Hu r r8 r91210

where forms the ground state (;' = 5/2), as listed in Table 12.3, and is the split-off level, which being a hole level, is more negative in energy. As discussed earlier, the Jahn-Teller effect is expected to give rise to a lower-symmetry molecule.

For calculation of the electronic levels of [12.34], the hHu configuration has orbital symmetries

as listed in Table 12.3. When combined with the spin singlet (5 = 0) and triplet (5 = 1) states, we find that the Pauli-allowed states for h8u have the following symmetries: lAg, 3Flg, lHg, (3F2g +3Gg), and (Hjg +1Hg) as shown in Table 12.3, corresponding to L values of 0,1,2,3,4, respectively. The effect of spin-orbit interaction on these levels is given in the far right-hand column.

Since the ground state multiplets for the half-filled hole configuration h5u and nearly half-filled h6u configuration have many constituents, the levels for these multiplets are not written in detail in Table 12.3. For the configurations hlu,...,h4u where the hu level is less than half filled with electrons, the same symmetries for the multiplets appear as for the less than half-filled hu hole states using the electron-hole duality, except that the level orderings are expected to be different for n holes as compared with n electrons.

12.5. Excitonic States for C60

Since the ground state multiplet h™ for the neutral C60 molecule consists of a single nondegenerate level, optically excited states are formed by promoting one of the electrons in the hu shell to a higher configurational multiplet. For C60 the lowest excited multiplet would correspond to the h9ufiu configuration and the next higher-lying excited multiplet to the h9uflg configuration, using the level ordering of the Hiickel calculation [12.4] or of a more detailed calculation [12.7,18]. The simple physical realization of these excited states is the electron-hole pair created by optical excitation through the absorption of a photon, as discussed further in §13.1.3, the hole corresponding to the h9u term and the electron to either the //H or //g term.

To find the symmetries of the lowest-lying exciton multiplet, we take the direct product of the excited electron (Flu) with that for the hole (Hu), to obtain Flu ® Hu = Hg + Gg + F2g + Flg, as listed under the fc9/,1 configuration in Table 12.4. Since both the electron and hole have a spin of 1/2, we can form singlet (5 = 0, Ag symmetry) and triplet (5 = 1, Flg symmetry) states. Including the spinorbit interaction, leads to Pauli-allowed singlet states lHg +1 Gg+ xF2g +1 Flg, and the corresponding triplet states are found by taking the direct product of Flg (for 5 = 1) with each of the orbital terms Hg + Gg + Flg + Fig yielding the triplet states given in Table 12.4 under Kf\u- For example, the levels arising from the triplet 3Hg state are found by taking the direct product Flg ® Hg = Hg + Gg + F2g + Flg, as listed

States for the neutral Cjjo molecule allowed by the Pauli principle.

Config.

Singlet

Triplet

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