Wavepacket bifurcation

D. J. Tannor Some years ago we looked at the photodissociation of the ozone molecule [/ Am. Chem. Soc. Ill, 2772 (1989)]. Ozone (O3) can be viewed in much the same way as Naa, in that it has an initial wavepacket with three lobes, one located in each of three symmetrically equivalent wells (see Fig. 1). In the excited state, at the same locations, the potential has saddle points. As a result, each of the component wavepackets bifurcates, and there is a change of alliances The left portion of one wavepacket joins up with the right portion of another wavepacket, and they tunnel into the same product arrangement channel. 

The failure to achieve selectivity in this model system can be traced to the dynamics on the anharmonic excited-state surface, and, in particular, the wavepacket bifurcation. This observation motivated us to explore more systematically the features of the excited-state potential energy surface and excited-state wavepacket dynamics that are compatible with the proposed selectivity scheme. 

After returning to the ground state, the wavepacket bifurcates again. From both conical intersections, pathways lead to CHD as well as to... 

Then in Chap. 5, we show that nonadiabatic transition and the associated wavepacket bifurcation are never a product of mathematical imagination but can be indeed observed experimentally. The direct observation of the instant of nonadiabatic transition (namely, the passage of a wavepacket across a nonadiabatic region) has been achieved only recently. This chapter also discuss a possible control of nonadiabatic transitions on the basis of the notion of wavepacket bifurcation. 

In conclusion, SET is quite useful and promising as a practical means to describe nonadiabatic transitions from the view point of electron dynamics. However the theoretical foundation is weak and therefore it is not easy to improve this theory so as to represent the phenomena of nuclear wavepacket bifurcation. 

SET is sometimes categorized as coherent limit in the sense we mentioned in the previous subsection the nuclear motion keeps on a single trajectory and the electronic state evolves coherently as superposition of various electronic state, which makes no sense in the asymptotic region after wavepacket bifurcation. (We will discuss what the coherence is all about later in this book.) Although being derived in rather intuitive manner, SET can also be derived from Pechukas formulation with an (often not correct) additional assumption. 

In contrast to the Pechukas formulation, Eq. (4.41) and the nuclear dynamics driven by Eq. (4.43) can be solved in an explicit manner since it has no futme time dependence (although both have to be solved stepwise in a self-consistent maimer). Despite of computational easiness, however, Eq. (4.43) leads to imphysical dynamics when the electronic wavepacket bifurcates into multiple states each of which asymptotes to a distinct channel the nuclear motion is driven by an unphysical superposition of these distinct channels. In order to see what is wrong, we recall the path-integral formulation. First observe that the mean-field force Eq. (4.43) can be derived from the Pechukas force in Eq. (4.31) by a special assumption that the final state (T) is given by a unitary transformation... 

These apparent defects, however, arises only when the wavepacket bifurcation occur. There is no obvious fault in using mean-field force in the region where quantum state superposition makes sense. Within such limitations, SET is known as a good approximation before possible occurrence of a wavepacket bifurcation. There have hence been proposed numbers of improved implementation of SET as we shall see in the next section. 

Yet, some theoretical problems are left to be discussed to seek for the ultimate and idealistic features as a nonadiabatic-transition theory Although a trajectory thus hopping plural times converges to run on an adiabatic potential surface asymptotically, the off-diagonal density matrix element Pij t) does not vanish practically, as in the original SET. This is ascribed to an incomplete treatment of the nuclear-electronic entanglement. This issue, often referred to as the problem of decoherence, is originated from the nuclear wavepacket bifurcation due to different slopes of potential surfaces, which will be discussed more precisely below. 

Direct Observation of the Wavepacket Bifurcation due to Nonadiabatic Transitions... 

Nuclear wavepacket bifurcation as observed with time-resolved photoelectron spectroscopy... 

In Sec. 3.3 we have shown how the nuclear wavepacket d3mamics on a single electronic state can be tracked in real-time with use of the time-resolved photoelectron spectroscopy. Here in this section we extend the theory to cope with nonadiabatic d3mamics and explore how the associated wavepacket bifurcation can be observed in this spectroscopy. 

Fig. 5.6 Photoelectron energy spectrum as the wavepacket bifurcates at the avoided crossing (first crossing). The probe polarization is perpendicular to that of the pump in Figs. 5.6 and 5.7. (Reprinted with permission from Y. Arasaki et al., J. Chem. Phys. 119, 7913 (2003)).

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