Evolution of the wavepacket

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. 

In deriving (4.11) we used (4.3) and (4.5) to represent (2.56) and (2.57) to evaluate the integral over t, and (2.70) for the total dissociation wavefunction. For the subsequent discussion we note the following important points  

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Figure Al.6.24. Schematic representation of a photon echo in an isolated, multilevel molecule, (a) The initial pulse prepares a superposition of ground- and excited-state amplitude, (b) The subsequent motion on the ground and excited electronic states. The ground-state amplitude is shown as stationary (which in general it will not be for strong pulses), while the excited-state amplitude is non-stationary. (c) The second pulse exchanges ground- and excited-state amplitude, (d) Subsequent evolution of the wavepackets on the ground and excited electronic states. Wlien they overlap, an echo occurs (after [40]).

As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schrbdinger equation. In the classical limit of the single surface (adiabatic) case, when effectively 0, the evolution of the wavepacket density... 

The form of the coupling that best fits the experimental features is quadratic, Eq. (9), where Qx is Qlig and Qy is Qco. This coupled potential surface includes the coupling constant as an adjustable parameter. It is shown for kxy = 0.25 in Fig. 12. The time evolution of the wavepacket on this surface is no longer independent along Qco and QIiB and therefore a calculation on the full two... 

S(t) reflects the evolution of the wavepacket in the excited electronic state and therefore it reflects the fragmentation dynamics and the forces —dV/dQi in the upper state. 

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).

Evolution of the wavepacket in the excited state with initial condition... 

The evolution of the wavepacket through the curve crossing is nicely reflected in the polarization of the photons emitted to the electronic ground state during dissociation (Lao, Person, Xayariboun, and Butler, 1990). The transition dipole moments of the two excited states, 3Qq and 1Q1, with the ground state are parallel and perpendicular to the C-I bond, respectively. The initial excitation is due to a parallel transition. The subsequent emission, however, involves both parallel and perpendicular transitions because the 1Qi state becomes populated during the breakup. 

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. 

First, we present the dynamics of the initial wavepacket a. Initially the system stands at the equilibrium position of the electronic ground X. The temporal evolution of the wavepacket Pe generated in the electronic excited state is shown in the left-hand column of Fig. 5.9. Apparently, tp originates in the Frank-Condon (FC) region, which is located at the steep inner wall of the electronically excited A state. The repulsive force of the potential l 0 the drives e(t) downhill toward the saddle point and then up the potential ridge, where Pe(t) bifurcates into two asymptotic valleys, with Ye = 0.495 in channel f. The excitation achieved using this simple quadratically chirped pulse is not naturally bond-selective because of the symmetry of the system. The role played by our quadratically chirped pulse is similar to that of the ordinary photodissociation process, except that it can cause near-complete excitation (see Table 5.1 for the efficiency). This is not very exciting, however, because we would like to break the bond selectively. 

From a dynamical (and/or spectroscopic) perspective, we may ask ourselves how to describe and predict the vibronic structures which are superimposed on many low resolution Abs. Cross Sections. These vibronic structures are deeply linked to the time evolution of the wavepacket, after the initial excitation, over typical times of a few hundreds of femtoseconds as discussed by Grebenshchikov et al. [31]. In ID, for a diafomic molecule, fhe fime evolufion is rafher simple when only one upper electronic state is involved. In contrast, for friafomic molecules fhe 3D character of the PESs makes the wavepacket dynamics intrinsically complex. So, for most of the polyatomic molecules, the quantitative interpretation of fhe vibronic structures superimposed to the absorption cross section envelope remains a hard task for two main reasons first because it requires high accuracy PESs in a wide range of nuclear coordinates and, second, it is not easy to follow fhe ND N = 3 for triafomic molecules) wavepackef over several hundred femtoseconds,... 

Fig. 2 a CASSCF/MR-CI Potential energy surface of the 1MLCT absorbing state of Mn(H)(CO)3(H-DAB) as a function of qa=[Mn-H] and qb=[Mn-COax]. b Time evolution of the wavepacket (solid lines) on the MLCT potential (dashed lines)... 

The symmetries of wavepackets viewed on a progressively finer scale offer a temperature-robust way of encoding several qubits of information [62-64], The encoding and often the full control over the quantum evolution of the wavepacket [65] can be implemented by alternating periods of free motion with phase kicks imposed by coordinate-dependent Stark shifts. Distinguishing odd from even wave forms is the essence of the decoding of the qubits of information encoded in the wavefunction by the dynamics of atoms in a trap. The calculations below demonstrate the possibility to distinguish between the even wave form/(°+) and the odd onef K... 

Fig. 3. Evolution of the wavepacket on the S surface for two sets of initial widths. After 20 fs both conical intersections are reached and the wavepacket has started to bifurcate. The arrows denote the locations of the two conical intersections (left Colnmin right C2-CoIn).

Time-resolved ionization offers several advantages as a probe of these wavepackets [41, 42, 343, 360]. For example, the ground state of an ion is often better characterized than higher excited states of the neutral molecule, particularly for polyatomics. Ionization is also universal and hence there are no dark states. Furthermore, ionization provides both ions and photoelectrons and, while ion detection provides mass and kinetic-energy resolution in time-resolved studies [508], photoelectron spectra can provide complementary information on the evolution of the wavepacket [22, 63, 78, 132, 201, 270, 271, 362, 363, 377]. Its utihty for real-time probing of molecular dynamics in the femtosecond regime has been nicely demonstrated in studies of wavepackets on excited states of Na2 [22], on the B state of I2 [132], and on the A state of Nal [201]. Femtosecond photoelectron-photoion coincidence imaging studies of photodissociation dynamics have been reported [107]. 

Figure 5.22 shows the time evolution of the wavepackets. Each column shows a projection of the amplitude of the wave function onto the (/ , ri) plane at times indicated at the top of the panels. The upper row shows the amplitudes of the wavepacket on state 2, and the lower row the amplitude for state 1. Potential energy contours are shown for 3.00 eV (outer contour) and 1.77 eV (inner contour). In each panel the conical intersection at V2 = 1.37 A is indicated with a cross (x). The initial wave function left unexcited on state 1 by the pump pulse has been removed for clarity in plotting Fig. 5.22. 

Fig. 5.29 Time evolution of the wavepacket formed by the pump pulse, (a) without and (b) with application of the control pulse. (Reprinted with permission from Y. Arasaki et ai, Phys. Chem. Chem. Phys. 13, 8681 (2011)).

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