The coupled-channel method

Here we will focus the attention on atomic treatments of the energy-transfer process. Thus, we will not consider solid-state effects such as intra-band transitions, collective excitations (bulk and surface plasmons) and the corresponding dynamic projectile screening. 

The theoretical formulation of atomic excitation and ionization processes is conveniently discussed by introducing the quantum-mechanical Hamilton operator. For a three-body system the Hamiltonian reads 

In the subsequent treatment the electron coordinate will be measured from the accelerated target nucleus and is the only dynamical variable. Thus the target system is the frame of reference [31,32], In such a noninertial system non-Newtonian forces arise. The corresponding Hamiltonian is 

It is reasonable to neglect the last term Vrecoii- By doing this transitions are excluded which are due to the interaction of the active electron with the recoiling target nucleus. This so-called recoil effect leads to insignificant contributions to total cross sections, but may be important for very close collisions (b 10 a.u.) [33]. Before the solution of equation (3) is 

With this Hamiltonian the classical equations of motion are solved. The last term in equation (7) was neglected because of its small influence on the motion of the target core in case of a strongly target-centered wave function 

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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. 

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. 

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/<. 

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. 

A little more complicated system is the de-excitation of He(2 P) by Ne, where the deexcitation is dominated by the excitation transfer and only a minor contribution from the Penning ionization is involved. The experimental cross section obtained by the pulse radiolysis method, together with the numerical calculation for the coupled-channel radial Schrodinger equation, has clearly provided the major contribution of the following excitation transfer processes to the absolute de-excitation cross sections [151] (Fig. 15) ... 

This remark is associated with the amount of calculation performed and is not intended as a criticism. This work provides a valuable quantum mechanical analysis of a three-dimensional system. The artificial channel method (19,60) was employed to solve the coupled equations that arise in the fully quantum approach. A progression of resonances in the absorption cross-section was obtained. The appearance of these resonances provides an explanation of the origin of the diffuse bands found... 

I. Bray, D.A. Konovalov, I.E. McCarthy, Convergence of an L2 approach in the coupled-channel optical-potential method for e-H scattering, Phys. Rev. A 43 (1991) 1301. 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics... 

The weak-coupling approximation (7.132,7.140) can be verified within the context of the coupled-channels-optical method. Equns. (7.123) may be solved with a particular channel, defined by the target state i), included in either P space or Q space. If it is in P space the channel i is fully coupled. The approximation is verified if the two solutions agree. In practice the lowest dipole-excited channels should be included in P space with the experimentally-observed channels, but the approximation is closely verified for higher channels in Q space. However, computation of (7.123) is not difficult and it is common to include all discrete channels in P space that are necessary for convergence. 

The comparison of theory and experiment in table 8.3 is somewhat unsatisfactory. The coupled-channels-optical and pseudostate calculations agree with each other and with the convergent-close-coupling calculation within a few percent, yet there are noticeable discrepancies with the experimental estimates. The convergent-close-coupling method calculates total ionisation cross sections in complete agreement with the measurements... 

Fig. 8.8 shows that the coupled-channels-optical method with the equivalent-local polarisation potential (McCarthy and Shang, 1992) gives a good semiquantitative description of the experimental data of Williams (1976 ) for elastic differential cross sections below the n—2 threshold. At energies just below the n=3 threshold the resonances affect the n=2 excitations. Fig. 8.9 shows the energy dependence of the integrated cross sections for the 2s and 2p channels. Since a resonance is a property of the compound system, not the channel, the resonances observed in... 

The equivalent-local form of the coupled-channels-optical method does not give a satisfactory description of the excitation of triplet states (Brun-ger et al, 1990). Here only the exchange part of the polarisation potential contributes. The equivalent-local approximation to this is not sufficiently accurate. It is necessary to check the overall validity of the treatment of the complete target space by comparing calculated total cross sections with experiment. This is done in table 8.8. The experiments of Nickel et al. (1985) were done by a beam-transmission technique (section 2.1.3). The calculation overestimates total cross sections by about 20%, due to an overestimate of the total ionisation cross section. However, an error of this magnitude in the (second-order) polarisation potential does not invalidate the coupled-channels-optical calculation for low-lying discrete channels. 

The example of magnesium at Eq = 40 eV illustrates the application of the coupled-channels-optical method to a two-electron atom with a core. It... 

Also shown in fig. 8.13 are comparisons of the differential cross sections of the coupled-channels-optical method with the experimental values of... 

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]. 

The coupled-channel calculations are used as benchmark results to check simple models of the impact parameter dependence of the electronic energy loss. A detailed description of such models (convolution approximation) may be found elsewhere [25,26]. Here we present only a short outline of the method. The electronic energy loss involves a sum over all final target states for each impact parameter. Usually this demands a computational effort that precludes its direct calculation in... 

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 final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

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