Ar0i arr

= <6T(6) + ho r(10) + <12 r(12) + • • •,

where $?/(i(i) is expressed in terms of expansion parameters and the lowest-order spherical tensor component T(6) is written in terms of the icosahedral harmonics Tt as

using Table 4.3, where it is noted that only icosahedral harmonics T£ with m = 0, ±5 couple in lowest order. Similar expressions can be written for the higher-order terms T(10), r(12), etc. [12.15]. It is of interest to note that the lowest-order icosahedral harmonic to enter Eq. (12.1) is for £ = 6 (see Table 4.3), since this is the lowest-order icosahedral harmonic with Ag symmetry, and the second nonvanishing term in the expansion is for I = 10. The large values for I which enter the expansion in Eq. (12.1) indicate the rapid convergence of the expansion and the validity of the spherical approximation as a good lowest-order approximation. The Hamiltonian [Eq. (12.1)] can thus be evaluated convergently using spherical harmonics as the basis functions. The diagonalization of Eq. (12.1) gives one-electron molecular orbital energy levels for the C60 molecule. Good agreement has been found for the lowest-energy states (up through I = 4) between the expansion of Eq. (12.1) using a three parameter fit [12.15] and first principles ab initio calculations [12.9,24-26], as shown in Table 12.1. To obtain a satisfactory fit at higher energies (with regard to the energy of the HOMO level and to the level ordering of the f2u and flg levels for i = 5 and t = 6, respectively) more terms in the expansion of Eq. (12.2) for icosahedral symmetry are needed.

The same basic Hamiltonian is applicable for all icosahedral fullerenes, while the Hamiltonians for the rugby ball-shaped fullerenes are fit using a perturbation in Eq. (12.1) = ¡%Dsh with DSh symmetry

XDJi) = t2T{2) + tj^ + t^lf + t^lf + • • • (12.4)

corresponding to the A\ entries in Table 4.16 (note that there are two distinct irreducible tensors of A\ symmetry for I = 6). We note that for fullerenes with lower symmetry, perturbation terms such as ffl'Dsh(i) enter in lower order as a function of I (i.e., I — 2,4), indicating that more terms may be necessary to carry out the perturbation expansion for fullerenes with lower symmetry. However, we can consider the symmetry-lowering perturbation for C70 to arise from two perturbations: first an icosahedral perturbation 3€ih given by Eq. (12.2), which is the dominant term, and a smaller perturbation 3€D given by Eq. (12.4) due to the elongation along the fullerene fivefold axis. Then more rapid convergence of the symmetry-lowering Hamiltonian in Eq. (12.1) may be achieved.

Table 12.1

77 energy states (in eV) of the Cu) phe-nomenological Hamiltonian model.

I Symmetry Model" Ab initiob

77 energy states (in eV) of the Cu) phe-nomenological Hamiltonian model.

I Symmetry Model" Ab initiob

Was this article helpful?

0 0

Post a comment