Curve crossing diabatic

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. 

Figure 1. Diabatic potential energy curves for Nal with an expanded view of the adiabatic potential curves, e, and Ej, near the diabatic curve crossing.

FIGURE 12. A plot of the (n P)" versus E for the curve crossing in IC1. P is the probability for passing the crossing diabatically and E is the excitation energy. The dots are the experimental points and the solid line is the least-squares fit to these points. The dashed and dotted lines were taken from references (215) and (216), respectively. The figure was taken from reference (201) with permission of Elsevier Science Publishers. 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

Formulated in another way, the metal (M)-level is suddenly pulled down from well above to well below the ligand (L) valence levels a distance larger than the exchange splitting 2H]2 as a consequence, the system cannot follow adiabatically but undergoes a diabatic (curve crossing) transition to an excited state of the ionic system. 

Fig. 14.4 A schematic display of a curve crossing reaction in the diabatic and adiabatic representations. See Section 2.4 for further details. (Note This is same as Fig. 2.3).

To return now to the semiclassical model of nonadiabatic behavior, one can describe reactions on the spin-state (diabatic) PESs as follows The system will move throughout phase space on the reactant PES until it reaches a point where the product PES has the same energy as the reactant one. At that point, it may either remain on the reactant PES or hop over onto the product one. The Landau-Zener formula for curve crossing in one-dimensional systems has often been used in a multidimensional context (10) as a useful approximation for the probability p with which this hop occurs, leaving (1 - p) oi the trajectories to continue on the initial PES (Fig. 1) ... 

In discussing the alternative theoretical approaches let us limit ourselves to those which have been applied directly to processes in which we are interested in this article, but first of all let us stress once more the importance of the work of Delos and Thorson (1972). They formulated a unified treatment of the two-state atomic potential curve crossing problem, reducing the two second-order coupled equations to a set of three first-order equations. Their formalism is valid in the diabatic as well as the adiabatic representation and also at distances of closest approach near Rc. Moreover the problem of the residual phase x(l) is solved implicitly. They were able to show that a solution of the three first-order classical trajectory equations is not sensitive to all details of the potentials and the coupling term, but to only one function which therefore can be used readily for modelling assumptions. The resulting equations should be solved numerically. Their method has been applied now to the problem of the elastic scattering of He+ + Ne (Bobbio et at., 1973) but unfortunately not yet to any ionization problem. 

Another PER exists that is based on an underlying electronic diabatic perspective (although aspects of the electronic coupling are included) the PER in the electronically adiabatic ET limit for a curve-crossing picture is [40]... 

The results obtained in our laboratory as well as by other experimentalists [3, 4] have inspired a considerable amount of theoretical work on this system [2, 5-8], Archirel and Levy [7] have calculated a set of potential energy surfaces for the states N2 (X) + Ar, N2(A) + Ar, and N2 + Ar+(2P) as well as the couplings between these surfaces using a novel computational technique. From their results they developed a set of diabatic vibronic potential energy curves, and they assumed that transitions could occur when two curves crossed. Cross sections were computed using either the Demkov or Landau-Zener formula, as appropriate, and good agreement was obtained with the experimental values in most cases. Nikitin et al. [8] have taken a somewhat similar approach to this system. They estimated the adiabatic vibronic interaction curves for this system, and they assumed that transitions... 

In Section 3.3 it will be shown that, to describe perturbations which result from neglected terms in the Hel + TN (R) part of the Hamiltonian, two different types of BO representations are useful. If a crossing (diabatic) potential curve representation is used, off-diagonal matrix elements of Hel appear between the states of this representation. If a noncrossing (adiabatic) potential curve representation is the starting point, the TN operator becomes responsible for perturbations. 

Figure 3.5 Diabatic and adiabatic potential curves. The diabatic curves (solid lines) cross at Rc and are defined by neglecting the part of Hel that causes the adiabatic curves (dotted lines) to avoid crossing by 2if at Rc-...

If the approximate deperturbed potential curves cross, they are diabatic curves. One can assume an interaction matrix element given by Eq. (3.3.5) and carry out a complete deperturbation. 

Nonadiabatic transitions play crucial roles in various fields of physics and chemistry [1, 2, 3 4. 5, 7, 8 9 10], and it is quite important to develop basic analytical theories so that we can understand fundamental mechanisms of various dynamics. The most fundamental models among them are the Landau-Zener type curve crossing and the Rosen-Zener-Demkov type non-curve-crossing. Furthermore, there is an interesting intermediate case in which two diabatic exponential potentials are... 

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