these dipole-allowed transitions may vary by several orders of magnitude [13.7,19]. This may be seen in Fig. 13.3, where, for example, the strength of the weak (h — 2)^l transition may be compared to that of the strong (h - 6) -> I transition.

The absorption coefficient for a dilute molecular system (i.e., gas phase or solution) may be calculated according to e(û>) = 1 - 4TTNX (13.5)

where e(w) is the optical dielectric function, N is the molecular density, and x is the molecular polarizability given by

In this expression, f} is the quantum mechanical oscillator strength, <o; is the transition frequency given by hwj — (Efj — E0j), and r, is the broadening parameter of the ;'th transition between initial and final electronic states with energies E0j and Efj, respectively. The absorption coefficient a(a>) can be calculated directly from Eq. (13.5) using the relation [13.25]

For a dense collection of molecules, Eq. (13.6) must be corrected using the Clausius-Mossotti relation (see §13.3.1) to account for local electric field contributions to the applied field [13.26].

Calculations [13.7,22] of the optical response of an isolated C60 molecule have shown that the spectrum is strongly affected by electron correlations. These correlations were found to lead to strongly screened oscillator strengths and renormalized transition energies. In Fig. 13.5 we show the results of the calculations of Westin and Rosén for the optical response of an isolated CM molecule for the case of: (a) free response (without screening) and (b) the screened response. The Westin-Rosén model employs a one-electron LCAO approach in the local density approximation, and the results were corrected for intramolecular screening using a random phase approximation (RPA) "sum over states" method. The transitions implied by curves (a) and (b) of Fig. 13.5 simulate the optical absorption coefficient a = 2(eo/c)Im[e1/2] for C60 in the gas phase or in solution, and the transitions implied by these curves are summarized in Table 13.2. The curves in Fig. 13.5 can be calculated according to Eqs. (13.5) and (13.6), using values for the resonance frequencies («,) and oscillator strengths (/;) found in Table 13.2. In calculating these curves, Tj was set equal to 0.3 eV to simulate line-broadening interactions, such as intramolecular vibrations and solvent-C60 interactions. The narrow vertical lines in Figs. 13.5(a) and (b) represent the respective oscillator strengths for the cases of free (fnm) and screened

(fnm) optical response [13.7], In Fig. 13.5(c), the experimental optical absorption data [13.17] for C60 in solution (hexane) are shown for comparison. In this figure, the calculations and the data are plotted vs. wavelength rather than photon energy. As stated above, the observed absorption band between 700 and 430 nm [which has been magnified in Fig. 13.5(c)] is associated with dipole-forbidden transitions between the HOMO (hu) and LUMO (i,„) levels and therefore does not appear in the calculated spectra.

The dramatic effect of screening on the electromagnetic response of C60 is shown in Fig. 13.5(a) and (b) and in Table 13.2, where it can be seen that the oscillator strength of the lowest-energy transition is screened by a factor of ~200. Some renormalization of the various transition frequencies in Table 13.2 also occurs. In this table the specific optical resonances are also identified with one-electron transitions. Comparing the calculated results in Fig. 13.5(b) for the screened response of C60 to the experimental data in (c), we can see that the number and relative strengths of the experimental absorption bands are qualitatively reproduced in the calculation. However, the calculated absorption band positions are red-shifted relative to the data. The calculated results of Bertsch et al. [13.22] are similar to those of Westin and Rosén [13.7] shown in Fig. 13.5, in that they also find significant screening of the low-frequency dipole-allowed transitions. However, their calculated band positions are in better agreement with the experimental data. Unfortunately, they only make assignments for their lowest three resonances to specify optical transitions. These three assignments, however, are in agreement with those of Westin and Rosén [13.7].

13.2.2. Photoluminescence o/C60 in Solution

The luminescence spectra discussed in this section relate to the recombination of excited excitons in the S{ exciton state with a transition to the ground 50 state. The conclusion by Negri et al. [13.9], that the experimental data favor the [Flg state as the lowest excited singlet (5,) state, is particularly important for the interpretation of luminescence data, since molecular systems most often emit radiation as they make a transition from the Si zero-vibration exciton state, even though the molecules may have been optically pumped to a much higher energy singlet state. This follows as a result of successive, rapid, nonradiative transitions down through the excited singlet state ladder to Sj [13.6], as shown in Fig. 13.2.

In the literature, the term photoluminescence (PL) refers to either luminescence or phosphorescence [13.6]. By definition, the "luminescence" is associated with radiative decay from an excited singlet state (usually 5,) and the "phosphorescence" with radiative decay from an excited triplet state (usually the T{ state, using the notation in Fig. 13.2). In the absence of significant spin-orbit coupling, the decay from a triplet state to the singlet ground state is slow but can be enhanced by a spin flip mechanism [possibly the magnetic dipole-dipole interaction of the exciton triplet state with a magnetic impurity such as a triplet (5 = 1) oxygen molecule]. Since the spin flip cannot be provided by the electric dipole operator, the radiative lifetimes for phosphorescence are typically much longer (> 1 ms) when compared to luminescence (< 1 ms). In C60, both luminescence [13.15,27,28] and phosphorescence [13.29] emission has been observed. By virtue of the nearly 100% efficient intersystem crossing [13.13,30,31] observed for a C60 molecule, electrons excited to the singlet states rapidly descend into the triplet states, unless they first leave the 5t excitonic state in a luminescence process. The high sensitivity of photomultiplier detectors allows detection of the luminescence from the 5j singlet excitonic state.

In Fig. 13.6(a), we display the luminescence spectra for C60 in a 6.6 x 10~5 molar methylcyclohexane (MCH) solution at 77 K using laser excitation wavelengths of Acx = 460 nm and Aex = 330 nm [13.15]. The corresponding optical absorption (OA) spectrum for the same solution, shown at the right in Fig. 13.6(a), is displayed in the wavenumber range between 15,000 cm-1 and 18,500 cm-1, while the PL spectra are displayed between 11,700 cm-1 and 16,000 cm-1 [13.15]. To make an identification between corresponding features in the PL and OA spectra of Fig. 13.6(a), we label the features M, and Me, respectively, in order of increasing frequency. First, it is found that the PL and OA features in this figure are only weakly sensitive to the solvent [13.15,17,32] (e.g., toluene or MCH), implying that the solution spectra probe the molecular states of C60 molecules, isolated from one another. Second, the energy of the vibronic features in the PL spectra depends only weakly on excitation energies, as shown by comparison of the two PL traces in Fig. 13.6(a) [13.15].

But most important, the assignments labeled with symbols from M(y to M9, in the observed OA spectrum are identified as mirror images of the vibronic transitions in the PL spectrum labeled with symbols from M0 to Mg, implying that transitions M, and M,, involve the same H-T active vibrational mode, consistent with the PL and OA processes described in Fig. 13.4. The vibrational frequencies wvjb are determined from the optical data as the difference between the observed position of the vibronic feature and the zero-phonon transition energy, E0fy, indicated by the solid vertical lines in Fig. 13.6(a). For a given sample and set of operating conditions, the electronic origins for the PL and OA processes in solution are determined, and for the data shown in Fig. 13.6(a), values of E^Jy = 15,200 cm-1 and _ 15 510 cm-1 are obtained. The resulting values obtained for <uvib (for features M0 to Mb in PL and M0. to M6, in OA) are generally in good agreement with first-order Raman [13.33] and infrared [13.34] mode

22 c

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