Electronic curve crossing

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

Lao, K.Q., Person, M.D., Xayariboun, P., and Butler, L.J. (1989). Evolution of molecular dissociation through an electronic curve crossing Polarized emission spectroscopy of CH3I at 266 nm, J. Chem. Phys. 92, 823-841. 

The wave packet is created by the pump laser at the inner turning point of the A state (first snapshot in Fig. 3.17). Energetically the wave packet is located about 0.1 eV above the electronic curve crossing. The ISC occurs for the first time at about 250fs (second snapshot in Fig. 3.17), when the center... 

State in the reverse direction. These results are the first dynamical experiments to extract the symmetries of the two electronic potentials involved in the curve crossing, and they demonstrate a powerful new way to investigate electronic curve crossing phenomena in general. 

RGENSEN - 1) Simulation of the step from t-BuCl to the contact ion pair is difficult because it corresponds to an electronic curve crossing in the gas phase. The ground state in the gas phase dissociates to t-Bu + CF, while a highly excited state correlates with the ion pair. The latter surface is, of course, preferentially stabilized by hydration. To model the initial step would consequently require results for both gas-phase surfaces and potential functions for water interacting with the substrate at all points along those surfaces. 

This situation arises when the electronic wave function of the transition state is described by the out-of-phase combination of the two base functions. If the electronic wave function of the transition state is described by the in-phase coinbination. no curve crossing occurs. 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

Let us finally discuss to what extent the MFT method is able to (i) obey the principle of microreversibility, (ii) account for the electronic phase coherence, and (iii) correctly describe the vibrational motion on coupled potential-energy surfaces. It is a well-known flaw of the MFT method to violate quantum microreversibility. This basic problem is most easily rationalized in the case of a scattering reaction occurring in a two-state curve-crossing system, where the initial and final state of the scattered particle may be characterized by the momenta p, and pf, respectively. We wish to calculate the probability Pi 2 that... 

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. 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

Fig. 2. Velocity dependence of the electronic stopping cross section for protons in propane. (—) prediction by the OLPA-FSGO treatment [42]. ( ) Oddershede and Sabin calculations [29]. (----) best fit curve to experimental data from Refs. [44,45].

The forbidden retro-[ls -I- 2s]-cycloaddition can now be treated using a simple curve-crossing model analagous to the Marcus-Hush theory of electron-transfer [11]. The ground state at the quadricyclane-like geometry is the... 

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