Coupled-channels equations

In matrix-vector fonn these coupled-channel equations are... 

The present derivation can easily be generalized to systems with an arbitrary number of internal degrees of freedom, and it leads to coupled channel equations identical with equation (A3.11.63). where the coupling temis (A3.11.62) are expressed as matrix elements of the interaction potential using states which depend on these internal degrees of... 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

The solution of the coupled-channel equations (5) leads to the total cross sections, i.e. for a transition from the initial target state i ( P(t - oc) = rtf) to any atomic state / when the collision is over... 

Using the orthogonality of the number states ]N) along with Eqs. (12.15), (12.33), and (12.34), we transform the Schrodinger equation [Eq. (12.24)] into at set of coupled differential equations, the so-called coupled-channels equations. ... 

An example for one of the first applications of the coupled-channel equations y with quantized fields for photodissociation problems is shown in Figure 12.1, where, the dissociation of the IBr molecule by a two-photon (visible + IR) process wasJS studied [388], The results of the calculations, shown in Figure 12.2, demonstrate 3 how the strong IR photon broadens the transition ( power broadening ) allowing th system to be dissociated even if the first photon is tuned substantially away from, 7 resonance. This illustrates how multiphoton transitions induced by strong fields [392] are less restricted insofar as they need not be very close to an intermediate T resonance, the situation described in Section 3.3. 

We here derive the coupled channel equations in Jacobi and hyperspherical coordinates. By using Eqs. (1) and (2). we can obtain the Hamiltonian in Jacobi coordinates for either channel as... 

An alternative mixed quantum-classical evolution scheme which does not suffer from this limitation is Surface Hopping [1]. This method, although based on heuristic arguments applied to the coupled channel equation more than on a rigorous classical limit, maintains a multiconfiguration picture of the system, and is able to describe complex non-adiabatic phenomena such as proton transfer in solution, or the branching into different product channels of photo-excited chemical reactions in clusters and condensed phase environments [1-7]. 

In order to integrate the coupled-channel equation (8) the time as well as the impact-parameter dependence of the matrix elements (equation (9)) have to be determined. For this purpose, the matrix elements V) (R(t)) are expanded in terms of the radial (7 ) and angular (K) parts of the internuclear vector R according to... 

With the matrix elements from equation (24) the coupled-channel equations are solved numerically in order to obtain the coefficients Oj after the collision (f —< oo). For instance, the probability of ionizing the target from the ground state to a continuum state of energy e, angular momentum I and projection m in a collision with impact parameter b is given by... 

The classification we are following is not strict. For example, coupled-channel equations appear also in computational modelling although other aspects are more important here. Furthermore, as the different approaches are refined, some of them should become equivalent. 

If this ansatz is introduced in equ. 2 one gets the coupled channel equations for the amplitudes ajj... 

D/f is a diagonal matrix. Wo have solved the coupled channel equations in Eq. (24) using the quasi-adiabatic log-derivative algorithm of Manolopoulos[29]. 

This results in a set of coupled hyperradial second-order ordinary differential equations (called coupled-channel equations) in the coefficients n"xlu(p p), which is cast as a second order differential equation in a matrix b7111 of the form... 

During the past few years we have observed an intensive development of many-channel approaches to the collision problem. In particular, the coupled-channels method is based on an expansion of the total wave fmiction in internal states of reactants and products and a numerical solution of the coupled-channels equations.This method was applied in the usual way to the atom-diatom reaction A + BC by MOR-TENSBN and GUCWA /86/, MILLER /102/, WOLKEN and KARPLUS /103/, and EL-KOWITZ and WYATT /101b/. Operator techniques based on the Lippmann-Schwinger equation (46.II) or on the transition operator (38 II) has also been used, for instance, by BAER and KIJORI /104/ The effective Hamiltonian approach( opacity and optical-potential models) and the statistical approach (phase space models, transition state models, information theory) provide other relatively simple ways for a solution of the collision problem in the framework of the many-channel method /89/<. 

In the next section (Sec. 2), we will develop the theory of the BCRLM. We discuss the solution of the coupled-channel equations in both natural collision coordinates " and hyperspherical coordinates. " Both coordinate systems are widely used to treat collinear reactive scattering processes. We will discuss the projection " of the hyperspherical equations on coordinate surfaces appropriate for applying scattering boundary conditions and review the definition of integral and differential scattering cross sections in this model. 

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