Franck-Condon wavepacket

Figure 1. Photoabsorption between two Born-Oppenheimer potential energy surfaces. (A) The Franck-Condon wavepacket, arising out of = nx L X >s shown on the lower surface, and /j>(t) on the upper], grazes (0) several times on the way to dissociation. The result is an absorption band with some limited vibrational structure. (B) Direct dissociation leading to a broad, featureless absorption band. (Reproduced, with permission, from Ref. 1.)...

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

Figure 1. Intramolecular vibrational density redistribution IVR of Na3

Figure 2. Franck-Condon windows lVpc(Gi, r, v5) for the Na3(X) - N83(B) and for the Na3(B) Na3+ (X) + e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(B), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx r cos

Long progressions of feature states in the two Franck-Condon active vibrational modes (CC stretch and /rani-bend) contain information about wavepacket dynamics in a two dimensional configuration space. Each feature state actually corresponds to a polyad, which is specified by three approximately conserved vibrational quantum numbers (the polyad quantum numbers nslretch, "resonance, and /total, [ r, res,fl)> and every symmetry accessible polyad is initially illuminated by exactly one a priori known Franck-Condon bright state. 

Nevertheless, this simple propagation method provides an intriguing picture of the evolution of the quantum mechanical wavepacket, at least for short times. It readily demonstrates that for short times the center of the wavepacket follows essentially a classical trajectory ( Ehrenfest s theorem, Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Figure 4.2 depicts an example the evolution of the two-dimensional wavepacket follows very closely the classical trajectory that starts initially with zero momenta at the Franck-Condon point. 

Figure 7.12 compares the measured and the theoretical absorption spectra for CH30N0(S i), the latter being obtained in a three-dimensional wavepacket calculation (Untch, Weide, and Schinke 1991a). The energy spacing of the main progression reflects the vibrational frequency of NO in the Si complex. The ratio of the peak intensities is mainly determined by the one-dimensional Franck-Condon factors... 

Manz and Romelt (1981). Rm and 7 hi are the two I-H bond distances. The heavy point marks the saddle point and the shaded area indicates schematically the Franck-Condon region in the photodetachment experiment. The arrow along the symmetric stretch coordinate (f Hi = -Rih) illustrates the early motion of the wavepacket and the two heavy arrows manifest dissociation into the two identical product channels, (b) The same PES as in (a) but represented in terms of hyperspherical coordinates (p, i9) defined in (7.33). The horizontal and the vertical arrows illustrate symmetric and anti-symmetric stretch motions, respectively, as indicated by the two insets. 

Figure 8.2 depicts a typical potential energy surface (PES) for a symmetric molecule ABA with intramolecular bond distances R and R2] the ABA bond angle is assumed to be 180° (collinear configuration). The PES is symmetric with respect to the line defined by Ri = R2 it has a saddle point at short distances and decreases monotonically from the saddle point out into the two identical product channels A + BA and AB + A (see also Figure 7.18). The shaded area indicates the Franck-Condon (FC) region accessed via photon absorption and the two arrows illustrate the main dissociation paths for the quantum mechanical wavepacket or, equivalently, a swarm of classical trajectories. Because no barrier obstructs dissociation, the majority of trajectories immediately evanesce in either one of the two product channels without ever returning to the vicinity of the FC point.

---