Interchannel couplings

If there are no interchannel couplings, R = 0, corresponding to the fact that the i channels have quantum defects of and occur when v, = 0. In general the R matrix is a real symmetric matrix with zeroes on the diagonal. 

From this starting point, the authors develop equations leading to the evaluation of the real symmetric K matrix to specify the scattering process on the repulsive surface and propose its determination by a variational method. Furthermore, they show explicitly the conditions under which their rigorous equations reduce to the half-collision approximation. A noteworthy result of their approach which results because of the exact treatment of interchannel coupling is that only on-the-energy-shell contributions appear in the partial linewidth. Half-collision partial linewidths are found not to be exact unless off-the-shell contributions are accidentally zero or (equivalently) unless the interchannel coupling is zero. The extension of the approach to indirect photodissociation has also been presented. The method has been applied to direct dissociation of HCN, DCN, and TCN and to predissociation of HCN and DCN (21b). 

C. A. Nicolaides, Th. Mercouris, Partial widths and interchannel coupling in autoionizing states in terms of complex eigenvalues and complex coordinates, Phys. Rev A 32 (1985) 3247. 

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

Here, dVjVi is the HF photoionization amplitude, F, (o>) is the effective interelectron interaction accounting for electron-electron correlation in the atom [55], i.e., the interchannel coupling matrix element, is the energy of a state v(, i]v = I for occupied states v in the atom, whereas i]v = 0 for vacant states v, the sum, when taken over continuum vacant states v, transforms into the integral over the energy ev of the vacant states, and f —> +0. 

Figure 18 Calculated [33] RPAE results for the Xe 5s photoionization cross section of Xe Cgo obtained in the A-potential model at the frozen-cage approximation level. (a) o 1" A iro), complete RPAE calculation accounting for interchannel coupling between photoionization transitions from the Xe 4d10, 5s2 and 5p6 subshells (b) 5 A ( >), the same as in (a) but with the 4d - f, p transitions being replaced by those of free Xe, for comparison purposes (c) o AA(

Figure 27 Relativistic RPAE calculated results [30] of the 6s dipole photoelectron angular distribution parameter j06s(eo) from free Hg and <3>Hg, The RRPA calculations included interchannel coupling...

Figure 28 Relativistic RPAE calculated results [30] of the 6s dipole photoelectron angular distribution parameter of Hg at two different levels of truncation with regard to RRPA interchannel coupling (a) including channels from the 6s2 subshell alone, Aa, and (b) including channels from the 6s2 and 5d10 subshells of d>Hg, as in Figure 27. Confinement effects were accounted for in the A-potential model at the frozen-cage approximation level.

Complex adiabatic energies (a) compared to resonance energies (b) for a model of a closed and an open channel described in reference (30). The zeroth order energy has been subtracted out. Unit cm , Interchannel coupling 200 cm . Crossing diabatic energy equal to the energy of the level with v = 18. 

Fig. 3.2. Plot of quantum defects for the bound states of two Rydberg series converging to different limits (a) for no interchannel coupling (b) for nonzero in ter channel coupling.

So far, we have discussed what is called empirical QDT, namely the fitting of experimental data using the finite number of adjustable parameters required to represent interchannel coupling. Unfortunately, while the procedure works well for simple cases, one eventually runs into two problems (1) for complex cases involving many parameters, it can turn out that more than one set is found which will represent the same experimental data equally well and (2) it may prove necessary to introduce energy-dependent parameters in order to represent observations. 

There is, of course, no objection in principle to the occurrence of energy-dependent parameters, provided this dependence is weak 2 after all, the constancy of p was the approximation in the first place, as the numbers in table 2.1, in the absence of perturbations, show quite clearly. However, the consequence of an energy dependence is that the variations in pi and p2 are no longer perfectly periodic, and it may then become unclear which part of the variation with energy is intrinsic (i.e. existed before interchannel coupling was turned on) and which part is produced by coupling between channels. 

In the usual applications of QDT (chapter 3), fi and 6 are assumed to be weakly varying functions of energy. Indeed, it is often argued that QDT, to be applicable at all, requires a weak variation of fi. Slow variations of /x and <5 were parametrised by Seaton [31], but situations where n varies rapidly as a function of energy in the absence of any interchannel coupling require an extension of the theory [217]. 

As just noted, autoionisation provides a clear example of an electron-electron correlation effect, since coupling between the discrete state of one channel and the continuum of another is excluded in the independent particle model. One sees over what range interchannel coupling acts, because it is a broadening effect, involving a continuous band of energies. The observed interference yields important information on the nature of electron-electron correlations. Its study reveals the interplay between single- and many-electron interactions. 

The strength of the avoided crossing (i.e. the magnitude of the interchannel coupling) is represented graphically by d (see the figure). It can be shown [276, 324] that the oscillator strength density distribution has the form... 

In addition to this important effect, the inner-shell excitations from the 6s and 5d subshells also overlap in energy with each other, giving rise to prominent interchannel coupling and examples of the q reversal effect which will be discussed in chapter 8. The 5d spectrum of Tl is thus unexpectedly rich and interesting. 

Similarly, quantum defect theory plays a very important role in modern descriptions of atomic physics, and should be included at a less rudimentary level than is found in most texts. Again, its modern developments provide an excellent illustration of many fundamental principles of scattering theory. The principles underlying the Lu-Fano graph are easily grasped, and provide excellent insight into an important aspect of the many-body problem, namely interchannel coupling. Likewise double- and inner-shell excitation are hardly discussed in textbooks, structure in the continuum receives little attention, etc, etc. 

The total width of the resonance is directly given by the resonance complex energy. In the case where many channels of autodetachment are open, the question of partial widths for the decay into individual channels arises. This always requires analysis of the wave fimction. The problem of obtaining partial widths from complex coordinate computation has been discussed by Noro and Taylor (39) and Bcicic and Simons (40), and recently by Moiseyev (10). However, these considerations do not seem to have found a practical application. Interchannel coupling for a real, multichannel, multielectron problem has been solved in a practical way within the CESE method by Nicolaides and Mercouris (41). According to this theory the partial widths, 7, and partial shifts to the real energy, Sj, are computed to all orders via the simple formula... 

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