Coupled-channel calculations

Let us briefly discuss the idea of and the results obtained from coupled-channel calculations with momentum eigenfunctions as it has been employed recently (Tenzer et al. 2000b). The collision is considered in a coordinate system, where the scattering ions have equal but opposite velocities v = voez. With respect to this system the total time-dependent Hamiltonian H is decomposed into the unperturbed (free) Dirac Hamiltonian 

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The numerical computations of the cross sections for molecular collisions in fields are demanding. Part of the complexity is due to the fact that most molecules of interest are open-shell radicals, so the basis for the coupled channel calculations must include multiple angular momentum states. In addition, the interactions of the molecules with an external field couple states with different total angular momenta. Note that in the presence of a single, axially symmetric field, the... 

Most of the recent literature on molecular collisions in external fields [1-3, 9, 10, 15, 16, 18, 19, 21, 23, 25, 27-84, 92] is focused on collisions of molecules at low and ultralow temperatures. As mentioned in the introduction, it is at temperatures < 10 K that strong electromagnetic fields are expected to have a noticeable effect on the scattering properties of molecules. The coupled-channel calculations... 

First of all, the theory presented is based on a few assumptions, which, while valid for the molecular systems considered in the literature so far, need to be care-fidly examined in every specific case. As mentioned in Section 8.3, we assume that the effects of external fields on the kinetic energy operator for the relative motion are negligible and that the interactions with electromagnetic fields are independent of the relative separation of the colliding particles. In addition, we ignore the nonadiabatic interactions that may be induced by external fields and that, at present, cannot be rigorously accounted for in the coupled channel calculations. 

Igarashi, A., Toshima, N. and Shirai, T. (1994). Hyperspherical coupled-channel calculation for antihydrogen formation in antiproton-positronium collisions. J. Phys. B At. Mol. Opt. Phys. 27 L497-L501. 

For more accurate estimation, complex-rotation calculations [80] and coupled-channel calculations [81, 82] with the correct relativistic and radiative splitting AE taken into account reveal that, for H Feshbach series below the H(n = 2) threshold, the number vmax of resonances for the 1Se series is 4 (and the same for the system e+H) [81], vmax = 3 for the 1P° series [80,82], and vmax = 4 for the 3P° series [82], The relativistic effects also mix different LS states. This affects the resonance positions only slightly, but the components... 

Clary, D.C. (1987). Coupled-channel calculations on energy transfer, photochemistry, and reactions of polyatomic molecules, J. Phys. Chem. 91, 1718-1727. 

Relative differential cross sections for the 3s and 3p channels at several energies have been measured by different groups. These are shown in figs. 8.10 and 8.11 in comparison with a coupled-channels-optical calculation for which P space consists of the 3s, 3p and 3d channels and the polarisation potential treats all Q space channels to convergence. A 3s, 3p, 3d coupled-channels calculation has been included to assess the effect of Q space. 

The coupled-channels-optical calculation converges at 15 channels in P space with polarisation potentials for the continuum included for all couplings in the first six channels. The effect of the inclusion of the continuum is shown by the 15-state coupled-channels calculation. The distorted-wave... 

Essentially-complete agreement with experiment is achieved by the coupled-channels-optical calculation. We can therefore ask if scattering is so sensitive to the structure details in the calculation that it constitutes a sensitive probe for structure. The coupled-channels calculations in fig. 9.3 included the polarisation potential (5.82) in addition to the frozen-core Hartree—Fock potential. Fig. 9.4 shows that addition of the polarisation potential has a large effect on the elastic asymmetry at 1.6 eV, bringing it into agreement with experiment. However, in general the probe is not very sensitive to this level of detail. 

What have we learned from coupled-channel calculations Acknowledgements... 

The present coupled-channel calculations also allow for the inclusion of projectile-centered states according to following expansion... 

The coupled-channel calculations allow for accurate calculations of higher order effects. At high energies the electronic energy loss may be expanded in terms of the projectile charge Zp according to... 

The following restrictions have been found to the application of coupled-channel calculations for the computation of pulsed-laser ionization. The dipole approximation restricts the photon energy to < 1 keV in the current treatment. This, however, does not pose a strict condition since a partial-wave expansion of the laser field may be used, similar to as in the case of screened Coulomb potentials. In comparison to ion/atom collisions, typical photon/atom interaction times are extremely long. An upper limit of the pulse width AZp = 100 fs at intermediate laser-power densities follows from the numerically restricted density of continuum states. 

High power densities (3>10 " W/cm") and small laser frequencies (A > 600 run) are related to extremely high orders of perturhation theory. This requires basis sets extending to high values of (/ 3> 15) and high ejected-electron energies (e 20 eV). With the help pf P/Q space methods [60] the range of validity of coupled-channel calculations may he extended in this case. 

The first coupled-channel calculations for total and differential energy losses were performed for very simple systems such as H on H, He [11,12,61]. Later theses calculations have been extended to more complex systems such as the inner-shells of Al and Si [22,24]. Good agreement with experimental data has been found and the remaining discrepancies have been attributed to multielectron processes. 

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... 

WHAT HAVE WE LEARNED FROM COUPLED-CHANNEL CALCULATIONS... 

We have presented a sample of resonance phenomena and calculations in reactive and non-reactive three-body systems. In all cases a two-mathematical dimensional dynamical space was considered> leading to a great simplification in the computational effort. For the H-K 0 system, low-energy coupled-channel calculations are planned in the future to test the reliablity of the approximations used here, i.e., the scattering path hamiltonian as well as the distorted wave Born approximation. Hopefully these approximations will prove useful in larger systems where coupled-channel calculations would be prohibitively difficult to do. Such approximations will be necessary as resonance phenomena will continue to attract the attention of experimentalists and theorists for many years. 

The study of quantum effects such as resonances In atom-molecule reactions has been largely confined to coupled-channel calculations for collisions constrained to colllnear geometries. Progress In quantum reactive scattering techniques Is reviewed periodically (1-4). A few 3D quantum calculations of simple reactions, some more approximate (5-17) than others (18-19), have been concerned with resonance features In the reaction dynamics, and with the Increasing sophistication and sensitivity of molecular beam experiments (20-23), It has become evident that the angular distribution of reaction products Is likely to be the most sensitive observable manifestation of resonant contributions to reaction mechanisms. 

It should be noted in passing that for this non-reactive collinear system a completely quantum mechanical (i.e. coupled channel) calculation may actually be no more difficult—i.e. require no more computer time—than these semiclassical calculations. Whether this is true or not is beside the point, of course, for the obvious interest in the semiclassical model is that it can be applied to physically realistic three-dimensional systems (see Section IV.C) for which coupled channel calculations are usually unreasonable unless simplifying approximations are introduced. The purpose for carrying out semiclassical calculations for collinear systems is to obtain definitive comparisons with reliable quantum mechanical values (which exist only for collinear systems). 

Preliminary results of calculations such as these have been reported by Miller and Raczkowski75 for the 0 - 1 vibrational excitation of H2 and He. Calculations have also been made for the 1 - 0 vibrational de-activation of H2 by Li +, and comparison with the quantum mechanical coupled channel calculations of Schaefer and Lester77 are quite encouraging. Fig. 12 shows... 

In a recent theoretical study on the Li+ — H2 system by Schaffer and Lester,109 coupled channel calculations of integral cross sections for rotational and vibrational excitation of H2 at a centre of mass energy 12 eV... 

This system U-Pb from Fig. 6 is the collision system which will be used soon for an experiment at GSI where they will try to observe MO X-rays as well as innershell X-rays of the arget Pb atoms in the solid. We hope to be able to provide coupled channel calculations soon in order to calculate the X-ray production of the innermost levels of the Pb target as well as MO calculations to give a prediction on the MO X-ray structure for this experiment. First theoretical attempts in this direction had already been made many years ago by Kirsch et al. 

An example of a calculation, performed on an IBM RS/6000-320 workstation, is the study of the collisions of argon and NH3 by van der Sanden et al. [122] with the use of an ab initio calculated Ar-NH3 potential [123]. The program Hibridon [124] was used to compute the elastic and rotationally inelastic scattering cross sections and the probability that the collisions with Ar invert the ammonia umbrella. A single (one collisional energy) coupled channel calculation on para NH3 colliding with argon took 241 CPU hours and was finished in about 2 weeks. On a mainframe this would have been a matter of months. 

A completely adiabatic correlation between nQ> and ljQ> states would assume that n equals j a result which unlike the partial 0-correlation is not borne out by coupled channel calculations. Physically this adiabatic correlation is untenable because the spatial character of free-rotor and bending wave-functions is quite different and so non-adiabatic coupling is certain to be large. Thus the change between the free-rotor and bending wavefunctions is better described by a sudden correlation. 

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