Wavepacket center

Because the initial vibrational state for absorption spectra often is v = 0, the vibrational nonstationary state typically produced initially is an only slightly distorted Gaussian wavepacket centered at R"g. Conservation of momentum requires that this approximately minimum-uncertainty wavepacket be launched at the turning point on the upper surface, R e = R"g, which lies vertically above R"g. [This is a consequence of the stationary phase condition, see Sections 5.1.1 and 7.6 and Tellinghuisen s (1984) discussion of the classical Franck-Condon... 

In an electron scattering or recombination process, the free center of the incoming electron has the functions Wi = ui U u, and the initial state of the free elechon is some function v/ the width of which is chosen on the basis of the electron momentum and the time it takes the electron to aiTive at the target. Such choice is important in order to avoid nonphysical behavior due to the natural spreading of the wavepacket. 

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

The fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

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... 

Figure 38. Time variation of the wavepacket population on the ground X state and the excited A state of NaK. The system is excited by a quadratically chirped pulse with parameters otm = 3.13 X lO eVfs, (5 = 1.76eV, and I = 0.20TWcm . The pulse is centered at r = 0 and has a temporal width t = 20 fs. Taken from Ref. [37].

Figure 39. Pump-dump control of NaK molecule by using two quadratically chirped pulses. The initial state taken as the ground vibrational eigenstate of the ground state X is excited by a quadratically chirped pulse to the excited state A. This excited wavepacket is dumped at the outer turning point at t 230 fs by the second quadratically chirped pulse. The laser parameters used are = 2.75(1.972) X 10-2 eVfs- 1.441(1.031) eV, and / = 0.15(0.10)TWcm-2 for the first (second) pulse. The two pulses are centered at t = 14.5 fs and t2 = 235.8 fs, respectively. Both of them have a temporal width i = 20 fs. (See color insert.) Taken from Ref. [37].

It seems quite natural to describe the extended part of a quantum particle not by wavepackets composed of infinite harmonic plane waves but instead by finite waves of a well-defined frequency. To a person used to the Fourier analysis, this assumption—that it is possible to have a finite wave with a well-defined frequency—may seem absurd. We are so familiar with the Fourier analysis that when we think about a finite pulse, we immediately try to decompose, to analyze it into the so-called pure frequencies of the harmonic plane waves. Still, in nature no one has ever seen a device able to produce harmonic plane waves. Indeed, this concept would imply real physical devices existing forever with no beginning or end. In this case it would be necessary to have a perfect circle with an endless constant motion whose projection of a point on the centered axis gives origin to the sine or cosine harmonic function. This would mean that we should return to the Ptolemaic model for the Havens, where the heavenly bodies localized on the perfect crystal balls turning in constant circular motion existed from continuously playing the eternal and ethereal harmonic music of the spheres. 

Wavepacket motion is now routinely observed in systems ranging from the very simple to the very complex. In the latter category, we note that coherent vibrational motion on functionally significant time scales has been observed in the photosynthetic reaction center [15], bacteriorhodopsin [16], rhodopsin [17], and light-harvesting antenna of purple bacteria (LH1) [18-20]. Particularly striking are the results of Zadoyan et al. [21] on the... 

In solution when iodine is excited to the bound B excited state, dissociation and recombination processes occur. The dissociation is the result of solvent-induced curve crossing to the dissociative a state, the recombination a result of momentum reversals arising from collisions with the surrounding solvent molecules. Eigenstates of the B state will decay in a continuous manner, whereas wavepackets—if the curve-crossing probability is less than unity—decay in a stepwise manner, giving rise to successive pulses of product. The B and a curves cross near the center of the B state, whereas the B state wavepacket is initially created near the left turning point thus there... 

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

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. 

Fig. 7.8. Snapshots of the two-dimensional time-dependent wavepacket evolving on the PES of the Si state of CH3ONO (indicated by the broken contours) only the inner part of 4>(f) is depicted. The Jacobi coordinates R and r denote the distance of CH3O from the center-of-mass of NO and the internal separation of the NO moiety, respectively. The heavy point marks the equilibrium in the So state where the evolution begins. The arrows indicate the evolution of the wavepacket. Adapted from Engel, Schinke, Hennig, and Metiu (1990).

Fig. 16.2. (a) Evolution of the wavepacket in the excited electronic state created by a laser pulse centered at to = 90 fs with a width of 50 fs. The times are given in femtoseconds, (b) Schematic illustration of the potentials in the lower and upper electronic states and of the excitation process. By courtesy of V. Engel. 

Thus, to the first order A(x) in (5.9) must be replaced by A(x) +tv VA(x), where v is the mean velocity of the wavepacket and At is the time delay measured from the pulse center. 

The time dependence of the wavepacket population on the X and B states are plotted in Fig. 5.3 for the case of a quadratically chirped pulse centered at tp = Ofs with a full temporal duration r = 20 fs. More than 86% of the initial state is excited to the B state within a few femtoseconds (see Fig. 5.3). After the excitation, Fe on the B state potential spreads rapidly due to the very light mass of LiH and the flatness of the potential. Finally, however,... 

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