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Variational calculations of monoatomic endofullerenes

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Endohedral fullerenes are an example of non-bonded complexes whose properties are dominated by strong quantum effects. Following the work of Bacanu et al., and Bacic and Mandziuk, He@C60 and Ne@C70 were studied. These systems exhibit quantisation of translational energies due to particle in a box type effects, and symmetry breaking lifting the degeneracy of energy levels. The energies, wavefunctions and spectra calculated very closely match the previously seen results. For He@C60 the potential was assumed to be radially symmetric and constructed from R2, R4 and R6 terms. This was then used to construct the Hamiltonian matrix in the finite basis representation which was diagonalised to calculate the translational and rotational energy levels with fundamental frequencies of 217.9 cm-1 and 96.7 cm-1 respectively. These were used to plot two low temperature predicted rotational infrared spectra which closely matched the experimental spectra. The wavefunctions show a strong resemblance to hydrogenic atomic orbitals because of the imposed spherical symmetry. For Ne@C70, the spherical symmetry was broken by elongation of the fullerene along the now unique z axis. The potential was taken to be a pairwise Lennard-Jones summation over atoms which could be modelled as a one dimensional anharmonic oscillator along the z direction, and a two dimensional isotropic anharmonic oscillator in the xy plane. The Hamiltonian matrix was constructed in the discrete variable representation and its eigenvalues calculated. The presence of an anisotropic direction lifted the (2l +1)-fold degeneracies into only singly and doubly degenerate states indexed by the angular momentum quantum number for the two-dimensional oscillator. The fundamental frequencies for the z and xy planes were found to be 9.92 cm-1 and 54.73 cm-1 respectively. While the wavefunctions do not have an intuitive analogue as for He@C60, they show a very simple and regular structure linked with the quantum number assignments.

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