Wavepacket time evolution

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 ... 

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...

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

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... 

A wavepacket is nothing other than a coherent superposition of stationary states, each being multiplied by the time-evolution factor In the present case, a most general time-dependent wavepacket is con-... 

Several accurate approximations of the time-evolution operator e tHdt/h and efficient algorithms for the propagation of the wavepacket have been developed in recent years. For a comprehensive overview and an extensive list of references see Gerber, Kosloff, and Berman (1986), Kosloff (1988), Leforestier et al. (1991), and Kulander (1991), for example. Here, we review only one of these methods. 

Fig. 10 The time evolution of the adiabatic wavepacket in the space of coordinates Q and Q2 for the 2D three-state model of the relaxation of Cr(CO)3 after formation by photodissociation superimposed on the lower adiabatic potential energy surface. The D,k conical intersection is at (0,0). Adapted from [17]...

Alternatively, Formulae (12 ), (27 ) or (29) can be used numerically to fit the ct(E)/E of triatomic molecules, even if the interpretation of the fitted parameters is not yet possible. The results presented below show that the numerical improvement obtained by using Formulae (12), (27) or (29) (all have 4 parameters and are able to describe the asymmetry of a a E)/E) is comparable with the improvement observed for CI2 (the Chi /DoF is reduced by typically up two orders of magnitude see Section 4). Here, it is essential to note that the reflection models are only able to describe the envelope of the XS, (corresponding to very short time evolution (f < 10 fs (femtosecond)) of the wavepacket after the photon absorption) and not the vibronic structures which are specific to each molecule and correspond to some vibrational (and or vibronic) oscillations at a time scale of several hiuidred femtoseconds. 

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,... 

The acoustic responses of the lossless fluid cylinders are shown in Figures 6-8, where I denote the incident wavepacket, R the wavepackets reflected by the cylinders and T labels the wavepackets transmitted by the cylinder along the axis of advance of the incident packet, normal to the cylinder axis. The acoustic wave patterns at various stages of time evolution of the interaction between the incident wave and cylinder are shown in the same figure for comparison. The subscripts indicate relative time. The incident wave pattern is displayed for reference purposes also in the figures. 

Time evolution of a one-dimensional free particle wavepacket... 

The designs of the previously mentioned selectivity schemes ignore the possibility of control of the evolution of excitation energy via exploitation of the coherence properties of the coupled matter-electromagnetic field system. Several schemes that do exploit the coherence of the time evolution of a wavepacket excitation have recently been proposed. This chapter is concerned with one of these schemes, namely, the use of coherent pulse sequences to control product formation in chemical reactions. We shall see that this scheme follows naturally from an understanding of the characteristics of time-delayed coherent anti-Stokes Raman spectroscopy (CARS) and of photon echo spectroscopy. 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

The quantum-mechanical description of the dynamics follows a very similar pattern. At the instant that the first photon is incident, the ground-state wavefunction makes a vertical (Franck-Condon) transition to the excited-state surface. The ground-state wavefunction is not a stationary state on the excited-state potential energy surface, so it must evolve as t increases. There are some interesting analytical properties of this time evolution if the excited-state surface is harmonic. In that case a gaussian wavepacket remains... 

An instructive description of the H + H2 reaction was provided by McCullough and Wyatt (1971a, b). They constructed a wavepacket and followed its time development on the Porter-Karplus surface. They introduced centre-of-mass coordinates appropriate for reactants and used these for all times. The wavefunction F at time t = n At was constructed, from the time-evolution operator U, in the form... 

Having all the eigenstates of the system calculated, we are now in a position to study the time evolution of an initial polariton excitation, which we choose in the form of a wavepacket k°) built out of the low-energy polariton states ) of the perfect system ... 

FlG. 10.9. Examples of the time evolution of spatially identical wavepackets built out of the polariton eigenstates of a perfect ID microcavity as in eqn (10.49) with the parameter /31/2 = 5 x 10 6 m. For panels (a) and (b), the initial packet has zero total momentum, ko = 0 for panel (c) the initial packet has a finite momentum determined by ko = 104 cm-1. Only the photon part p 2 of the polariton wave-function is displayed. The initial packets are shown by long-dashed lines, results of the evolution after indicated times t are shown by solid lines for the disordered system and by short-dashed lines for the perfect microcavity (except in panel (a), where the latter practically coincides with the initial packet.) Reprinted with permission from Agranovich et al. (22). Copyright 2007, American Physical Society. 

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)... 

In a strict sense, the time-evolution generated by the operator (2) is acausal A wavepacket that is initially strictly localized in a finite region of space instantaneously spreads over the whole space. Even for the Dirac equation there are some problems with causality and localization (see, e.g., [5]), but since the propagator of the Dirac equation (the time-evolution kernel) has support in the light-cone, distortions of wave functions and wave fronts can at most propagate with the velocity of light. 

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