The Line Shape Function for Radiative Transitions

In addition to the study of nonradiati ve decay probability (of exdted electronic states), the consideration of the decay of excited electronic states has many additional 

The line shape function is given for transition in emission from excited state s) to a ground state 0) in the form [92] 

The energies E m and Ei of the vibrational levels in each electronic manifold are measured from the zeroth level of that manifold. Thus, 

The function (f) may be related to the generating function introduced in Chapter 3 for calculating the probability of radiationless transitions. If for the matrix elements in Equation 7.6 we write 

The operator Vso(q) for the radiation matter interaction may be expanded about some reference configuration q along some nontotally coordinates of the electronic ground state 

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The Line Shape Function for Radiative Transitions 165 TaWe 7.2 The MNDO-optimized molecular structural parameters of p-terphenyl. 

Fig. 5. tyNe(6 — 5) transitions. The line shape is identical with the spectrometer response function because of the small radiative width of 10 meV. A line width of 26 or 550 meV is achieved for a silicon crystal of 95 mm in diameter... 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

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