Effective Hamiltonians for the guests in endohedral complexes

With the help of perturbation theory, the ground state energy of the inclusion complex can be expressed as 

This allows one to calculate (G) as the ground state eigenfunction of an effective nonlinear Hamiltonian 

One quickly recognizes the endohedral potential, Eq.(l), in the first term of Eq.(32). Since, as discussed in Section 3.4, the electrostatic potential is almost constant in the vicinity of the cage center, for small guest atoms, ions, or molecules one can neglect the higher-order multipole terms in Eq.(32) and write 

Disappearance of the dipole term is also the consequence of the high symmetry of the Ceo cluster. By the same token, 

In summary, the first-order correction to the Hamiltonian in vacuo is just a constant equal to the product of the endohedral potential and the charge of the guest. This means that, within the first order of approximation, f(G) is the same as the ground state wavefunction of the noninteracting guest. Therefore, the complexation energy, defined as 

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