Wavepacket nuclear

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

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. 

Finally, Gaussian wavepacket methods are described in which the nuclear wavepacket is described by one or more Gaussian functions. Again the equations of motion to be solved have the fomi of classical trajectories in phase space. Now, however, each trajectory has a quantum character due to its spread in coordinate space. 

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 fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

In Section III.A, it was shown that the nuclear wavepacket can be represented by a packet associated with each electronic state, %i. Each of these packets can... 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Takeuchi T, Tahara T (2005) Coherent nuclear wavepacket motions in ultrafast excited-state intramolecular proton transfer sub-30-fs resolved pump-probe absorption spectroscopy of 10-hydroxybenzo[h]quinoline in solution. J Phys Chem A 109 10199-10207... 

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. 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

Under even more intense photoexcitation ( 10mJ/cm2), the coherent A g and Eg phonons of Bi and Sb exhibit a collapse-revival in their amplitudes (Fig. 2.10) [42,43], This phenomenon has a clear threshold in the pump density, which is common for the two phonon modes but depends on temperature and the crystal (Bi or Sb). At first glance, the amplitude collapse-revival appears to be analogous to the fractional revival in nuclear wavepackets in molecules [44,45]. However, the pump power dependence may be an indication of a polarization, not quantum, beating between different spatial components of the coherent response within the laser spot [46],... 

By making use of classical or quantum-mechanical interferences, one can use light to control the temporal evolution of nuclear wavepackets in crystals. An appropriately timed sequence of femtosecond light pulses can selectively excite a vibrational mode. The ultimate goal of such optical control is to prepare an extremely nonequilibrium vibrational state in crystals and to drive it into a novel structural and electromagnetic phase. 

In this section, by applying the heterodyne interferometry to a mixed gas of H2 and D2 molecules, we probe attosecond dynamics of nuclear wavepackets in the molecules. We find that not only the single molecule responses but also the propagation effects of harmonics differ between the two isotopes and that to discuss dynamics of molecules in the single molecule responses, the propagation effects need to be excluded from the raw harmonic signals. The measured relative phase as well as intensity ratio are found to be monotonic functions of the harmonic order and are successfully reproduced by applying... 

R. de Vivie-Riedle and J. Manz Prof. Neumark s question about detecting the hole burning in the nuclear wavepacket of the electronic ground state is very stimulating. In this context, we have developed a scheme for detecting the hole in the wavepacket by a femtosecond chemistry laser experiment that involves two laser pulses Our explanation will be for the specific system K2, but more general applications for other systems are obvious ... 

In this chapter Prof. Rice has advocated two techniques that should be useful for evaluations of optimal fields for laser control of chemical reactions (i) reduced space of eigenstates for representations of nuclear wavepackets and (ii) the use of effective reaction coordinates. Both techniques have already been used for efficient evaluations of reaction probabilities in model reactions. See, for example, Ref. 1 for the prediction of population inversion and Ref. 2 for the demonstration of rather strong deviations of chemical reactions from the reaction path, specifically in the case of hydrogen transfer reactions. 

J. Manz I would like to suggest that one should go even beyond Prof. R. D. Levine s suggestion to put the nuclear wavepacket on spe-... 

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)... 

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