Close-coupling method

Solution of this set for F R) represents tire adiabatic close-coupling method. The adiabatic states are nomrally detennined (via standard computational teclmiques of quanUim chemistry) relative to a set of axes (X, Y, Z ) with the Z- axis directed along the nuclear separation R. On transfomring to this set which rotates during the collision, then /(r, / ), for the diatomic A-B case, satisfies... 

Alternatively, one can use close-coupling methods. These methods are easiest to understand for single arrangement problems (i.e. when both the AB + C and AC + B product arrangements are very high in energy so that only the A + BC reactant arrangement can be accessed). Then one writes... 

The fits of Janev et al. [12] stem from a compilation of the results obtained with different theoretical approaches (i) semi-classical close-coupling methods with a development of the wave function on atomic orbitals (Fritsch and Lin [16]), molecular orbitals (Green et al [17]), or both (Kimura and Lin [18], (ii) pure classical model - i.e. the Classical Trajectory Monte Carlo method (Olson and Schultz [19]) - and (iii) perturbative quantum approach (Belkic et al. [20]). In order to get precise fits, theoretical results accuracy was estimated according to many criteria, most important being the domain of validity of each technique. 

A. Igarashi, N. Toshima, Application of hyperspherical close-coupling method to antiproton collisions with muonic hydrogen, Eur. Phys. J. D 46 (2008) 425. 

Fig. 3.3. Measured (Barts and Halpern 1989) and calculated rotational state distributions following the photolysis of C1CN. The energy in the excited state is 2.133 eV. The quantal results are calculated with the close-coupling method described in Section 3.2 and the classical distribution has been obtained by Barts and Halpern using the ultrasimple model which we shall present in Section 6.3. Reprinted from Schinke (1990).

Fig. 5.4. Comparison of the quantum mechanical and the classical absorption spectra for H2O in the second continuum. The quanta result is calculated by means of the time-independent close-coupling method and the classical curve is obtained in a Monte Carlo simulation. Both cross sections are normalized to the same area. The arrow indicates the threshold for H + OH(2E). Reproduced from Weide and Schinke (1989).

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

I. Bray, Convergent close-coupling method for calculation of electron scattering on hydrogenlike targets, Phys. Rev. A 49 (1994) 1066. 

The first exact quantum calculations of integral and differential cross sections on the adiabatic state were reported in 2001 by Honvault and Laimay [15,81]. They have carried out quantum reactive scattering calculations of the title reaction on the DK PES within the Time Independent Quantum Mechanical (HQM) framework using the hyperspherical close-coupling method. 

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. 10.14 shows that the convergent-close-coupling method describes the total ionisation cross section within experimental error for the whole energy range above total energy = 4 eV. Just above threshold it underestimates the cross section by up to 30%. 

This work shows, in a first approach, the possibility to extend the close-coupling method, well-adapted to the treatment of ion-atom collisions, to much more complex systems, in particular of biological interest. The model presented in the case of the Cq+-Uracil collision system, although very simple, could provide results compa-rable to experimental data. Such an approach, of course, does not take into account all the motions of the biomolecule in particular, the orientation of the biomolecule with respect to the angle of attack of the ion should be considered and orientation-average calculations be performed. Nevertheless, this first approach appears very encouraging and further developments are in progress. 

A study of reactive H + H2 collisions has also been carried out by Wu and Levine (1971) using the close-coupling method and natural collision coordinates. They employed a Porter-Karplus (1964) surface and investigated the collision energy range between 9 and 35 kcal/mole. For the interaction they used... 

Until recently, another problem with the so-called correlation quantum numbers was that they did not enable the most accurate calculations of doubly-excited spectra to be performed. With the work of Tang and Shi-mamura [330] on the hyperspherical close-coupling method, this situation has changed, and there is renewed interest in this approach since the situation is still evolving, we merely summarise developments in this area in section 7.11. 

Even more recently, the experimental resolution has been still further enhanced, and both partial cross sections and photoelectron angular distributions have been determined [332]. The data shown provide fine examples of interfering autoionising resonances (see chapter 8) and have been analysed in remarkable detail using the hyperspherical close-coupling method they represent a critical test of the dynamics of double excitation in He. 

The hyperspherical close-coupling method is well adapted to atoms with two external electrons, but should also be capable of treating heavier systems such as the alkaline earths, in which the singly- and doubly-excited spectra interact strongly. So far, not much progress has been accomplished in extending it to systems in which interactions with the valence- or inner-shell spectra become important. 

As is well-known, the two-electron system e -F H has been attracting theoretical attention for decades, with a large number of reports on the identification and nature of its resonances. Hence, it is possible to compare the CESE results of Ref. [123,124] with those published by other groups, who applied large scale R-matrix methods [130-132], or the CCR method [133-136], or specially improved close-coupling methods [137], or implementation of Feshbach s formalism. The related references are cited in Refs. [123,124] and below. 

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