Theoretical studies on the structure of the endohedral complexes Na@${\mathrm{xaC}}_{{}_{60}{}^{}{}_{}{}^{}}$ and Na@${\mathrm{xaC}}_{{}_{70}{}^{}{}_{}{}^{}}$

On the basis of the Su-Schrieffer-Heeger model and the analysis of the physical pictures of the so-called endohedral complexes Na@${\mathrm{xaC}}_{{}_{60}{}^{}{}_{}{}^{}}$ and Na@${\mathrm{xaC}}_{{}_{70}{}^{}{}_{}{}^{}}$, which consist of ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ molecules and a sodium ion or atom trapped inside, we have introduced two reasonable terms in the Hamiltonian to describe the interaction between the metal ion and the fullerene cage. From this simple Hamiltonian, we can reproduce the results of other first-principles calculations. Using the simplex optimization method, we have calculated the lattice distortions when all carbon atoms relax, and their effect on the potential-energy curve. Our results show that this effect is large and important. We discuss the physical meaning of our results.