Wavepacket stationary states

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

In the experiments described thus far, the pump laser simply populates the intermediate excited state. The consequence is that the experiment becomes a means to study that excited state. Often we are more concerned with learning about the ground state than about excited states. For this purpose, it is useful to prepare a vibrational wavepacket of that ground state. One useful means to do this is to excite the species of interest to an allowed excited state and then to down-pump from that excited state back to the ground state, with a pulse that generates a packet rather than a stationary state. The simplest way to do this currently seems to be to raise the power level of the pulsed pump laser [26]. This process is shown schematically in Fig. 7. 

Thus, in order to derive quantitatively meaningful wavepacket dynamics of molecular motion (and also quantitative stationary-state wave functions) from the effective Hamiltonian, it is necessary to carry out a further step in the analysis, which is highly nontrivial. We have carried out such analyses for a number of systems and I think an adequate understanding of the problem does exist nowadays and is accepted by the subcommunity most interested in this question. I like to illustrate this in the scheme from molecular spectra to molecular motion [5] (see Scheme 1). 

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-... 

Fig. 4.1. Schematic illustration of the evolution of a one-dimensional time-dependent wavepacket in the upper electronic state. The wavepacket is complex for t > 0 only its real part is shown here. Note that the upper horizontal axis does not correspond to a particular energyl The wavepacket is a superposition of stationary states corresponding to a broad range of energies, which are all simultaneously excited by the infinitely short light pulse indicated by the vertical arrow.

The time-independent and the time-dependent approaches are completely equivalent. Equation (4.11) documents this correspondence in the clearest way the time-dependent wavepacket f(t), which contains the stationary states for all energies, and the time-independent wavefunction tot(Ef), which embraces the entire history of the fragmentation process, are related to each other by a Fourier transformation between the time and the energy domains,... 

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... 

This shows that the time evolution is exactly like that of a stationary state of a fime-independenf Hamiltonian, provided the probing is limited to T, or any multiple of T. Since, (0)) is equal fo 4> (0)), Eq. (27) also shows fhaf exp ( - iEiT/h) is an eigenvalue of fhe evolution operator over one period of the field. Suppose now thaf we wish to follow fhe developmenf in fime of an arbitrary initial wavepacket rj 0). We can expand it over the complete set of Floquet eigenfunctions of a given Brillouin zone af fime f = 0 ... 

There are two classes in applications of quantum nuclear dynamics one is the stationary-state scattering theory to treat reactive scattering (chemical reactions), and the other is time-dependent wavepacket method. Here... 

As illustrated above, nonadiabatic dynamics exhibits vividly how electrons move in and between molecules. Complex natural orbitals, in particular SONO in the present case, clearly illustrate how the electronic wavefunction evolves in time. In addition to the time scale, the driving mechanism for the electron migration has also been illustrated. By clarifying such complex electron behavior, not available using stationary-state quantum chemistry, our understanding of realistic chemical reactions is greatly enhanced. As a result, it has clearly been shown that nonadiabatic electron wavepacket theory is invaluable in the analysis of non-rigid and mobile electronic states of molecular systems. 

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]).

In a time-dependent picture, resonances can be viewed as localized wavepackets composed of a superposition of continuum wavefimctions, which qualitatively resemble bound states for a period of time. The unimolecular reactant in a resonance state moves within the potential energy well for a considerable period of time, leaving it only when a fairly long time interval r has elapsed r may be called the lifetime of the almost stationary resonance state. 

One expects the timescale of the nonadiabatic transition to broaden for a stationary initial state, where the nuclear wavepacket will be less localized. To mimic the case of a stationary initial state, we have averaged the results of 25 nonstationary initial conditions and the resulting ground-state population is shown as the dashed line in Fig. 8. The expected broadening is seen, but the nonadiabatic events are still close to the impulsive limit. Additional averaging of the results would further smooth the dashed line. 

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. 

The wavepacket /(t), on the other hand, is constructed in a completely different way. In view of (4.4), the initial state multiplied by the transition dipole function is instantaneously promoted to the excited electronic state. It can be regarded as the state created by an infinitely short light pulse. This picture is essentially classical (Franck principle) the electronic excitation induced by the external field does not change the coordinate and the momentum distributions of the parent molecule. As a consequence of the instantaneous excitation process, the wavepacket /(t) contains the stationary wavefunctions for all energies Ef, weighted by the amplitudes t(Ef,n) [see Equations (4.3) and (4.5)]. When the wavepacket attains the excited state, it immediately begins to move under the influence of the intramolecular forces. The time dependence of the excitation of the molecule due to the external perturbation and the evolution of the nuclear wavepacket /(t) on the excited-state PES must not be confused (Rama Krishna and Coalson 1988 Williams and Imre 1988a,b)... 

The great asset of the time-dependent picture rests on the fact that no reference is made to the (many) stationary continuum states in the excited electronic state. The propagation of a single wavepacket, which contains the entire history of the dynamics in the upper state, thus yields the absorption cross section, all Raman cross sections, and the final state distributions of the photofragments. 

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... 

The situation is much more complicated for a vibrational wavepacket launched from a r" 0 initial eigenstate (see Fig. 9.7). However, a simple picture based on the classical Franck-Condon (stationary phase) principle (see Section 5.1.1) captures the essential details of the wavepacket produced at to on the electronically excited potential surface. First, there is the limiting case of an excitation pulse sufficiently short that an exact replica of the electronic ground state vibrational eigenstate, (R v" / 0), is created at to on the excited potential surface,... 

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