Time-dependent view

H(Q) is the nuclear Hamiltonian in the corresponding electronic state at short distances it describes the motion of the complex and at large intermolecular separations it describes the free fragments. The matrix elements (3.1) are needed for the calculation of photodissociation cross sections. In this chapter we discuss numerically exact and approximate methods that are directly based on the solution of (3.2). The complementary time-dependent view follows in the next chapter. 

Which approach offers the better interpretation of the photodissociation dynamics in general and of the dissociation cross sections in particular The answer to this question depends very much on the particular system and cannot be decided in a unique way. There are cases which are better described by the time-independent picture while other systems may be better understood within the time-dependent view. In the course of this monograph we will always attempt to combine both views in order to reveal all facets of a particular system. 

The classical picture of photodissociation closely resembles the time-dependent view. The electronic transition from the ground to the excited electronic state is assumed to take place instantaneously so that the internal coordinates and corresponding momenta of the parent molecule remain unchanged during the excitation step (vertical transition). After the molecule is promoted to the potential energy surface (PES) of the upper electronic state it starts to move subject to the classical equations... 

Zhang, J. and Imre, D.G. (1989). Spectroscopy and photodissociation dynamics of H2O Time-dependent view, J. Chem. Phys. 90, 1666-1676. 

Dynamical, time-dependent view of molecular theory... 

Chapter 2 - Dynamical, time-dependent view of molecular theory, Pages 9-40,... 

Figure 2 Time-dependent view of the luminescence transition in Figure 1. Only the potential energy curve of the final state is shown. Time-dependent wave functions are given for times of 0,10, and 50 fs. The bottom panel shows the absolute value of the autocorrelation function, visualized as the overlap between the time-dependent wavefunction (t) and the wavefunction at time zero. The first recurrence at 95 fs occurs after a single vibrational period of the 350 cm vibrational frequency used to define the harmonic potential energy curve.

L. The time-dependent view of spectroscopy (Heller, 1981). Suppose that at time t = 0 we make a Franck ondon transition to an upper electronic state. This takes the initial vibrational wave function up in energy and onto a different potential for the motion of the nuclei. This initial state is no longer stationary and it starts evolving in time. The first thing that will happen is tiiat the wave function will depart from the Franck-Condon region. What we want to know is how quickly it will do so. If the molecule is a diatomic, the initial state will periodically revive. But for a polyatomic, IVR will, over time, reduce tiie revival. Hence for a polyatomic we also want to know where the wave-packet goes to. This... 

The temporal width ATq is determined by the speed with which the wave packet moves out of the FC position and therefore by the slope of the PES near the FC point the steeper the potential, the more rapidly the wave packet leaves its place of birth, the broader is the spectrum and vice versa. The maximum of the spectrum is determined by the vertical excitation energy, i.e the difference between the energy of the initial state and the (potential) energy in the upper state evaluated at the ground-state equilibrium. In the time-dependent view the general shape of the absorption spectrum can be explained by the reflection principle. ... 

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