Wavepacket evolution

In order to show the main features of the reaction dynamics after EP, it is relevant to follow the wavepacket evolution in time, and in Eigs.5 and 6 two different snapshots are shown for t= 7.75 and 15.25 fs. The dynamics leads rapidly to LiE products at short times because of the node present in the r coordinate. At longer times, however, there is a relative small proportion of the wavepacket that remains... 

M. Quack Prof. Zewail and Gerber, when you make an interpretation of your femtosecond observations (detection signal as a function of excitation), would it not be necessary to try a full quantum dynamical simulation of your experiment in order to obtain a match with your molecular, mechanistic picture of the dynamics or the detailed wavepacket evolution Agreement between experimental observation and theoretical simulation would then support the validity of the underlying interpretation (but it would not prove it). The scheme is of the following kind ... 

L. Woste In stationary spectroscopy ZEKE certainly provides spectroscopic results at an impressive resolution. Using femtosecond pulses one can certainly not excite specific states as compared to ZEKE. The Fourier transform of the wavepacket evolution, however, exhibits also spectral resolution that easily reaches and even exceeds what we see in ZEKE spectra. For this reason, I do not see any disadvantage in using femtosecond NeNePo to probe states of a prepared molecule. 

In both cases, la and lb, the total photodissociation cross section is completely determined by the short-time dynamics in the Franck-Condon region. In contrast, the partial cross sections, which determine the vibrational, rotational, and electronic-state distributions of the products, involves longer time dynamics. To obtain all of the relevant information about the reaction, the wavepacket evolution must be followed out into the product region of the potential energy surface and projected onto the various different vibrational and rotational states of the fragments. The partial cross section for scattering... 

Figure 13. The (a) ground and (b) excited- electronic-state Bom-Oppenheimer potential surfaces. The n/2 pulse moves half of the initial amplitude, y(O), from a surface a to surface b. After the pulse the nuclear wavefunctions of the ground-state and excited-state surfaces are denoted x Ui) and r (ti (b) Wavepacket evolution of xb. Motion of the wavepacket causes the overlap of the ground-state wavefunction and the excited-state wavefunction to decay, resulting in free induction decay. x remains in place in coordinate space, (c) The n pulse exchanges the amplitude of surface a with surface b. (

The sum is taken over all the discrete vibrational levels if of state g>. Vr (f) is the component of the wavepacket on the g channel evolved up to time t from the field-free vibrational state v > prepared at time f = 0. Note that Pbound(y if) actually represents the total bound state population at any time after tj, since no further decay is then possible, the laser being turned off at such a time. It is clear that Eq. (71) gives a useful approximation for the result of a full time-dependent wavepacket evolution, [Eq. (73)], only if the assumption of an adiabatic transport of Floquet states is valid. 

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].

Figure Al.6.20. (Left) Level scheme and nomenclature used in (a) single time-delay CARS, (b) Two-time delay CARS ((TD) CARS). The wavepacket is excited by cOp, then transferred back to the ground state by with Raman shift oij. Its evolution is then monitored by tOp (after [44])- (Right) Relevant potential energy surfaces for the iodine molecule. The creation of the wavepacket in the excited state is done by oip. The transfer to the final state is shown by the dashed arrows according to the state one wants to populate (after [44]).

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]).

This section is divided into two sections the first concerned with time-dependent methods for describing the evolution of wavepackets and the second concerned with time-independent methods for solving the time independent Sclirodinger equation. The methods described are designed to be representative of what is in use. 

The methods described here are all designed to detennine the time evolution of wavepackets that have been previously defined. This is only one of several steps for using wavepackets to solve scattering problems. The overall procedure involves the following steps ... 

Central to the description of this dynamics is the BO approximation. This separates the nuclear and electionic motion, and allows the system evolution to be described by a function of the nuclei, known as a wavepacket, moving over a PES provided by the (adiabatic) motion of the electrons. 

If the PES are known, the time-dependent Schrbdinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

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 evolution of a wavepacket representing the H -b H2 scattering reaction for a particular set of inifial conditions is plotted on Figure 2 as a series of snapshots. To display the three-dimensional (3D) wavepacket on a two-dimensional (2D) plot, the reduced density... 

The evolution of the nuclear wavepacket is also traced by a number of snapshots of the absolute values of the wavepacket, again integrating over the... 

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. .. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

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