Wavepacket simulations

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

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. .. 

To return to the simple picture of vertical excitation, the question remains as to how a wavepacket can be simulated using classical trajectories A classical ensemble can be specified by its distribution in phase space, Pd(p,Q), which gives the probability of finding the system of particles with momentum p and position q. In conUast, it is strictly impossible to assign simultaneously a position and momentum to a quantum particle. 

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 ... 

The question is then, first, how often has such a complete match between experiment and theoretical simulation been achieved Second, are there good examples where complete simulations have been carried out but lead to two or more equally acceptable models to interpret the experimental results I refer to this question of ambiguity also in relation to a very similar problem arising in the interpretation of nontime-resolved high-resolution spectroscopy data [1,2], which provided in fact, the first experimental results on nontrivial three-dimensional wavepacket motion on the femtosecond time scale [3]. 

The wavepacket calculation for the femtosecond pump-probe experiment presented in Fig. 16 (bottom) is the result of the first consistent ab initio treatment for three coupled potential-energy surfaces in the complete three-dimensional vibrational space of the Naa molecule. In order to simulate the experimental femtosecond ion signal, the experimental pulse parameters were used duration A/fWhm = 120 fs, intensity I - 520 MW/cm2, and central... 

Figure 2. Franck-Condon windows lVpc(Gi, r, v5) for the Na3(X) - N83(B) and for the Na3(B) Na3+ (X) + e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(B), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx r cos

The use of wavepacket spectroscopy to follow the solvent-induced dissociation of iodine in solution has been described in detail by Scherer, Jonas, and co-workers [18, 28, 30]. Recently the role of the solvent in inducing the curve crossing has been examined by simulation [29], Remarkably, the experiments show that the wavepacket survives the solvent-induced curve crossing and appears intact (i.e., the atoms are separating ballistically) up to at least 4 A separation [28], The simulations imply that destruction of the wavepacket by the solvent cage (polarizable Ar atoms in this case) occurs between I-I separation of 5-6 A [29]. 

Thus simulation and experiment provide a consistent picture of the dissociation The wavepacket, whose motion is influenced by both the intramolec-... 

In order to check our imaging procedure we have to first stimulate the fluorescence emitted by excited polarized (and unpolarized) Na2 wavepackets. In these simulation we assume that the molecule, which exists initially in a (Xvg,jg) Na2 (X1 5 ) vib-rotational state, is excited by a pulse to a superposition of (xs) vib-rotational states belonging to the Na2(B IIu) electronic-states. 

Fig. 1.9a—d. The pi p>2 correlation maps obtained with a pump pulse only (a), and with At = 300 fs (b), 600 fs (c) and 1 ps (d). Red solid lines indicate results from the free-rotor simulation for the sequential process via the formation of metastable CS2+. The component corresponding to the nuclear wavepacket of CS2+ dissociating towards the symmetric stretching coordinate is visible along the diagonal Pi = P2 line (orange broken circle)... 

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