Electronic states nuclear dynamics

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

The import of diabatic electronic states for dynamical treatments of conical intersecting BO potential energy surfaces is well acknowledged. This intersection is characterized by the non-existence of symmetry element determining its location in nuclear space [25]. This problem is absent in the GED approach. Because the symmetries of the cis and trans conformer are irreducible to each other, a regularization method without a correct reaction coordinate does not make sense. The slope at the (conic) intersection is well defined in the GED scheme. Observe, however, that for closed shell structures, the direct coupling of both states is zero. A configuration interaction is necessary to obtain an appropriate description in other words, correlation states such as diradical ones and the full excited BB state in the AA local minimum cannot be left out the scheme. 

In this way r is singled out as the fast electronic coordinate. In the absence of prior knowledge we can, equally well, designate R as the fast coordinate. This gives rise to an alternative adiabatic breakdown of the Schrodinger equation, and exemplifies the nonuniqueness associated with the adiabatic separation of bound-state nuclear dynamics. 

The photoelectron angular distributions from the B state of LiH (H-state) and all the others considered (X, A, C states having S symmetry) are qualitatively different from each other. Therefore future incorporation of geometry- and orientation-dependent photoionization matrix elements should enhance the understanding of information that may be obtained from such a study of the electron and nuclear dynamics. 

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. 

Therefore, we present here our semiclassical Field-Induced Surface Hopping (FISH) method [59] for the simulation and control of the laser-driven coupled electron-nuclear dynamics in complex molecular systems including all degrees of freedom. It is based on the combination of quantum electronic state population dynamics with classical nuclear dynamics carried out on the fly . The idea of the method is to propagate independent trajectories in the manifold of adiabatic electronic states and allow them to switch between the states under the influence of the laser field. The switching probabilities are calculated fully quantum mechanically. The application of our FISH method will be illustrated in Sect. 17.6 on the example of optimal dynamic discrimination (ODD) of two almost identical flavin molecules. 

The discussion in the previous sections assumed that the electron dynamics is adiabatic, i.e. the electronic wavefiinction follows the nuclear dynamics and at every nuclear configuration only the lowest energy (or more generally, for excited states, a single) electronic wavefiinction is relevant. This is the Bom-Oppenlieimer approxunation which allows the separation of nuclear and electronic coordinates in the Schrodinger equation. 

The simplest approach to simulating non-adiabatic dynamics is by surface hopping [175. 176]. In its simplest fomi, the approach is as follows. One carries out classical simulations of the nuclear motion on a specific adiabatic electronic state (ground or excited) and at any given instant checks whether the diabatic potential associated with that electronic state is mtersectmg the diabatic potential on another electronic state. If it is, then a decision is made as to whedier a jump to the other adiabatic electronic state should be perfomied. 

V is the derivative with respect to R.) We stress that in this formalism, I and J denote the complete adiabatic electronic state, and not a component thereof. Both /) and y) contain the nuclear coordinates, designated by R, as parameters. The above line integral was used and elaborated in calculations of nuclear dynamics on potential surfaces by several authors [273,283,288-301]. (For an extended discussion of this and related matters the reviews of Sidis [48] and Pacher et al. [49] are especially infonnative.)... 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

Knowledge of the underlying nuclear dynamics is essential for the classification and description of photochemical processes. For the study of complicated systems, molecular dynamics (MD) simulations are an essential tool, providing information on the channels open for decay or relaxation, the relative populations of these channels, and the timescales of system evolution. Simulations are particularly important in cases where the Bom-Oppenheimer (BO) approximation breaks down, and a system is able to evolve non-adiabatically, that is, in more than one electronic state. 

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... 

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