Localized state early time evolution

The Eq. (9.1.16) alternative form for Ivn io) is mathematically equivalent to the standard Eq. (9.1.9) form, but in many cases Eq. (9.1.16) is both more convenient to use and provides deeper insights into how local features of the e-state potential energy surface affect the dynamical processes that are encoded in the absorption spectrum (see Figs. 9.1 and 9.2). The keys to these insights are (i) that k, (0)) is the initially localized state-function that would be produced by a sufficiently short excitation pulse and (ii) that the early time evolution of ti) generates the most prominent features of the absorption spectrum, Iv<< u)). 

Molecules are complex, many-body systems. But, especially for a short time after a specific, experimentalist-designed, localized excitation is created, the evolution of the excitation in the molecule is simple, mechanistic, visualizable, and describable, not stochastic, statistical, or chaotic. The vast majority of inter-particle and inter-state interactions negligibly affect the early time intramolecular dynamics. 

Experimental and theoretical interest in USCSs has existed since the early days of quantum mechanics. For example, a textbook picture of such an unstable state is that of the one-dimensional potential with a local minimum and a finite barrier that is used to explain, in terms of quantum mechanical tunneling, the instability of a nucleus, the concomitant emission of an alpha particle, and ifs energy. Another textbook example of basic importance is the formal construction of a wave packet from a superposition of a complete set of stationary states and the determination, at least for simple one-dimensional motion, of its time evolution. Finally, another example often presented in books is the appearance of structures ("peaks") in the energy-dependent transition rates (cross sections) over the smoothly varying continuum characterizing a physico-chemical process, which are normally called resonances and which are associated with the transient formation of USCSs. 

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