Nonadiabatic coupling elements

Now, recalling the definition of Equation 4.13 we can rewrite the nonadiabatic coupling element between adiabatic states K and L along the nuclear coordinate s ... 

The vibrationally adiabatic approximation is coordinate dependent. However one may formulate the quantal adiabatic approximation using the coordinate system defined by pods. One can then show that up to terms of order that if at Uq there exists a pods v/ith action (n+l/2)h, and energy (0 ) then also quantally Uq is an adiabatic barrier or well of the n-th vibrationally adiabatic potential energy surface.Furthermore, the quantal adiabatic frequency for motion perpendicular to the pods is excellently approximated by the adiabatic frequency of the pods. Finally, one can show, that to order fi, all quantal nonadiabatic coupling elements are identically zero at Uq In other words, one should expect that just as in the classical case, the coordinate system defined by the pods is also quantally, the optimal coordinate system for the vibrationally adiabatic approximation. One should also expect that the semiclassical barrier and well energies and frequencies computed via the pods are an excellent approximation to the quantal energies and frequencies., ... 

Since we have emphasized second-order perturbation theory in this section, it might be useful to point out that the effect of the Bi j i(s) nonadiabatic coupling elements could also be included by second-order perturbationtheory, using a procedure analogous to that of Barton and Howard.87... 

To handle the complex reactive process, we first focus on the dynamics on the Si surface to study how the system evolves towards the conical intersections. Therefore we introduce a reduced set of reactive coordinates, develop the corresponding Hamiltonian and study the time evolution of the system by means of wavepacket propagations on the calculated ah initio potential reaction surface. In the following steps, we include the nonadiabatic coupling elements as well as the laser-molecule interaction to describe the complete relaxation process. The final aim is to drive the reaction systematically through either one or the other of the two conical intersections and thus to influence the resulting product distribution. 

In a next step we now want to simulate the complete transfer to the ground state, for which both Si and So potential surfaces together with their nonadiabatic coupling elements are needed. 

Thus, we are forced to stick to the adiabatic representation, which raises other problems. As the complete nuclear Schrodinger equation is solved for both coupled states, all quantum effects like interferences or phase effects are included (see Sec. 7), but one needs to keep track of the phases of the electronic wavehmctions while computing the nonadiabatic coupling elements (NAC). Additionally, we are faced with the strong localization of the NACs, which requires many grid points for the wavepacket propagation and makes the calculations quite time consuming. 

Fig. 4. Nonadiabatic coupling elements / 2(top) and / (bottom) in the vicinity of Coinmin- The plots extend over Ar = 0.02 A, A95 = 0.42°.

Thereby it became evident that the description of such a reactive system absolutely needs ab initio ingredients such as reaction surfaces and nonadiabatic coupling elements to set a realistic stage for the subsequent wavepacket propagation. Furthermore, it was necessary to adapt the adiabatic approach for the coupled dynamics in order to describe simultaneously the transfer through multiple conical intersections. On this basis it was possible to obtain a microscopic understanding of the relaxation process. 

R 3 Bohrs, which is pretty far from the avoided crossing region. The asymmetric spatial distribution of the nonadiabatic coupling element Xap having the peak aroimd R 3 Bohrs accounts for this deviation. However, the truly remarkable featme of (A A ) is that the crossing takes... 

We next study the branching in the three-state model. The successive bifurcation of the paths has been made with the following automatic procedure. We first define a threshold value of nonadiabatic coupling element D,... 

Fig. 6.13 Potential energy curves versus the coordinate R. (a) Adiabatic representation. The solid curves denote APEC Cj, while the dashed curve does nonadiabatic coupling element Xi2. (b) Diabatic representation. The solid curves represent the DPEC (ascending to the right Vii, descending V22) and the dashed curve shows the electronic coupling element Vi2- (Reprinted with permission from K. Yamamoto et al, J. Chem. Phys. 140, 124111 (2014)).

This system was artificially modeled based on the ab initio potential curves of LiH molecule. [493] The spatial distribution of the nonadiabatic coupling element is superposed on the two potential curves. In this system,... 

We will now briefly review some of the methods used to calculate the nonadiabatic coupling elements. As can be easily siumised, the computational scheme of nonadiabatic coupling elements depends heavily on the methods of electronic state calculation such as configuration interaction, multi configurational self consistent field (MCSCF) method, and so on. 

The nonadiabatic coupling element was found to be sharply peaked in the vicinity of the avoided crossing between the two A adiabatic surfaces. These results indicate that nonadiabatic... 

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