Resonant absorption and Lorentzian line shapes

In analogy to Equation (4.3) we expand the time-dependent wavepacket created by the delta-pulse in the excited electronic state in terms of the bound-state wavefunctions 4,1/. Using (4.5) we obtain  

In the first line of (7.20) we have exploited the symmetry relation (4.10) and the constant C is defined in Section 2.5 as C = pn/TteoC with p = (27rfi)-1. The absorption spectrum is discrete as mandatory for a bound-bound transition. 

If we switch on the coupling to the continuum at t = 0 the excited bound states begin to decay with the consequence that the wavepacket and therefore the autocorrelation function decay too. In order to account for this we multiply, according to Equation (7.14), each term in (7.18) by 

At short times, when all states u) are still significantly populated, S(t) will show a complicated time dependence. As time elapses, however, states with shorter lifetimes decay, the number of contributions in (7.21) decreases, and the behavior of S(t) gradually becomes simpler. At very long times when only the state with the smallest rate is still populated S(t) becomes a simple function of time and its modulus decays exponentially to zero. 

Inserting (7.21) into (7.20) yields, after some straightforward algebra, the following expression for the absorption spectrum, 

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