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..h16 3 15 18c

2F2g ..ft •A/ 3 Gg 3 Hg

A c

5-.../115 |l4fc

4r,_

2 Gf

3 pc A p Uc % Ht

3-...h" 1 9

2rs- ..fl 317

Ul IVe 2fIu r6-,r8-2HU r8, r9

2— ... hn 5 5 10c

3 pc AC p LI

1-...Ä" y 0 yc

Ti fg "/lu %u 1*6 ' Fg r9-

"Assumes spherical symmetry.

''Only state of maximum multiplicity S = n/2 is listed. cHund's rule ground state. The ordering in this table is according to the J values for the free molecule. dAll Pauli-allowed states are listed.

states associated with these multiplets for the negatively charged free ion states, while in §12.4.3 we discuss the corresponding states for the positively charged Q,, ions, and in §12.5 we discuss excited states formed by excitation of electrons to higher-lying electronic configurations (e.g., by excitation across the HOMO-LUMO gap), such as would occur in the optical excitation of neutral C60.

The simplest example of an excited state within the ground state configuration comes from the spin-orbit splitting of the ground state configuration

Various Pauli-allowed states associated with the ground state configurations for the icosahedral

Cw cations.

0 .../i10 5 5 0C Ag ...hl° 'Agc A/

1+...A» I 9

1 9

2+...A8 4 12 8C

>Hg Hs

3+... h1 1 14

r6, r8, r, 2[2//J 2[r8,r-] cf2jgu) r9-,(r7-,r,-) (%JGuy (r8,r9,r7-, r9 ), (r8-, r ,)< ecu,2Hu) (r7-,r9),(r8-,r9) (2P\JF2JHU) (r8,r,-),(r6-,r8,r9)

>H<g Ag,Fig,Hg,(F2g + GgUGg+Hgy

5+ ...hs § 15 fc

r8- '• u \ 6au r9c 3r-

"Assumes spherical symmetry.

''Only state of maximum multiplicity is listed, i.e., S = n/2.

cHund's rule ground state. The ordering in this table is according to the J values for the free molecule.

dAll Pauli-allowed states for the valence configuration (Val.) are listed for n = 0,1+, 2+,3+. For n = 4+,5+ only selected states are listed.

h^fl, which arises from the addition of a single electron to C60 (see Table 12.2). Information about this unpaired electron can be obtained from EPR studies (see §16.2.2) The SU (2) notation in Table 12.2 denotes the electronic configuration in spherical symmetry. Physically, the electron in the flu configuration is delocalized on the shell of the C60 molecule and interacts with the other 60 it valence electrons of C60. The orbital part of this electronic configuration transforms as /lu and the spin (5 = 1/2) as r6+ so that on taking the direct product, we obtain the symmetries for the j states

where Tg forms the ground state (;' = 1/2) and r8 forms the excited state (j = 3/2) of the multiplet, as listed in Table 12.2. The label HOMO for the last column in Table 12.2 denotes levels which include the spinorbit interaction in icosahedral Ih symmetry; i.e., C^ is described by double group representations for configurations with an odd number of electrons.

A more interesting case is that found for the ground state multiplet for CIq which has an h]uaf2u electronic configuration. Here the orbital part of the multiplet is described by the states contained in the direct product

These orbital states must be combined with spin states through the spinorbit interaction to form Pauli-allowed singlet (5 = 0) and triplet (5 = 1) states, for which the total wave functions are antisymmetric under interchange of any two electrons. Since the Flg state is orbitally antisymmetric (L = 1), whereas the Ag and Hg states are symmetric (L = 0,2, respectively), the only Pauli-allowed states for which the total wave functions are antisymmetric under interchange of electrons are the 'Ag, 3Flg, and 'Hg states indicated in Table 12.2 under C50 • States in this multiplet are split in energy by relatively small interactions (perhaps ~0.2 eV). Extending the Hund's rule arguments to icosahedral symmetry, we argue that the ground state is the one where 5 is a maximum (5 = 1), suggesting 3Flg as the ground state. Spin-orbit interaction splits the 3Flg level into a multiplet, via the direct product of spin and orbit (Flg <g> Flg = Ag + Flg + Hg), corresponding to j = 0,1,2,. where the minimum j value is identified with the Ag ground state for a less than half-filled shell. The singlet lHg state is not split by the spin-orbit interaction since 5 = 0 for this state (see Table 12.2).

The three-electron configuration for C^ corresponding to h\afl has symmetries for its ground state multiplet that are found by taking the direct products

Flu ® Flu ® Flu -+AU + 3Flu + F2u + Gu + 2Hu. (12.9)

This configuration is important since it corresponds to M3Q0, which is a relatively high-rc superconductor for M = K or Rb (see §15.1). Then considering the spin wave functions for three electrons, we can have S = 3/2 and S = 1/2 spin states yielding the Pauli-allowed states for this configuration: 4AU, 2Flu, and 2HU, which are listed in Table 12.2. Since the 4AU state has the maximum S value of 3/2, it is likely to be the ground state, via extended Hund's rule arguments. This level transforms as the fourfold irreducible representation r8 of the point group Ih. The 2Fiu level corresponding to L = 1 transforms under the spin-orbit interaction as FXu 0 = Tg + T8, resulting in a level splitting. Finally, the 2HU level under spin-orbit interaction transforms as //u<8>r^ = r8+T9, also resulting in a spin-orbit splitting. In addition to the ground state multiplet associated with the configuration flu, many higher-lying multiplets are created by exciting one (or more) of the three electrons into a higher-lying state, as discussed below in §12.5.

As the flu shell becomes more than half full (see Table 12.2), as for example for CJ?0, then the extended Hund's rule for the state j of the multiplet suggests that the maximum j value should be selected for the ground state configuration. In general, the ground state levels for a more than half-filled state can be considered as hole levels. These hole levels correspond to the same structure as the corresponding electron state of each multiplet under the electron-hole duality, except for the ordering of the levels in energy. Thus the multiplet levels for C^ and C^ are closely related, as are the multiplet levels for Cg0 and C]60. In the following section (§12.4.3) we consider hole multiplets for the HOMO states, in contrast to the hole multiplets considered in this section for the LUMO states.

12.4.3. Positive Molecular Ions C^

The electronic states for the positive ions Ccan be treated similarly to the negative ions discussed in §12.4.2, and the Pauli-allowed states for Q0+ are summarized in Table 12.3 for n < 5. Thus for positively charged C^, the configuration in Ih symmetry is h9u, which corresponds to one hole in the HOMO level and thus has fivefold orbital degeneracy and a spin of 1/2. When spin-orbit interaction is taken into account, the hyperfine splitting in icosahedral symmetry corresponds to the level splitting

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