Nuclear wave packets

Coherent optical phonons can couple with localized excitations such as excitons and defect centers. For example, strong exciton-phonon coupling was demonstrated for lead phtalocyanine (PbPc) [79] and Cul [80] as an intense enhancement of the coherent phonon amplitude at the excitonic resonances. In alkali halides [81-83], nuclear wave-packets localized near F centers were observed as periodic modulations of the luminescence spectra. 

We denote the vibrational wave packet associated with electronic state i by (7,0 and fi2l is the transition dipole moment. Initially the system is in the vibrational ground state on Vt and treating the interaction with the field E[ v (t) within first-order perturbation theory gives the following expression for the nuclear wave packet on Vi-... 

The CICD rate as a function of the Kr-Kr distance was used in the nuclear wave packet simulation of the process within the Born-Oppenheimer picture... 

To monitor fast events in time, firstly, a precise zero of time must be established. To that end, a femtosecond pulse is used to excite the system and initiate the dynamics. This pulse is called a pump pulse, and a nuclear wave packet is created as described... 

Recent progress in laser technology has led to the widespread use of ultrafast lasers with pulse widths shorter than the vibrational periods of most chemical bonds. A localized state, called a nuclear wave packet, is created on a potential surface by exciting a molecule with ultrashort pulses of radiation. The time-evolution of such wave packets can be directly utilized to observe the transition states of chemical reactions. This development is one of the major accomplishments of femtosecond chemistry. 

Although we interpreted our experimental results with the motion of nuclear packets on the light-dressed ground state of the parent ions, a pump-probe scheme could also explain the pulse width dependence. When two successive pulses are interacted with molecules with a proper interval, the second pulse can be adjusted to achieve synchronicity with the motion of the nuclear wave packet of an excited state. This results in energy transitions to... 

One has to emphasize that Eqs. (82) and (83) do not involve the Born-Oppenheimer approximation although the nuclear motion is treated classically. This is an important advantage over the quantum molecular dynamics approach [47-54] where the nuclear Newton equations (82) are solved simultaneously with a set of ground-state KS equations at the instantaneous nuclear positions. In spite of the obvious numerical advantages one has to keep in mind that the classical treatment of nuclear motion is justified only if the probability densities (R, t) remain narrow distributions during the whole process considered. The splitting of the nuclear wave packet found, e.g., in pump-probe experiments [55-58] cannot be properly accounted for by treating the nuclear motion classically. In this case, one has to face the complete system (67-72) of coupled TDKS equations for electrons and nuclei. 

In the many-dimensional case, the situation is more complicated (Figure 6.5). The arrival of a nuclear wave packet into a region of an unavoided or weakly avoided conical intersection (Section 4.1.2) still means that the jump to the lower surface will occur with high probability upon first passage. However, the probability will not be quite 100%. This can be easily understood qualitatively, since the entire wave packet cannot squeeze into the lip of the cone when viewed in the two-dimensional branching space, and some of it is forced to experience a path along a weakly avoided rather than an unavoided crossing, even in the case of a true conical intersection. 

The system is to be prepared in a coherently pseudorotating nuclear wave packet in the ground electronic state. Specifically, the wave packet is a moving displaced vibrational ground state of the two-dimensional oscillator with the form... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

We wish to describe the preparation of a nuclear wave packet corresponding to coherent molecular pseudorotation in the electronic ground state by a sequence of nonresonant light pulses [30], The system is driven by a pair of vibrationally abrupt pulses... 

Fig. 1 Potential energy curves and sketch of the relevant nuclear wave-packet motion of Nal. After an ultrashort excitation, Nal is vertically excited from the X1X+ ground state to the A1X+ first excited state, which exhibits a shallow minimum. The nuclei then vibrate on the excited state surface with bond distances ranging from approximately 3-12 A. Through the avoided crossing around 7 A, parts of the wave-packet can make a transition to the electronic ground state, and dissociate. Adapted from [21]...

Figure 9.10. Diabatic free energy curves illustrating (a) photoinduced ET reaction and (b) back ET reaction in a ID solvation coordinate system. A resonant optical pulse brings a stationary nuclear wave packet from the ground potential surface to the donor surface, where it relaxes toward equilibrium with concomitant ET to the acceptor state.

Nuclear Wave-Packet Simulation Within the Bom-Oppenheimer Approximation... 

Nuclear Wave-Packet Simulation for Nonadiabatic Dynamics... 

Interference Between Nuclear Wave Packets Through Nonadiabatic Coupling... 

The first approximation made in the Ehrenfest method is thus the factorisation of the total wavefunction into a product of electronic and nuclear parts. One deficiency of the ansatz (2) is the fact that the electronic wavefunction does not have the possibility to decohere the populated electronic states in P(r,t) share the same nuclear wave-packet x(R, t) by definition of the total wavefunction. Decoherence here is defined as the tendency of the time-evolved electronic wavefunction to behave as a statistical ensemble of electronic states rather than a coherent superposition of them [26]. The neglect of electronic decoherence could lead to non-physical asymptotic behaviors in case of bifurcating paths. It is not expected to be a problem here as we are interested in relatively short timescale dynamics. 

Note Uncertainties correspond to a 95% confidence interval. Columns a and correspond to two choices on the widths of the Gaussian function mimicking nuclear wave packets. 

As mentioned above, vibronic-coupling model Hamiltonians constructed by low-order Taylor expansions of the diabatic PE functions in terms of normal coordinates are particularly suitable for the calculation of low-resolution spectra of polyatomic molecules. In resonance Raman spectroscopy, for example, the usually extremely fast electronic dephasing in polyatomic s tems limits the time scale for the exploration of the excited-state PE surface by the nuclear wave packet to about 10 jjj... 

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