Center of the wavepacket

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Hamiltonian, the wavepacket immediately starts to move away from its origin. When it has reached the asymptotic region where the potential is zero, the center of the wavepacket travels with constant velocity to infinity. The oscillations in i -space reflect the momentum gained during the breakup. 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

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 3.2 shows the same trajectory as in Figure 5.1 in a different representation together with the time-independent total dissociation wave-function StotiR,7)- The energy for the trajectory is the same as for the wavefunction. It comes as no surprise that, on the average, the wave-function closely follows the classical trajectory. Another example of this remarkable coincidence between quantum mechanical wavefunctions on one hand and classical trajectories on the other hand is shown in Figure 4.2. There, the time-dependent wavepacket (f) follows essentially the trajectory which starts at the equilibrium geometry of the ground electronic state. This is not unexpected in view of Equations (4.29) which state that the parameters of the center of the wavepacket obey the same equations of motion as the classical trajectory provided anharmonic effects are small. Figures 3.2 and 4.2 elucidate the correspondence between classical and quantum mechanics, at least for short times and fast evolving systems ...

Within the short-time approximation, the center of the wavepacket remains at Re while its center in momentum space, V t, moves outward with constant velocity Vr = —dV/dR. 

It should also be noted that the LiH molecule is one of the most difficult systems to apply the present method to, because the mass of LiH is very light at 0.875 amu and the gradient of potential difference is relatively large (V A —0.473 eV/ao) at the center of the wavepacket. All of these difficulties have been nicely overcome by employing fast quadratic chirping. This fact guarantees the usefulness of the present method. 

An additional result that emerges from our study concerns the extent to which wavepacket control is possible using coherent pulse sequences. In a two-level system one can exchange the phases of the two levels with a 7t pulse and, in effect, achieve time reversal of the state of the system. In a multilevel system the extent of control is much more restricted. The center of the wavepacket evolves according to the Franck-Condon principle and Hamilton s equations of motion, which in turn are dictated by nature s potential energy surfaces. What can be controlled by the experimenter is the instant at which the wavepacket changes surfaces. This concept forms the basis for a scheme for controlling the selectivity of a reaction,24,25 which we discuss in the next section. 

Figures 37a and 37b show the wavefunction on the excited state potential surface at t = 200 a.u. and 400 a.u without the locking pulse. Figures 37c and lid show the wavefunction at the corresponding times with the locking pulse. The motion of the center of the wavepacket is greatly reduced. More important, with respect to selectivity, there is almost no wavepacket spreading. This example suggests that strong fields may be used in conjunction with the carefully tailored waveforms we have described above to achieve selectivity of reaction.

In the phase-coherent, one-color pump/probe scheme (see Section 9.1.9) the wavepacket is detected when the center of the wavepacket returns to its to position, (x)to+nT — (x)to, after an integer number of vibrational periods. The pump pulse creates the wavepacket. The probe pulse creates another identical wavepacket, which may add constructively or destructively to all or part of the original pump-produced wavepacket. If the envelope delay and optical phase of the probe pulse (Albrecht, et al, 1999) are both chosen correctly, near perfect constructive or destructive interference occurs and the total spontaneous fluorescence intensity (detected after the pump and probe pulses have traversed the sample) is either quadrupled (relative to that produced by the pump pulse alone) or nulled. As discussed in Section 9.1.9, the probe pulse is delayed, relative to the pump pulse, in discrete steps of At = x/ojl- 10l is selected by the experimentalist from within the range (ljl) 1/At (At is the temporal FWHM of the pulse) to define the optical phase of the probe pulse relative to that of the pump pulse and the average excitation frequency. However, [(E) — Ev ]/K is selected by the molecule in accord with the classical Franck-Condon principle (Tellinghuisen, 1984), also within the (ojl) 1/At range. When the envelope delay is chosen so that the probe pulse arrives simultaneously with the return of the center of the vibrational wavepacket to its position at to, a relative maximum (optical phase at ojl delayed by 2mr) or minimum (optical phase at u>l delayed by (2n + l)7r) in the fluorescence intensity is observed. 

Figure 3 shows the absolute values of the autocorrelation functions for three different offsets AQ, defining three different initial positions for 4> on the final state potential energy surface in Figure 2. The slope of the potential surface at the initial position determines the decrease of the autocorrelation function from its initial value of 1, and it depends on the offset AQ between the minima of the potentials along the normal coordinate in Figures 1 and 2. For an offset AQ of zero, the center of the wavepacket (f> encounters a flat potential surface. No decrease of the absolute value of the autocorrelation is expected with time, as the overlap remains 1 at all times. The slow decrease seen for the solid line in Figure 3 is therefore caused by the damping factor F and the calculated spectrum is narrow. For an offset AQ of 1, the decrease at short times is faster, due to the nonzero slope of the potential surface at Q = l, and the calculated spectrum shows a large overall bandwidth. This trend is even more pronounced for the larger offset Ag = 3, the...

Without the control pulse [Fig. 5.29(a)], there is no transfer between diabatic states at the center of the wavepacket (ri = T2) and the center of the wavepacket goes through the conical intersection, where the interaction potential between diabatic states is zero (V12 = 0 at the conical intersection). Thus, population transfer from the initially excited state 2 to state 1 occurs away from the position of the conical intersection and symmetrically for ri < T2 and ri > r2- There are thus two regions of state 1 population at 8 fs. After passage through the region of conical intersection, the state 2... 

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