where the rotational and nuclear terms that appear in Eq. (11.3) are assumed to have Ag symmetry, i.e., no rotation or isotopic symmetry breaking is considered here, and therefore the corresponding terms in the total wave function do not appear in Eq. (13.2). The important term in the Hamilto-nian that gives rise to the H-T coupling is the electronic-vibrational mode coupling, denoted by $"ei-vib> which has Ag symmetry because of the general invariance of the Hamiltonian under symmetry operations of the group. There is, however, an admixture of other electronic symmetries associated with the group theoretical direct product rd <g> Tvib which is brought about by the $?ei-vib term 'n the Hamiltonian that turns on the H-T terms in the optical absorption.
To discuss optical transitions we must investigate symmetry-allowed matrix elements of the type (i/r(r,)|p • A\ip(rf)) where T, and denote the symmetries of the initial and final states, respectively, and p • A is proportional to the electric dipole matrix element. To illustrate the calculation of matrix elements for vibronic transitions explicitly, we select the transition li4g;0 —»•' Flg; i>(Au), which denotes an absorption from a singlet ground state with no vibrational excitations and Ag symmetry to an excited vibronic singlet state with Flg symmetry for the electronic state and Au symmetry for the vibration. This vibronic transition is allowed since Flg ® Au = Flu so that the final state couples to Ag via an electric dipole transition.
The matrix element for the 'Ag; 0 -t1 Flg;v(Au) vibronic transition is written in terms of the electric dipole interaction pa and the electron-vibration (electron-phonon for a crystalline solid) interaction
The same interaction that couples the initial and final states also gives rise to a shift in the energy for the transition given by
In some cases a "dipole-forbidden" transition, which occurs via an H-T coupling, can show a larger coupling than a higher-energy "dipole-allowed"
transition, because the admixed state [see Eq. (13.3)] can contribute a larger matrix element.
We now present a more explicit discussion of the application of the Herzberg-Teller theory to C60, reviewing the description of the ground state and the excited excitonic state, before discussing the vibronic states in more detail. As discussed in §12.1.1, the ground state electronic configuration for C60 is h1", and because all the states in /i"1 are totally occupied, the h1° many-electron configuration has Ag symmetry and is a singlet ground state 'Ag with quantum numbers S = 0 and L = 0. This ]Ag state is also commonly labeled S0 by chemists (see Figs. 13.2 and 13.4). The higher-lying excited singlet states of various symmetries are referred to by chemists as Su S2, . ■.. The lowest group of excited states stem from the promotion of one of the hu electrons to the higher-lying tlu state, leaving the molecule in an excited state configuration h9ut\u. This orbital configuration then describes an exciton with one electron (tlu) and one hole (hu), each with spin 1/2. As described in §13.1.1, this excited exciton state can be either a singlet state Sj (S = 0) or a triplet state Tt (5 = 1), where 5 is the spin angular momentum quantum number. In Fig. 13.4, we show a schematic energy level diagram of singlet and triplet molecular excitonic states of C60 built from the h9ut\u configuration (bold horizontal lines) which lie above the Ag ground state. When spin-orbit coupling is considered, then the "spin" and "orbital" parts of the wave function determine the symmetry of the coupled wave functions through the direct product rspin(g>rorbit, where rspin = Ag denotes the symmetry for the singlet state and rspin = Flg for the triplet state (see §12.5). Figure 13.4 shows that the triplet states are generally lower in energy than the singlet states formed from the same molecular configuration. Because of the Pauli principle, two electrons in a triplet state will have different orbital wave functions, and because of wave function orthogonality considerations, these two electrons will tend to repel each other. Thus, the effect of the Coulomb repulsion in the triplet state is reduced relative to that of the singlet state where the orbital wave functions are the same. Excitonic states can also be formed by promoting an electron from the hg° HOMO-1 level (see Fig. 12.1) to the tlu LUMO level, as indicated in the figure legend to Fig. 13.4.
Since the electric dipole operator does not contain a spin operator in the absence of spin-orbit coupling, the selection rule AS = 0 is in force, so that optical transitions from the singlet lAg ground state must therefore be to an excited singlet state (and not to an excited triplet state). The construction of excitonic triplet and singlet states is discussed in §12.5. In Fig. 13.4 the singlet and triplet states have been separated (by convention) to the left and right in the diagram, respectively. The dotted arrows represent the important radiationless intersystem crossings from the singlet manifold to
Fig. 13.4. Schematic diagram for two of the lowest excitonic levels (boldface lines) associated with the h\t\u excitonic configuration of C«,. Associated with each excitonic level are a number of vibronic levels some of which couple to the ground state by an absorption or emission (luminescence) process. Also shown schematically are intersystem crossings between the singlet and triplet vibronic levels, as well as transitions from a triplet exciton state to the electronic ground state in a phosphorescence transition. The S0 ground state together with its associated vibronic levels are shown across both left and right columns for convenience. The lFlu\ Sq and 3FU; Tq states in the figure are associated with the h9eh^t\u excitonic configura tion of the HOMO-1 and LUMO exciton and represent states for allowed optical transitions (heavier full vertical lines). The transition to the 'Flu state is expected to occur in C«, at about
the triplet manifold [13.13]. The superscript on the symmetry label indicates whether the state is a singlet (e.g., lF2g for antiparallel spins of the electron and hole) or a triplet (e.g., 3Flg for spins parallel to each other).
Intersystem crossings (see Fig. 13.4) involve a singlet-triplet transition. The singlet and triplet states are coupled by the spin-orbit interaction which in carbon is very small (see Table 3.1). Since $fs_0 has Ag symmetry, the spin flip transition couples states of the same symmetry. The triplet spin state (5 = 1), however, transforms as Flg, which means that the singlet state with orbital symmetry Ts will make an intersystem crossing to those triplet states which contain the symmetry in the direct product Fig ® rs. Likewise, a phosphorescent transition from the lowest triplet state 3Flg(Ti) would couple weakly to a vibronic level in the lAg ground state manifold if its symmetry is contained in the direct product Fig ® Flg ® Flu between the orbital (Flg) and spin (Flg) wave functions and the electric dipole matrix element (Flu).
Using group theory, the symmetries of the excitonic states built from the configuration h9ut\u can be worked out from the direct product describing the hole and the electron: Hu <g> Flu = Flg + F2g + Gg + Hg (see §4.2, §12.1, and §12.5). These four symmetries apply to both singlet and triplet excitonic states that are formed from the h9ut\u configuration in CM [13.14], The symmetries of the two lowest-lying singlet excitonic levels have been identified as Flg and F2g [13.9,15], as shown in Fig. 13.4, and the two excitonic levels lie very close to each other in energy. The energy separation between these levels is not drawn to scale in the figure in order to show the vibronic states discussed below. Theoretical calculations of these excitonic energy levels and the associated oscillator strengths for the free molecules have been made by Negri et al. [13.9] using a quantum mechanical extension to 7r-electrons of the "consistent force field method" (QCFF/PI method). They find that the JF,g and xFlg excitonic states are nearly degenerate and are the lowest-lying singlet states, but to explain magnetic circular dichro-ism measurements in C60 [13.16], Negri et al. [13.9] have placed the lFlg level slightly lower in energy than the lF2g level.
Since all the states in the initial h™ state and final h9utlu excitonic configurations have gerade parity, electric dipole transitions between these singlet excitonic states and the Ag ground state are forbidden. This conclusion follows whether using the many-electron configuration notation or the one-electron notation given above. Thus configurational mixing, by itself, is insufficient to explain the weak optical absorption observed in C60 below 2.6 eV [13.17], denoting the threshold for allowed dipole transitions (see §13.2). However, with a sufficiently strong electron-vibration interaction, it is possible to admix enough ungerade parity into the excitonic state wave function to form a "vibronic" state to which optical transitions can occur, and whose energy is, to lowest order [Eq. (13.1)], the sum of an electronic contribution (£,) and a molecular vibration energy (hoj). Thus, we associate with each excitonic level a manifold of vibronic levels, for which the excitonic level is the zero vibration state (see Fig. 13.4).
Shown in Fig. 13.4 are the vibronic manifolds for the Ag ground state (hl°) and for the various excitonic states. The vibronic manifold for the ground state is involved in the photoluminescent transitions from the vi-brationless exciton states shown in Fig. 13.4 by the solid downward arrows [13.6]. The absorption transitions between the vibrationless ground state and selected vibronic excited states are indicated in Fig. 13.4 by solid upward arrows. The dashed and heavy horizontal solid lines, respectively, indicate the positions of vibronic states (Ej+htoj) and zero-vibration electronic states (£,), where E, denotes both the ground state (E0) and the excitonic states. The phosphorescent transitions from the Tt triplet state to the S0 ground vibronic states are indicated by vertical downward dotted arrows. The S0 ground vibronic state in Fig. 13.4 is shown as extending across the two columns, showing singlet and triplet excited states, respectively.
The symmetries of vibrations which couple the ground state S0 to a vibronic excited state are given in Table 13.1. As an example of using this table, consider the symmetries of the vibronic states arising from an Hu vibrational mode coupling the ground state (Ag) to a lFlg symmetry excitonic state. The symmetries of these vibronic states are found by taking the direct product Flg <8> Hu = Flu + F2u + Gu + Hu and examining the irreducible representations contained in this direct product. If the direct product contains the representation for the electric dipole operator (Flu), as it does in this case, then the electric dipole transition between the 'Ag ground state and the vibronic states \lF2g;v[Hu(j)]) is allowed for each of the j — 1,..., 7 vibrational modes with Hu symmetry listed in Table 11.1. Therefore, as listed in Table 13.1, vibrations with Hu symmetry activate a transition between the lF2g excitonic state and the Ag ground state either in an absorption process or in a luminescence process. In Table 13.1 are tabu-
Symmetries of vibrations which activate an electronic transition between the ground state with ag symmetry and an excitonic state with symmetry T, for icosahedral symmetry." The symmetry of the vibronic states is found from the direct product Tex ® Tvjb for the exciton re„ and the normal mode vibration Tvib. Listed are those vibrational modes which contain flu symmetry in the direct product re> ® rvib.
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