Coupled-channels method

A powerful way of achieving this goal uses the coupled-channels expansion, a method widely used in calculations of scattering cross sections [6]. In the context of quantized matter-radiation problems, the coupled-channels method amounts to expanding E, n, N ) in number states. Concentrating on the expansion in the /th mode, we write E, n, N ) as... 

A review is given on the application of the coupled-channel method for the calculation of the electronic energy loss of ions as well as ionization in matter. This first principle calculation, based on the solution of the time-dependent Schrodinger equation, has been apphed to evaluate the impact parameter and angular dependence of the electronic and nuclear energy losses of ions as well as the ionization due to high-power short laser pulses. The results are compared to experimental data as well as to other current theoretical models. 

The chapter is organized as follows. The principle of the coupled-channel method is reviewed in detail in Section 2. The results are discussed in connection to higher order terms in Section 3. The application to multiphoton ionization is described in Section 4. Comparisons with measurements are provided in Section 5. A simple model for the electronic energy loss is... 

Here we apply the coupled-channel method to calculate photo ionization of atomic hydrogen by short (femtosecond) laser pulses at high power densities (up to 5 X 10 " W/cm ). A classical electro-dynamical field approximates the laser/atom interaction, according to (in the Coulomb gauge)... 

A direct measurement of the electronic energy loss as a function of the impact parameter is a hard task to be performed from the experimental point of view and only a few experiments have been performed for fast light ions. Experiments in gas targets under single collision condition provide a more direct and precise comparison of the theoretical results with the experimental data. Here we compare the results of the coupled-channel method for collisions of protons with He as a function of the projectile scattering angle. 

Complex-Coordinate Coupled-Channel Methods for Predissociating Resonances in van der Waals Molecules... 

Coupled channel methods for colllnear quantum reactive calculations are sufficiently well developed that calculations can be performed routinely. Unfortunately, colllnear calculations cannot provide any Insight Into the angular distribution of reaction products, because the Impact parameter dependence of reaction probabilities Is undefined. On the other hand, the best approximate 3D methods for atom-molecule reactions are computationally very Intensive, and for this reason. It Is Impractical to use most 3D approximate methods to make a systematic study of the effects of potential surfaces on resonances, and therefore the effects of surfaces on reactive angular distributions. For this reason, we have become Interested In an approximate model of reaction dynamics which was proposed many years ago by Child (24), Connor and Child (25), and Wyatt (26). They proposed the Rotating Linear Model (RLM), which Is In some sense a 3D theory of reactions, because the line upon which reaction occurs Is allowed to tumble freely In space. A full three-dimensional theory would treat motion of the six coordinates (In the center of mass) associated with the two... 

The coupled-channels method may be developed within the language of wave-mechanics, or more formally (and more compactly) by means of operator equations. The common feature of both approaches is that the total scattering state is expanded in internal states of reactants and products. The nature of the colliding particles and the quantum numbers of the interna] states define the reaction channel index c = a, b,. We begin with the wave-mechanical approach, some of whose features have been presented in the section on statistical theories. For the total wavefunction [pa of reactants in channel a, with relative wave vector ka, we can write... 

Elementary substitution reactions of type I + R2R3 -> R1R2 + R3. with Rk a molecular group, have been described in the context of the coupled-channel method by Brodsky and Levich (1973). These authors introduced distortion potentials for reactants and products and a parametrized, isotropic potential coupling. In practice, transition amplitudes were calculated... 

We would like to complete this section by briefly describing some of the recent developments on electronically non-adiabatic reactions. From the standpoint of the coupled-channels method, there is in principle no added difficulty in treating more than one electronic state of the reactive system. This may be done, for example, by keeping electronic degrees of freedom in the Hamiltonian and expanding the total scattering wavefunction in the electronic states of reactants and products. In practice, however, some new difficulties may arise, such as non-orthogonality of vibrational states on different electronic potential surfaces. There is at present a lack of quantum mechanical results on this problem. 

To develop coupled-channel methods to solve the Schrodinger equation, we first transform the Hamiltonian (A3.11.81) to hyperspherical coordinates, yielding ... 

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/<. 

There are two general approaches to the calculation of bound states of multichannel systems coupled-channel methods and radial basis set methods. 

Hutson, J.M., Coupled-channel methods for solving the hound-state Schrodinger equation, Comput. Phys. Commun., 84, 1-18,1994. 

Cold alkali metal atoms have a variety of magnetically tunable resonances that have been exploited in a number of experiments to control the properties of ultracold quantum gases or to make cold molecules. For the most part, experiments have succeeded with species that either do not have inelastic loss channels, or, if they do, the loss rates are very small. Thus, for practical purposes, we can set the resonance decay rate yc = 0 in examining a wide class of magnetically tunable resonances. While general coupled channel methods can be setup to solve the multichannel Schrodinger equation [1], we will use simpler models to explain the basic features of tunable Feshbach resonance states. 

The stationary wavefunction of an atom pair, 4 (r, ), can be determined using the coupled-channels method [29,31,55]. To this end, 4 (r, ) is expanded in terms of basis-set components lra(r, ) associated with the channel states defined in Equation 11.9. Using the radial wavefunctions. 

As in the collinear case, several approaches are possible. Around 1976, three coupled-channel methods had been used in cross-section calculations for 3-PD systems. One of them, developed by Elkowitz and Wyatt [50, 51], used natural collision coordinates (NCC) and local hindered asymmetric-top-vibrator basis sets [41]. Another, developed by Kuppermann and Schatz [106], used asymptotic free rotor and local vibrator basis sets, and different coordinates in different regions of configuration space, similar to those described... 

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