism types connected to the subsequent inclusion of cages inside the other cages and the division of the space into the three-dimensional onion-like convex (or otherwise) layers. However, here, we conclude reflections about isomerisms at one of the simplest: inversion in a cage isomerism.

Computational Example of the Inversion in a Cage Isomers
Consider a computational example of the inversion in a cage isomerism of three-component molecules composed of an acid A = HF or HCl, a base B = NH 3 , and a cage C = C 60 ( Figure 1). Encapsulation of HF and NH 3 in C 60 was already achieved with the molecular surgery i.e., synthesis of an open-cage fullerene and placing HF or NH 3 inside and closing the cage [61][62][63][64][65]. For HCl, analogous synthesis seems to be feasible as well [66][67][68][69]. Although the XH· · · NH 3 complex is likely to be stable in the C 60 cage [70], here, we focus on the ic isomerism rather than on the A, B, C system in full generality.
which some of the (n-1)-components exchange their in-out positions, while the cage remains unchanged.

Definition 5. The cage-configuration in-out isomerism occurs when configuration of cage C in two isomers is different while the cage-connectivity remains unchanged.
Let us also remember that recently, we introduced [41] Definition 6. The cage isomerism of the endohedral molecules is the phenomenon whereby the same internal individuum is captured in two or more cages of differing cage connectivity.

Definition 7. The endohedral isomerism is the phenomenon whereby an internal individuum captured in a cage can occupy more than one stable position without changing the cage connectivity.
At the end of this section, one should remark that the onion-like fullerene and nanotube structures [52][53][54][55][56][57] offer a chance for the existence of a huge number of the isomerism types connected to the subsequent inclusion of cages inside the other cages and the division of the space into the threedimensional onion-like convex (or otherwise) layers. However, here, we conclude reflections about isomerisms at one of the simplest: inversion in a cage isomerism.

Computational Example of the Inversion in a Cage Isomers
Consider a computational example of the inversion in a cage isomerism of three-component molecules composed of an acid A = HF or HCl, a base B = NH3, and a cage C = C60 (Figure 1). Encapsulation of HF and NH3 in C60 was already achieved with the molecular surgery i.e., synthesis of an open-cage fullerene and placing HF or NH3 inside and closing the cage [61][62][63][64][65]. For HCl, analogous synthesis seems to be feasible as well [66][67][68][69]. Although the XH•••NH3 complex is likely to be stable in the C60 cage [70], here, we focus on the ic isomerism rather than on the A, B, C system in full generality.

A•••(B@C) B•••(A@C)
A = HF, HCl, B = NH3, C = C60 Look at the energetics of the studied systems (Tables 1, A1 and A2). Hereafter, we will comment only on the BSSE corrected values obtained at the ωB97XD/Def2TZVP level (Table 1), while the other energy values are given for the sake of comparison only, unless stated otherwise (Tables 1, A1 and A2).
We have grouped rows in Table 1 into seven Blocks numbered with Roman numbers. The first Block, I, ("the acid-base interaction"), E(HX· · · NH 3 )-E(HX)-E(NH 3 ), contains interaction energies (and their components) determined for the two free XH· · · NH 3 systems interacting in the vacuum. Block II ("the external interaction with the empty cage"), E(M· · · C 60 )-E(M)-E(C 60 ), contains interaction energies for the XH acids or NH 3 base interacting with the empty C 60 cage from the outside which is denoted as M· · · C 60 (where M = XH or NH 3 ). The Block III ("the guest-cage iternal interaction"), E(M@C 60 )-E(M)-E(C 60 ), shows the energies for the XH or NH 3 molecules interacting from the inside with the C 60 cage denoted as M@C 60 . The Block IV ("the ternary interaction") presents gain in energy in the ternary systems, E(M 1 · · · (M 2 @C 60 ))-E(M 1 )-E(M 2 )-E(C 60 ), where M 2 is inside while M 1 is outside the cage in the M 1 · · · (M 2 @C 60 ) system (M 1 , M 2 = XH or NH 3 and M 1 M 2 ; and always either M 1 = NH 3 or M 2 = NH 3 ). This stabilization energy contains contributions from all three components. Block V ("the external interaction with the filled cage"), E(M 1 · · · (M 2 @C 60 ))-E(M 1 )-E(M 2 @C 60 ) contains interaction energies for the molecule interacting from the outside with the C 60 cage, which contains the partner molecule. They are similar to that in Block II, yet now C 60 is not empty. Block VI ("the acid-base through-the-cage interaction"), E(M 1 · · · (M 2 @C 60 ))-E(M 1 · · · M 2 )-E(C 60 ) contains interaction energies between the XH and NH 3 molecules where one is outside and the other is inside the C 60 cage. They correspond to the energies in Block I, yet now the cage wall disturbs the interaction. The last Block, VII, ("the guest interaction with the cage and surrounding"), E(M 1 · · · (M 2 @C 60 ))-E(M 1 · · · C 60 )-E(M 2 ) contains interaction energies for the molecules inside the cage with the cage interacting from the outside with yet another molecule. They correspond to the energies in Block III, but now a third molecule interferes with the interaction between the guest and the cage.
First, comparison of the energy difference between A· · · B@C and B· · · A@C (Block IV, Table 1 and ∆E = E(A· · · B@C)-E(B· · · A@C), Table A2) reveals that the systems with the HX acid outside (X = F, Cl) and the NH 3 base inside the cage, XH· · · NH 3 @C 60 , are more stable than their ic isomers, NH 3 · · · HX@C 60 , by about 4-8 kcal/mol. The difference depends on the acid and the kind of energy compared (Block IV, Table 1 and E, E ZPE , G, total, zero-point-corrected, and Gibbs free, respectively, Table A2).
Greater stability of the ic isomers with NH 3 inside is a consequence of the greater stabilization of ammonia than HX in the cage, while very similar stabilization energies occur when HX or NH 3 interact from the outside of the cage. Indeed, inside the cage, the base is stabilized by about 16 kcal/mol, whereas the acid is stabalized by about 10 kcal/mol (Block III, Table 1). On the other hand, all binary systems with the molecule outside C 60 are similarly stabilized by about 2.3 kcal/mol (Block II, Table 1). Both the C 60 and heterofullerene cages have π-electron systems which can be donating or accepting [41,71].
However, comparison of the Li@C 60 and F@C 60 molecules indicates that C 60 is a better π-electron acceptor than π-electron donor [71]. This explains why a free-electron-pair donor is more stabilized inside C 60 than an electron acceptor. Despite the fact that NH 3 and HX are both electron donors (free-electron pairs on N, F, and Cl) and electron acceptors (H atoms with a positive partial charge), ammonia is a much stronger electron donor then HX, and HX is a much stronger electron acceptor than NH 3 is.
Second, notice that interaction from the outside with the filled cage is only a bit stronger (about 2.5 vs. about 2.3 kcal/mol, Block V vs. II of rows, in Block V the E(X· · · Y@C) values are separated into the (X) and (Y@C) subsystems, Table 1). On the other hand, the interaction energies of the molecule inside with the cage and a molecule outside exhibit about 6.5 ± 1.0 kcal/mol greater stabilization of NH 3 than HX inside the cage (Block VII, Table 1).
Third, the energy of the acid-base interaction through-the-cage would probably be the most interesting, if it could be calculated accurately (Block VI, Table 1). But, (i) ∆E CP is much more negative than ∆E, while for common binary systems it is the opposite. In the BSSE calculations, the presence of the basis functions of the interaction partner additionally stabilizes the system, its energy is decreased, and thus, the ∆E CP is more positive than ∆E. (ii) The deformation energies (Equation (4)) are much higher than in the other Blocks. This is because the energies are referenced to the freely interacting XH· · · NH 3 systems. Inserting the cage surface between XH and NH 3 causes an enormous change in geometry of both HX and NH 3 which increases the deformation contribution to the interaction energy. (iii) The BSSE correction in Block VI is also relatively large and exceeds the ∆E 7 interaction energy. All these problems would be avoided if the systems could be calculated at the basis set saturation limit for which BSSE vanishes. However, for the C 60 endohedral fullerenes, this is hardly possible. As a result of meticulous BSSE calculations and comparison of ∆E 7 and ∆E values in Blocks I and VI, we can only say that (1) the through-the-cage acid-base interactions are weaker than the direct acid-base ones and (2), in the case of the FH· · · NH 3 interaction, weakening of the interaction is quite significant whereas for the ClH· · · NH 3 system, it is less evident.
However, in the light of the recent computational studies [72], conclusion (1) is not as meaningless as it seems. The analysis of the charge distribution between an endohedral species and the cage revealed that, unlike in a common lone electron pair (LP) LP-π bonding, a unique LP-π(cage) interaction pattern displays a charge-depletion in the bonding region. As a result, the HF bond inside the cage is elongated, and HF appears to be more acidic inside the cage [72]. It was shown that this effect has a significant impact on hydrogen bonding inside the cage [72], yet, the through-the-cage interaction behaves differently.

Acids and Bases in the Inversion in a Cage Isomer
Let us discuss the acid-base properties of the studied systems within the frame of the Lewis concept of acids and bases. A Lewis acid is an electron-pair acceptor, while a Lewis base is an electron-pair donor [73]. The acid and the base interact to form a Lewis adduct by sharing the electron pair furnished by the Lewis base [73]. It is clear that to call a molecule a Lewis base, the donated negative charge should not literally be an electron-pair; the Lewis base can simply supply a partial negative charge to the Lewis acid. In fact, all components of the studied systems, an acid A, a base B, and a cage C, have an amphoteric Lewis character. Actually, different parts of the HX, NH 3 , and C 60 molecules can either accept or donate a negative charge depending on how strong a donor or an acceptor directly interacts with them.
Indeed, the gas phase deprotonation enthalpies of the studied species increase from NH 3 through HF to HCl, whereas the proton affinities decrease in different order, from NH 3 through HCl to HF [74][75][76][77]. These strikingly demonstrate differences between proton donating and proton accepting abilities of the studied species. In the gas phase, HCl is the strongest acid, while HF is the weakest base. On the other hand, the gas phase proton affinity of C 60 is 860 kJ/mol [78], is slightly larger than that of NH 3 [74][75][76][77], and makes the C 60 H + cation probably the most abundant fullerene derivative in interstellar environments [79,80]. Moreover, the C 60 gas phase basicity is 827.5 kJ/mol and is again a bit larger than that of NH 3 [74,75]. Thus, in the studied set of compounds, C 60 is located at the beginning of the two series. However, the known data for C 60 refer to the properties of the fullerene outer surface of an empty molecule. The endohedral species interacts with the inner fullerene surface which has slightly other protonation and deprotonation enthalpies. Moreover, the situation varies when an additional molecule is simultaneously present inside or outside of C 60 . The two situations change both the inner and the outer surfaces. Although, C 60 seems to be a relatively simple molecule, consideration of the gas phase basicity and proton affinity in the case of endohedral species or exohedral species, when the other molecule is present on the other side of the cage requires meticulous studies devoted exclusively to this very problem.
Taking into account the approximative sense of the acid-base terminology and the stabilization energies collected in Table 1, we can state that in both binary and ternary systems, the HX molecule can always be classified as an acid and the NH 3 molecule can always be classified as a base. However, the HX interaction partner molecules are always the bases, and the NH 3 interaction partner molecules are always acids. Therefore, C 60 is the base in XH· · · C 60 and HX@C 60 interactions, while it is the acid in H 3 N· · · C 60 and NH 3 @C 60 interactions. Also, in the ternary systems, in the interactions with NH 3 and HX (be they external or internal species) the (HX@C 60 ) and (HX· · · C 60 ) moieties are the acids while the (H 3 N@C 60 ) and (H 3 N· · · C 60 ) moieties are the bases respectively. Eventually, in the ternary systems separated into three components, C 60 is amphoteric because it is simultaneously an acid and a base.
Let us finally add that heteroatoms, N or B for example, or a selected functional group introduced into the fullerene structure, can produce substantial change in the charge distribution of the internal and external surface of the parent fullerene. If there is more than one such a heteroatom or substituent, the number of possible constitutional isomers rapidly grows [8]. In consequence, a heteroatom or a substituent can stabilize more than one endohedral and exohedral position, making isomeric equilibria more complex. However, in the case of compounds exhibiting useful properties, the complexity opens a possibility for the construction of a molecular switch, a diode, or a machine. Another important modification of the studied systems could be based on the use of stronger acids or bases such as, primary, secondary, or tertiary amines. In such a case, the fullerene cage should be larger to accommodate the larger molecule. However, to do this, the art of fullerene synthesis has to be shifted to a higher level of effectiveness. Yet, this can be done quickly only if the expected usefulness of the new systems promises a breakthrough in the new applications. At the moment, computational methods are the most effective way to study the potential of the new type of isomerism.

Conclusions
In this paper, we have considered the possibility of the existence of the inversion in a cage isomerism (ic) which was not previously studied, although a few of its examples were computationally studied for different purposes. We define the inversion in a cage isomerism as a three-component system of molecules X, Y and a cage C, in which one of the components is located inside and the other outside the cage: X@C· · · Y www.mdpi.com/journal/symmetry

gh-the-Cage Interaction as an ion in a Cage Isomerism
Ostrowski ology, 16 Dorodna Street, 03-195 Warsaw, Poland; .pl 2020; Published: date n in a cage isomerism (ic): X@C•••Y ₪ic Y@C•••X, (₪ is the n the three-component system of molecules X, Y, and a cage cated inside and the other outside the cage. The ic isomerism occurs only if either the interior or exterior of C is empty. By terior nor the exterior of C is empty. We also discuss the other are also discussed. Calculations of the XH•••NH3@C60 and med at the ωB97XD/Def2TZVP level. The calculated energies the HX acid outside (X = F, Cl) and the NH3 base inside the than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. d inside the cage than HX (a matter of 6.5 kcal/mol). In the X molecules are Lewis acids and the NH3 molecule is always X inside or outside the cage is also an acid for the NH3 base On the other hand, the C60 cage is truly amphoteric because it merism; acid-base interaction w materials, and their new properties are among the primary y and set theory can provide a flash of inspiration for the ause without a name a thing does not exist in our minds, and topological and set theory relations between the objects is One can find new molecules through modifications of known , finding new natural products, but also through a search for is the phenomenon of the existence of molecules of identical ifferent properties during the time characteristic for a given bit quite specific types of isomerism connected to both tuting the cage and the presence of a well-defined interior, ism occurs at the boundary and is due to the arrangement of lygons (mostly pentagons and hexagons) [4]. For example, 60 (non-isomorphic graphs of) fullerenes composed of only in three-dimensional space, out of 1812 structures, 1508, 189, D2, D3, and D5 point symmetry groups, respectively [5]. Thus nantiomeric forms, which makes the existence of 3532 C60 nd 20 hexagons possible. Heterofullerenes are fullerenes with ic Y@C· · · X, where the is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the ems and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always . The C60 molecule with HX inside or outside the cage is also an acid for the NH3 base utside or inside the cage. On the other hand, the C60 cage is truly amphoteric because it ously an acid and a base. ndohedral fullerenes; isomerism; acid-base interaction n g for new compounds, new materials, and their new properties are among the primary istry. Geometry, topology and set theory can provide a flash of inspiration for the ew molecules. This is because without a name a thing does not exist in our minds, and a plethora of geometric, topological and set theory relations between the objects is y known in mathematics. One can find new molecules through modifications of known scovery of new syntheses, finding new natural products, but also through a search for isomerism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical mula but demonstrating different properties during the time characteristic for a given . es form cages and exhibit quite specific types of isomerism connected to both between the atoms constituting the cage and the presence of a well-defined interior, d exterior. A cage isomerism occurs at the boundary and is due to the arrangement of composing the cage in polygons (mostly pentagons and hexagons) [4]. For example, sibly as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only d hexagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, elong to the chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus ullerenes occur in two enantiomeric forms, which makes the existence of 3532 C60 posed of 12 pentagons and 20 hexagons possible. Heterofullerenes are fullerenes with symbol stands for isomerism relation and the superscript, ic, specifies the kind of isomerism. The name of the isomerism originates from the resemblance of an interchange of X with Y and Y with X with fixed C with an inversion in a sphere or spherical inversion, where R 3 \{0} is transformed onto itself, meaning every internal point of a sphere, except the origin, is transformed on one and only one external point of the sphere, keeping the sphere fixed. Discussion of the isomerism classification shows the similarity of the ic isomerism with the endo-exo one which occurs only if either the interior or exterior of C is empty, while ic occurs only if neither the interior nor exterior of C is empty. The other closely related types of isomerisms are also discussed. However, the possibility of the existence of a new type of isomerism can always have implications on many chemical fields. In the case of the inversion in a sphere isomerism, we suppose that, first of all, it can be important for material science chemistry wherein modifications of the inner and outer space of a cage can significantly transform the properties of the entire solid phase.
In the computational part of the study, which illustrates the new type of isomerism rather than carefully describes details of the system, we calculated the XH· · · NH 3 @C 60 and NH 3 · · · HX@C 60 ic isomers, and all of their components, at the ωB97XD/Def2TZVP level. Comparison of the calculated energies demonstrated that the systems with the HX acid outside (X = F, Cl) and the NH 3 base inside the cage, XH· · · NH 3 @C 60 , are more stable than their ic isomers, NH 3 · · · HX@C 60 , by about 4-8 kcal/mol. This is mainly because NH 3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol), while from the outside they interact with the C 60 cage with a similar strength of about 2.3 kcal/mol. As the theme of this special issue is "Symmetry in Acid-Base Chemistry," let us only emphasize that despite the fact that different parts of the HX, NH 3 , and C 60 molecules can either accept or donate a negative charge and are amphoteric in the sense of the Lewis acid-base concept, the HX molecules are acids and the NH 3 molecule is always a base in all studied systems. The C 60 molecule with HX inside or outside the cage is also an acid for the NH 3 base positioned outside or inside the cage. The analogous is true for the C 60 molecule with NH 3 inside or outside the cage. Eventually, in the ternary systems separated into three components, the C 60 cage is truly amphoteric because it is simultaneously an acid and a base.

Appendix A
As we deal with chemical systems, we obviously assume that

Appendix B
In this set of molecules, isomerism is an equivalence relation: The ic isomerism ne, which occurs only if either the interior or exterior of C is empty. By ither the interior nor the exterior of C is empty. We also discuss the other omerisms are also discussed. Calculations of the XH•••NH3@C60 and ere performed at the ωB97XD/Def2TZVP level. The calculated energies ems with the HX acid outside (X = F, Cl) and the NH3 base inside the ore stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. re stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the tems, the HX molecules are Lewis acids and the NH3 molecule is always cule with HX inside or outside the cage is also an acid for the NH3 base e the cage. On the other hand, the C60 cage is truly amphoteric because it nd a base.
erenes; isomerism; acid-base interaction ounds, new materials, and their new properties are among the primary ry, topology and set theory can provide a flash of inspiration for the This is because without a name a thing does not exist in our minds, and geometric, topological and set theory relations between the objects is thematics. One can find new molecules through modifications of known syntheses, finding new natural products, but also through a search for ]. Isomerism is the phenomenon of the existence of molecules of identical nstrating different properties during the time characteristic for a given and exhibit quite specific types of isomerism connected to both oms constituting the cage and the presence of a well-defined interior, ge isomerism occurs at the boundary and is due to the arrangement of cage in polygons (mostly pentagons and hexagons) [4]. For example, as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only . However, in three-dimensional space, out of 1812 structures, 1508, 189, iral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus r in two enantiomeric forms, which makes the existence of 3532 C60 entagons and 20 hexagons possible. Heterofullerenes are fullerenes with The ic isomerism hich occurs only if either the interior or exterior of C is empty. By he interior nor the exterior of C is empty. We also discuss the other sms are also discussed. Calculations of the XH•••NH3@C60 and erformed at the ωB97XD/Def2TZVP level. The calculated energies ith the HX acid outside (X = F, Cl) and the NH3 base inside the able than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. bilized inside the cage than HX (a matter of 6.5 kcal/mol). In the the HX molecules are Lewis acids and the NH3 molecule is always ith HX inside or outside the cage is also an acid for the NH3 base age. On the other hand, the C60 cage is truly amphoteric because it ase. s; isomerism; acid-base interaction s, new materials, and their new properties are among the primary ology and set theory can provide a flash of inspiration for the s because without a name a thing does not exist in our minds, and etric, topological and set theory relations between the objects is atics. One can find new molecules through modifications of known eses, finding new natural products, but also through a search for erism is the phenomenon of the existence of molecules of identical ting different properties during the time characteristic for a given exhibit quite specific types of isomerism connected to both onstituting the cage and the presence of a well-defined interior, merism occurs at the boundary and is due to the arrangement of in polygons (mostly pentagons and hexagons) [4]. For example, 12 C60 (non-isomorphic graphs of) fullerenes composed of only ever, in three-dimensional space, out of 1812 structures, 1508, 189, , C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus The ic isomerism -exo one, which occurs only if either the interior or exterior of C is empty. By y if neither the interior nor the exterior of C is empty. We also discuss the other of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and ers were performed at the ωB97XD/Def2TZVP level. The calculated energies e systems with the HX acid outside (X = F, Cl) and the NH3 base inside the are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the subsystems, the HX molecules are Lewis acids and the NH3 molecule is always molecule with HX inside or outside the cage is also an acid for the NH3 base inside the cage. On the other hand, the C60 cage is truly amphoteric because it acid and a base. al fullerenes; isomerism; acid-base interaction compounds, new materials, and their new properties are among the primary ometry, topology and set theory can provide a flash of inspiration for the cules. This is because without a name a thing does not exist in our minds, and ra of geometric, topological and set theory relations between the objects is in mathematics. One can find new molecules through modifications of known f new syntheses, finding new natural products, but also through a search for [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical demonstrating different properties during the time characteristic for a given cages and exhibit quite specific types of isomerism connected to both the atoms constituting the cage and the presence of a well-defined interior, r. A cage isomerism occurs at the boundary and is due to the arrangement of g the cage in polygons (mostly pentagons and hexagons) [4]. For example, many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only ns [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, he chiral C1, C2, D2, D3, and D5 point symmetry groups, respectively [5]. Thus The ic isomerism ccurs only if either the interior or exterior of C is empty. By erior nor the exterior of C is empty. We also discuss the other re also discussed. Calculations of the XH•••NH3@C60 and ed at the ωB97XD/Def2TZVP level. The calculated energies e HX acid outside (X = F, Cl) and the NH3 base inside the han their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. inside the cage than HX (a matter of 6.5 kcal/mol). In the X molecules are Lewis acids and the NH3 molecule is always X inside or outside the cage is also an acid for the NH3 base n the other hand, the C60 cage is truly amphoteric because it erism; acid-base interaction materials, and their new properties are among the primary and set theory can provide a flash of inspiration for the use without a name a thing does not exist in our minds, and topological and set theory relations between the objects is ne can find new molecules through modifications of known finding new natural products, but also through a search for is the phenomenon of the existence of molecules of identical ifferent properties during the time characteristic for a given it quite specific types of isomerism connected to both uting the cage and the presence of a well-defined interior, m occurs at the boundary and is due to the arrangement of lygons (mostly pentagons and hexagons) [4]. For example, 0 (non-isomorphic graphs of) fullerenes composed of only in three-dimensional space, out of 1812 structures, 1508, 189, e, which occurs only if either the interior or exterior of C is empty. By ther the interior nor the exterior of C is empty. We also discuss the other merisms are also discussed. Calculations of the XH•••NH3@C60 and ere performed at the ωB97XD/Def2TZVP level. The calculated energies ms with the HX acid outside (X = F, Cl) and the NH3 base inside the re stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. e stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the tems, the HX molecules are Lewis acids and the NH3 molecule is always ule with HX inside or outside the cage is also an acid for the NH3 base the cage. On the other hand, the C60 cage is truly amphoteric because it d a base. renes; isomerism; acid-base interaction ounds, new materials, and their new properties are among the primary y, topology and set theory can provide a flash of inspiration for the This is because without a name a thing does not exist in our minds, and eometric, topological and set theory relations between the objects is thematics. One can find new molecules through modifications of known syntheses, finding new natural products, but also through a search for . Isomerism is the phenomenon of the existence of molecules of identical nstrating different properties during the time characteristic for a given and exhibit quite specific types of isomerism connected to both ms constituting the cage and the presence of a well-defined interior, ge isomerism occurs at the boundary and is due to the arrangement of cage in polygons (mostly pentagons and hexagons) [4]. For example, as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only However, in three-dimensional space, out of 1812 structures, 1508, 189, as an isomerism in the three-component system of molecules X, Y, and a cage of the molecules is located inside and the other outside the cage. The ic isomerism endo-exo one, which occurs only if either the interior or exterior of C is empty. By rs only if neither the interior nor the exterior of C is empty. We also discuss the other types of isomerisms are also discussed. Calculations of the XH•••NH3@C60 and c isomers were performed at the ωB97XD/Def2TZVP level. The calculated energies at the systems with the HX acid outside (X = F, Cl) and the NH3 base inside the C60, are more stable than their ic isomers, NH3•••HX@C60, by about 4-8 kcal/mol. NH3 is more stabilized inside the cage than HX (a matter of 6.5 kcal/mol). In the and subsystems, the HX molecules are Lewis acids and the NH3 molecule is always e C60 molecule with HX inside or outside the cage is also an acid for the NH3 base de or inside the cage. On the other hand, the C60 cage is truly amphoteric because it ly an acid and a base. hedral fullerenes; isomerism; acid-base interaction r new compounds, new materials, and their new properties are among the primary y. Geometry, topology and set theory can provide a flash of inspiration for the molecules. This is because without a name a thing does not exist in our minds, and lethora of geometric, topological and set theory relations between the objects is nown in mathematics. One can find new molecules through modifications of known ery of new syntheses, finding new natural products, but also through a search for erism [1][2][3]. Isomerism is the phenomenon of the existence of molecules of identical a but demonstrating different properties during the time characteristic for a given orm cages and exhibit quite specific types of isomerism connected to both een the atoms constituting the cage and the presence of a well-defined interior, terior. A cage isomerism occurs at the boundary and is due to the arrangement of posing the cage in polygons (mostly pentagons and hexagons) [4]. For example, y as many as 1812 C60 (non-isomorphic graphs of) fullerenes composed of only xagons [4]. However, in three-dimensional space, out of 1812 structures, 1508, 189, Z transitivity (A11) Appendix C