Thomas-Fermi Theory of Fullerenes

We study C60 with the use of Thomas-Fermi theory. A spherical shell model is invoked to treat the nuclear potential, where the nuclear and core charges are smeared out into a shell of constant surface charge density. The valence electron distribution and the electrostatic potential are efficiently computed by integration of the Thomas-Fermi equation, subject to the shell boundary conditions. Total energy is numerically calculated over a range of shell radii, and the mechanical stability of the model is explored, with attention to the constraints of Teller’s theorem. The calculated equilibrium radius of the shell is in good agreement with experiment. PACS Numbers: 31.15.Bs, 36.40.Qv, 31.10.+z Typeset using REVTEX 1 The highly symmetrical structure of C60 has motivated geometrical approximations which have previously been invoked to study electronic and optical properties of the molecule. While consideration of the icosahedral structure of the molecule is necessary for detailed comparison with experiment, previous studies have had success in describing some of the properties of C60 within the continuum approximation, where a system of free electrons are constrained to moving on the surface of a sphere [1–4]. While the peak electron density should be found on the shell, electrostatic consideration of the mechanical stability of the entire system requires that a sizeable fraction of the total number of valence electrons be inside the shell. Motivated by this observation, we study here a generalization of the previously considered continuum model: we allow the valence electrons to move in three dimensions in the external potential generated by a spherical shell of constant surface charge density. We call such an artificial molecule whose nuclear potential has spherical shell symmetry “spherene.” We treat spherene by the Thomas-Fermi (TF) method. While TF results are typically rather rough, it is often used to efficiently generate starting potentials for more exact selfconsistent field methods. We have had success [5] using the resulting TF potential in this fashion. Of course, TF theory has historically had value in its own right. In this Letter, we use the TF results to discuss the stability of C60. This is rather subtle business, as it is well-known that local density methods such as TF are cases for Teller’s theorem [6] which states that molecules in TF theory will unbind. We prove Teller’s theorem in the context of a spherical shell, and we circumvent it by considering the true point charge distribution in the calculation of the “nuclear” energy. The approach used here was previously employed by N. March [7] to investigate molecules with the form of a central atom with tetrahedrally or octahedrally coordinated ligands, such as in the case of CH4 and SF6. We closely follow March’s work, applying it to the case of C60, where there is no central atom and an icosahedral arrangement of “ligands.” The absence of a central atom changes the boundary condition of March at the origin. The case of C60 would seem to be ideally suited to this approach, given its high “coordination” number; and 2 the case of an endohedrally-doped fullerene, such as La@C60, is of the exact form considered by March. We start with a positively charged spherical shell of radius R and charge Ze. The shell charge arises from the sum of the positive nuclear charges with the core electrons of the constituent atoms. We take for the case of C60, Z = 60; thus, the valence electrons are the remaining 60 π-electrons. These valence electrons interact with the shell via a spherically symmetric cut-off Coulomb potential, given by