Curve crossing problems

Sawada S and Metiu H 1986 A multiple trajectory theory for curve crossing problems obtained by using a Gaussian wave packet representation of the nuclear motion J. Chem. Phys. 84 227-38... 

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... 

The advantages of this generalized TSH method can be summarized as follows (1) both types of transitions in the potential curve crossing problems. 

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. 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix... 

A. H. Zewail If the curve-crossing problem of I2 in hexane is similar to Nal, then it is not surprising that the wavepacket survives the crossing for up to -1 ps in Nal it is up to -10 ps. If the interaction with the solvent is weak, coherence persists, as is certainly the case for rhodopsin and others. 

To belabor this point, let us consider in more detail a simple case, Refs. [78, 79], where the bound states of the Coulomb potential, through successive switching of a short-range barrier potential, becomes associated with resonances in the continuum. The simplicity of the problem demonstrates that resonances have decisively bound state properties, yields insights into the curve-crossing problem, and displays the tolerance of Jordan blocks. The potential has the form... 

Broeckhove, J., Feyen, B., Lathouwers, L., Arickx, F., and Van Leuven, P. (1990). Quantum time evolution of vibrational states in curve-crossing problems, Chem. Phys. Lett. 174, 504-510. 

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. 

We see that PW86 includes both the correct uniform gas limit and the GEA by having the right s2 dependence. It still has curve-crossing problems, but not as severe as LM. It violates the Lieb-Oxford bound, Eq. (66), and the exact condition... 

The theory of nonadiabatic transition dates back to 1932, when the pioneering works for curve-crossing and noncrossing problems were published by Landau (13), Zener (14), and Stiickelberg (15) and by Rosen and Zener (28), respectively. Since then numerous papers by many authors have been devoted to these subjects, especially to curve-crossing problems (1,2,19,29-33). In this section a brief survey to around 1991 is made, which is informative and gives a good introduction to further developments described in subsequent sections. 

Landau discussed the curve-crossing problem by using the complex contour integral method (13,34). In general, the transition probability p in first-order perturbation theory is given by... 

In this section the mathematical procedure is briefly outlined for deriving the exact expressions of the reduced scattering matrices and of the Stokes constants in the linear curve-crossing problems and for devising new semiclassical approximations to them. 

The most accurate method for multilevel curve crossing problems is, of course, to solve the close-coupling differential equations numerically. This is not the subject here, however instead, we discuss the applications of the two-state semiclassical theory and the diagrammatic technique. With these tools we can deal with various problems such as inelastic scattering, elastic scattering with resonance, photon impact process, and perturbed bound state in a unified way. The overall scattering matrix 5, for instance, can be defined as... 

C. Zhu and H. Nakamura, Theory of nonadiabatic transition for general two-state curve crossing problems. I Nonadiabatic tunneling case, J. Chem. Phys. 101 10630 (1994). 

C. Zhu, H. Nakamura, N. Re, and V. Aquilanti, The two-state linear curve crossing problems revisited. I Analysis of Stokes phenomenon and expressions for scattering matrices, J. Chem. Phys. 97 1892 (1992). 

We refer to Chapter 4 for a detailed discussion on the definition and explicit construction of diabatic states. The diabatic representation is generally advantageous for the computational treatment of the nuclear dynamics if the adiabatic potential-energy surfaces exhibit degeneracies such as conical intersections. Moreover, the diabatic representation often reflects more clearly than the Born ppenheimer adiabatic representation the essential physics of curve crossing problems and is thus very useful for the construction of appropriate model Hamiltonians for polyatomic systems. 

It should be stressed that for multidimensional curve crossing problems the low-order Taylor expansions (8), (9) and (19) are justified only in the diabatic electronic representation. In the adiabatic representation, curve crossings generally lead to rapid variations of potential-energy functions and transition dipole moments, rendering a low-order Taylor expansion of these functions in terms of nuclear coordinates meaningless. 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix formulation yields a consistent treatment of electronic populations and coherences, and the momentum changes associated with an electronic transition can be directly derived from the formalism without the need of ad hoc assumptions. Employing a Monte-Carlo sampling scheme of local classical trajectories, however, we have to face two major complications, that is, the representation of nonlocal phase-space operators and the sampling problem caused by rapidly varying phases. At the present time, the... 

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