Orbital collapse

The limit cycle found in the previous section holds only for 103 — 031 small. Obviously, once the limit cycle exists, it can be continued, either globally or until certain bad things happen such as the period tending to infinity or the orbit collapsing to a point. It is very difficult to show analytically that these events do not occur. Moreover, the computations necessary to actually prove the asymptotic stability of the bifurcating orbit are very difficult. We discuss briefly some numerical computations, shown in Figure 8.1, which suggest answers to both these problems. 

In the weak interaction limit (Fig. 4.3e) the orbitals collapse. The localization energy cannot be overcome and the surface electron structure is only weakly disturbed. 

With the Coulomb and exchange parts of the MP discussed so far, the core-like solutions of the valence Fock equation would still fall below the energy of the desired valence-like solutions. In order to prevent the valence-orbitals collapsing into the core during a variational treatment and to retain an Aufbau principle for the valence electron system, the core-orbitals are moved to higher energies by means of a shift operator... 

It has since been discovered that a more accurate equation, which takes account of radial correlations the g Thomas-Fermi equation, does allow d-orbital collapse to occur. Thus Mayer was actually not far from her objective... 

In early literature [208], the term wavefunction collapse was frequently used to describe this property. We avoid it here it causes confusion with wavefunction collapse in the quantum theory of measurement. For this reason, we prefer the term orbital collapse, following [196]. 

The understanding of orbital collapse remained in a rather unsatisfactory state until around 1969 when, as a result of considerable progress in numerical methods and the advent of fast computers, it became a relatively simple matter to solve the radial Schrodinger equation in the Hartree-Fock and related schemes, which take better account of the shell structure of the atom. 

Fig. 5.4. The double-well radial potential for 4/ elements close to the point at which orbital collapse occurs. Note the highly nonlinear scales on both axes, designed to show as much detail as possible for both wells (after D.C. Griffin et al. [208]).

For the d electrons, the situation is similar, but not as dramatic the centrifugal term in this case is smaller, and the potential, instead of a positive barrier between two wells, exhibits only a knee, as shown in fig. 5.5. In spite of this, orbital collapse occurs, but over intervals of more than one atomic unit. 

Fig. 5.6. Orbital collapse in an extended homologous sequence (see text for details) (after J.-P. Connerade [205]).

Fig. 5.7. The trend of effective quantum numbers for a Rydberg manifold of high angular momentum, showing the discontinuity in all the quantum numbers as a result of orbital collapse (after D.C. Griffin et al [208]).

We turn now to the mechanism of orbital collapse, which is not immediately obvious from the numerical calculations of the various authors quoted above. It has been explained by Connerade [210, 211] using analytic potentials and elementary quantum theory. The key feature to note is the difference in nature between different kinds of potential in quantum mechanics. This is illustrated in fig. 5.9(a). First, we have the familiar Coulomb well or long range potential. This, as we have seen in chapter 2, gives rise to Rydberg series containing an infinite number of... 

Thus far, it may have surprised the reader that the double well was always considered as a bistable situation, leading either to orbital collapse into the inner well or to its expulsion into the outer reaches of the atom. The possibility has not been considered that the wavefunction might be split evenly between both wells, with comparable probabilities of the particle being on either side of the centrifugal barrier. In fact, although such a circumstance is rare, there is nothing in principle to prevent its occurrence. 

Centrifugal barriers have a profound effect on the physics of many-electron atoms, especially as regards subvalence and inner shell spectra. One aspect not discussed above is how energy degeneracies arising from orbital collapse can lead to breakdown of the independent electron approximation and the appearance of multiply excited states. Similarly, we have not discussed multiple ionisation (the ejection of several electrons by a single photon) enhanced by a giant resonance. Both issues will be considered in chapter 7. 

Fig. 7.9. Multiple ionisation in the energy range of the giant resonances of lanthanides, showing the different behaviour in different parts of the lanthanide sequence (a) before orbital collapse, where multiple ionisation dominates and (b) after collapse, where single ionisation becomes the dominant process (after P. Zimmermann [353]).

Since orbital collapse for unexcited atoms begins before the onset of the long periods, the implication is that the 5d orbital, which in the presence of an unexcited core would lie well outside the 6s orbital (which is filled), moves inwards to a radius closer to that in Ba once the core is excited. Similarly, if the 5p6 subshell of Ba is excited, the 5d orbital... 

Another property of atoms which is sensitive to the conditions in the outer reaches of the atomic field is of course orbital collapse, which can be controlled as described in section 5.23. This has led Golovinskiy et al. [480] to consider whether a strong laser field could be used to precipitate orbital collapse, and to propose an experiment in which dynamic collapse at the Rabi frequency could be detected by X-ray spectroscopy of the irradiated sample. 

The strongest pieces of evidence complex atoms provide in favour of independent electron modes and simple Bohr-Sommerfeld quantisation are (i) the existence of Rydberg series and (ii) the regularity of the periodic table of the elements. As a corollary, we should look for quantum chaos (if it occurs) in atoms for which there is some breakdown in the quality of the shell structure, combined with prolific and heavily perturbed overlapping series of interacting levels. These conditions are most readily met, as will be shown below, in the spectra of the alkaline-earth elements, as a result of d-orbital collapse. 

A still more dramatic size variations occurs as a result of the centrifugal barrier effects discussed in chapter 5 since orbital collapse results in deep filling within the atom, in which the outermost electrons are not involved, it is possible for excited states and resonances involving collapsed orbitals in the final states to survive in the solid. 

We have emphasised the distinction between localised and itinerant states. Under certain circumstances (governed by atomic properties) a given orbital is poised at the critical point where it can become either one or the other for small changes in the environment of the atom. In solid state physics, this gives rise to a first-order Mott transition. In the present context, such a situation is closely related to the problem of controlled orbital collapse (section 5.23) if a solid is built up from free atoms with a double well potential and the corresponding orbitals in the outer well, these may hybridise easily, the external part of the orbital going into itinerant states. If one forms a solid from atoms with collapsed orbitals, then they remain localised. 

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