| Summary: | We define a new inversion in a cage isomerism (<i>ic</i>): X@C∙∙∙Y <i>₪<sub>ic</sub></i> Y@C∙∙∙X, (<i>₪ </i>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 <i>ic</i> isomerism is similar to the endo-exo one, which occurs only if either the interior or exterior of C is empty. By contrast, <i>ic</i> 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∙∙∙NH<sub>3</sub>@C<sub>60</sub> and NH3∙∙∙HX@C<sub>60</sub><i>ic</i> 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 NH<sub>3</sub> base inside the cage, XH∙∙∙NH<sub>3</sub>@C<sub>60</sub>, are more stable than their <i>ic</i> isomers, NH<sub>3</sub>∙∙∙HX@C<sub>60</sub>, by about 4–8 kcal/mol. This is because NH<sub>3</sub> 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 NH<sub>3</sub> molecule is always a Lewis base. The C<sub>60</sub> molecule with HX inside or outside the cage is also an acid for the NH<sub>3</sub> base positioned outside or inside the cage. On the other hand, the C<sub>60</sub> cage is truly amphoteric because it is simultaneously an acid and a base.
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