O

13.1. Optical Response of Isolated Cm Molecules 5.0

Fig. 13.3. Oscillator strength of allowed transitions according to Westin and Rosén [13.7, 8] and to Negri et al. [13.9], In the diagram, the one-electron transitions are labeled by I and h for the LUMO and HOMO levels, respectively, while h — 2 refers to two levels below the HOMO level and I + 3 refers to the one-electron band three levels above the LUMO level.

Energy (eV)

Fig. 13.3. Oscillator strength of allowed transitions according to Westin and Rosén [13.7, 8] and to Negri et al. [13.9], In the diagram, the one-electron transitions are labeled by I and h for the LUMO and HOMO levels, respectively, while h — 2 refers to two levels below the HOMO level and I + 3 refers to the one-electron band three levels above the LUMO level.

do not. In the many-particle formalism discussed in §12.3, the electronic ground state for a C60 molecule has Ag symmetry, and the electric dipole transitions are only allowed from this ground state to excited state configurations with Tlu symmetry. In the single-particle approach, an electron in a filled molecular orbital is excited by the photon to another, higher-lying empty orbital, whose parity must be different from that of the initial state. The many-particle calculations of Negri et al. show that the electronic eigen-energies are stable after reaching a configuration size of 14 (or 196 singly excited configurations). However, an accurate determination of the oscillator strengths required a larger configuration [13.9], The dipole-allowed transitions for C60 in solution are further discussed in §13.2.1.

13.1.3. Optical Transitions between Low-Lying Herzberg-Teller Vibronic States

In a one-electron electronic structure model (see Fig. 12.1), a C60 molecule in the ground state has two spin states for each of the five degenerate hu symmetry orbitals (HOMO) [13.10], The lowest unoccupied molecular orbitals (or LUMO) involve the threefold degenerate tlu one-electron states, and each orbital tiu state can hold a spin-up and a spin-down electron. Since both the hu and tlu states have the same odd [or ungerade («)] parity, an electric dipole transition between these one-electron states (hu -*■ tiu) involving the absorption of a single photon is forbidden by parity considerations alone. The first (lowest energy) allowed electric dipole optical absorption would then be hu -> flg (to the "LUMO + 1 state," as shown in Fig. 12.1). Of course, transitions from other filled ungerade (gerade) to empty gerade (ungerade) one-electron molecular orbitals are allowed, subject to satisfying the selection rules for electric dipole transitions (e.g., hu ->■ ag is forbidden, even though the initial and final states are of opposite parity). The theoretical selection rules for a "dipole-allowed" transition from an initial state with symmetry T, require that the decomposition of the direct product r, ® Flu contains the symmetry of the excited state level . The magnitude of the nonvanishing matrix element can often be inferred by invoking the "/-sum rule" [13.11]. This rule relates to the physical fact that wave functions for the allowed states which are close to the initial state in energy will have nearly the same number of nodes as the initial state and hence have the largest matrix elements and show the strongest optical transitions. This argument would say that the transitions between the angular momentum states I = 4 and I — 5 (or gg tXu and hg —> tXu in icosahedral symmetry) would have a larger optical oscillator strength than the I = 5 to I — 6 transitions (hu tlg and hu —> t2g) [13.7,8,12], because the former transitions have fewer nodes in the wave functions and hence a greater overlap in phase between the wave functions for the initial and final states.

To assign the weak absorption band at the absorption edge to transitions between HOMO- and LUMO-derived states, a symmetry-breaking interaction must be introduced which alters the symmetry of either the initial or the final state, so that electric dipole transitions can occur. Similar to the behavior of other aromatic molecules [13.6], this symmetry-breaking mechanism can be supplied by the electron-vibration interaction, and this effect is sometimes referred to as "Herzberg-Teller" (H-T) coupling [13.6], The coupled electron-vibration state is called a "vibronic" state. Group theory can be used to identify the symmetry of these vibronic states, and the optical selection rules follow from the symmetries of the vibronic states and that of the dipole operator. The first step in the process is to identify the symmetries of the many-electron states which result from a configurational mixing of the one-electron states, and the second step is the determination of the particular vibrational mode symmetries which can participate in H-T coupling to these many-electron molecular states.

The mathematical formulation of this problem follows from the arguments given in §11.3.3. The Herzberg-Teller states (vibronic states) are written as

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