Time evolution electron nuclear dynamics

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... 

Following a description of femtosecond lasers, the remainder of this chapter concentrates on the nuclear dynamics of molecules exposed to ultrafast laser radiation rather than electronic effects, in order to try to understand how molecules fragment and collide on a femtosecond time scale. Of special interest in molecular physics are the critical, intermediate stages of the overall time evolution, where the rapidly changing forces within ephemeral molecular configurations govern the flow of energy and matter. 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

This level of theory outhned above is implemented in the ENDyne code [18]. The explicit time dependence of the electronic and nuclear dynamics permits illustrative animated representations of trajectories and of the evolution of molecular properties. These animations reveal reaction mechanisms and details of dynamics otherwise difficult to discern, making the approach particularly suitable for the study of the subtleties of contributions to the stopping cross section. 

The central question in liquid-phase chemistry is How do solvents affect the rate, mechanism and outcome of chemical reactions Understanding solvation dynamics (SD), i.e., the rate of solvent reorganization in response to a perturbation in solute-solvent interachons, is an essential step in answering this central question. SD is most often measured by monitoring the time-evolution in the Stokes shift in the fluorescence of a probe molecule. In this experiment, the solute-solvent interactions are perturbed by solute electronic excitation, Sq Si, which occurs essenhaUy instantaneously on the time scale relevant to nuclear motions. Large solvatochromic shifts are found whenever the Sq Si electroiuc... 

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 integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

In this chapter, we turn to problems of quantum chemistry and of many-electron atomic and molecular physics for which fhe desideratum is the quantitative knowledge and easy conceptual understanding of dynamical processes and phenomena thaf depend explicifly on time. We focus on a theoretical and computational approach which computes q>(q,t) by solving nonperturbatively the many-electron TDSE for unstable states of atoms and small molecules. The time evolution of fhese states is caused either by the time-independent Hamiltonian, Ham ( -g-/ case of time-resolved autoionization—see below) or by the time-dependent Hamiltonian, H t) = Ham + Vext(f), where Vext(f) is the sum of the identical one-electron operators that couple the field of a strong pulse of radiation to the electronic and nuclear moments of N-electron atomic or molecular states of inferest, thereby producing, during and at the end of the interaction, final stafes in the ionization or the dissociation continua. 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

In the present framework, all the dynamical information, including that of electrons, is formally contained in the nuclear wavefunctions Xn(R,t) and Xc,k R,t), the coefficients of the stationary electronic wavefunctions in Eq. (3.51). After numerically solving Eq. (3.69) for the time-evolution of the nuclear wavefunctions, x R t) Xkim R, t), we may obtain the total ion population at any time for any given pump probe delay time tpr,... 

At our most fundamental level of description we consider molecular systems to be composed of atomic nuclei and electrons, all obeying quantum mechanical laws. The question of the kinetics of a physicochemical event is therefore related to the time evolution of such composite systems. In the first sub-section we recall the basic quantum mechanical equation-of-motions relevant in this context. We then consider approximations that can be operated to simplify the nuclear-electronic dynamics, leading to the derivation of the mixed quantum-classical rate constant expression. 

The measured signal can be e.g. the total fluorescence from state 2) to an energetically lower electronic state or an ion signal, as discussed below. The idea behind this kind of experiments is that the pump/probe signal which is connected to the wave function 1 2) 2) depends on the time-evolution in the intermediate state 1), that is, on the nuclear dynamics described by the wave function 7pi t)). 

As the evolution of the electronic phase is much faster than the dynamics of the nuclei, the time step Af needs to be adjusted relative to the time step At of the nuclear dynamics. The time-dependent expectation value of the nuclear coordinates R(t) = irtot(R,t) R ktot(R,t)) is evaluated using the solution of Eq. (8.3). Subsequent quantum chemical calculations are performed at the nuclear geometries R(t) to obtain the electronic wavefunctions /,o(r R t))- The quantities ai(t) aj(t) xlfi(R,t) xlfj(R,t))R are also obtained from Eq. (8.3). The integrals over the electronic wavefunctions /[Pg.223]

Fig. 8.9 Summary of the coupled electron and nuclear dynamics during the dissociation. The black vertical line indicates the time of recollision, 1.7 fs after ionization at the maximum electric field, (a) Temporal evolution of the electric field, (b) Time-dependent populations of the E n, and iPn states of CO+ after recoUision excitation ( solid CEP = 0, dotted CEP = 7r). (c) Temporal evolution of the probability of measuring a C+ fragment Pc+ for the dissociative ionization of CO+ after recoUision (black CEP = 0, gray CEP = it). Reprinted from [66] with copyright permission of APS...

Due to the NAC, population is switched between the intersecting electronic states. Thereby, a superposition state and hence an electronic wavepacket is formed. In the vicinity of Coins, the time scales of the electron and nuclear dynamics are well synchronized. The energy difference between the coupled electronic states becomes very small slowing down the dynamics of the usually faster electrons to the time scale of the nuclear dynamics and below. The motion of the electronic density in the vicinity of a Coin is visualized in Fig. 8.14 for CoIn-1 of the cyclohexadiene/all-d -hexatriene system (Fig. 8.2). The underlying electronic wavepacket is created as the normalized superposition of the CASSCF-wavefunctions of ground and tirst excited state, keeping the nuclear geometry fixed. To describe the temporal evolution we take into account the time-dependent phase of both components. 

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