Wavepacket initial

Figure Al.6.14. Schematic diagram showing the promotion of the initial wavepacket to the excited electronic state, followed by free evolution. Cross-correlation fiinctions with the excited vibrational states of the ground-state surface (shown in the inset) detennine the resonance Raman amplitude to those final states (adapted from [14].

Typically, the ratio of this to the incident flux detennines the transition probability. This infonnation will be averaged over the energy range of the initial wavepacket, unless one wants to project out specific energies from the solution. This projection procedure is accomplished using the following expression for the energy resolved (tune-independent) wavefunction in tenns in tenns of its time-dependent counterpart ... 

The time-dependent approach is thus just one teclmique for evaluating the action of the Green s fiinction on the initial wavepacket. 

The denominator in Eq. (13) can be interpreted as an average value over the momentum distribution from the initial wavepacket, that is,... 

If V(R) is known and the mahix elements ffap ate evaluated, then solution of Eq. nO) for a given initial wavepacket is the numerically exact solution to the Schrddinger equation. 

In the reactant channel leading up to the transition region, motion along represents the FI atom approaching the molecule, while motion along / is the vibrational motion of the atom. The initial wavepacket is chosen to represent the desired initial conditions. In Figure 2, the FI2 molecule is initially in the ground... 

The time-dependent Schrddinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the paiticles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

Figure 35. Electronic excitation of a LiH wave packet from the outer classical turning point 6ao) of the ground X S" state. The X —> B transition is considered. The initial wavepacket is the shifted ground vibrational state. Taken from Ref. [37].

Photodissociation has been referred to as a half-collision. The molecule starts in a well-defined initial state and ends up in a final scattering state. The intial bound-state vibrational-rotational wavefunction provides a natural initial wavepacket in this case. It is in connection with this type of spectroscopic process that Heller [1-3] introduced and popularized the use of wavepackets. 

The time-dependent quantity in the integrand of Eq. (4.24), (,(r, R, f = 0) ,(r, R, f)), is called the autocorrelation function. It is the integral over all space of the product of the initial wavepacket with the wavepacket at time t. Rama Krishna and Coalson [86] have shown that the Fourier transform over time in Eq. (4.24) can be replaced by twice the half-Fourier transform where the time integral mns from f = 0 to f = oo. Using this result we obtain the final expression ... 

Eigure 4.1 shows a schematic of an excited-state potential energy surface and superimposed on it is a wavepacket, which is the initial wavepacket for a photodissociation process. An analysis line is drawn perpendicular to the contour lines in the asymptotic region of the surface, where there is no longer any substantial interaction between the separating fragments and the contour... 

Figure 4.1. Schematic diagram of an excited-state potential energy surface showing an initial wavepacket for a photodissociation calculation and indicatng its path toward the dissociation products. The line marked Roo is the analysis line. ...

Figure 4.3. The initial wavepacket superimposed on an Li+HF potential energy surface. Also shown (in a schematic manner) is the analysis line, marked R o, in the product channel. In an actual calculation the analysis line would be placed at a much larger value of the product scattering coordinate.

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