Continuum dissolution

Because of the negative-energy continuum in the spectrum of the one-electron Dirac Hamiltonian already a two-electron equation constructed from 

The next two sets are mixtures of bound and continuum states. Here, one electron is in a bound state and the other is in either a positive- or negative-energy continuum state. The former set, spans the range 

If the electron-electron interaction is switched on, we will face either continuum or autoionizing states. In the autoionizing states, a bound state couples to the continuum, which would lead to its decay. As a consequence, the Dirac-Coulomb model is not considered a useful physical Hamiltonian. However, it is the most widely applied Hamiltonian as projection on the square-integrable one-electron bound states yields remarkably accurate results, despite their dependence on the choice of the projection operators (see the next sections for a further discussion of these issues). 

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Apart from these one-particle effects , additional complications arise in the case of two electrons, which may dissolve from a bound state to the positronic and electronic continua. A mathematically rigorous approach to avoid this so-called continuum dissolution is to use as basis functions the relativistic Coulomb Sturmians... 

More serious is the fact that neither or Urg are permissible approximations to an effective potential, in tW context of a Dirac-spinor description of spin-1/2 particle wave functions. This can be most easily seen by including a third particle "3". The use of a sum such as U(l,2)+U(l,3)+U 2,3) to describe the interaction is disastrous from the start, because of the problem of continuum dissolution [4,5]. Analysis shows that a theoretically well-founded choice for the second-order potential is neither Ugg nor Ugg, but either of the two operators Vgg or Vgg, defined by... 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

A natural way to generalize the non-relativistic many-body Schrodinger equation is to combine the one-electron Dirac operators and Coulomb and Breit two-electron operators. However such an equation would have serious defects. One of them is the continuum dissolution first discussed by Brown and RavenhaU [36]. This means that the Schrodinger-type equation has no stable solutions due to the presence of the negative energy Dirac continuum. A constrained variational approach to the positive energy states becomes therefore necessary. 

There are two particular aspects of radiative corrections in atomic physics that will be emphasized here. One has to do with the correct implementation of QED to many-electron atoms and ions, a subject also discussed by Labzowsky and Goidenko in Chapter 8 of this book. While QED has been tested quite stringently over the years, and is unlikely to be fundamentally incorrect, to actually carry out bound state calculations is a highly nontrivial task. Even the introduction of relativity has raised serious questions about the stability of atoms, referred to as the Brown-Ravenhall wasting disease [2] or continuum dissolution [3]. While the problem is, in a practical sense, still open for neutral systems, we will show that a particular way of applying QED to atoms, use of the Furry representation [4], allows a consistent and accurate treatment of these questions for highly charged ions. 

These diagrams lie at the heart of the continuum dissolution problem... 

In principle problems of relativistic electronic structure calculations arise from the fact that the Dirac-Hamiltonian is not bounded from below and an energy-variation without additional precautions could lead to a variational collapse of the desired electronic solution into the positronic states. In addition, at the many-electron level an infinite number of unbound states with one electron in the positive and one in the negative continuuum are degenerate with the desired bound solution. A mixing-in of these unphysical states is possible without changing the energy and might lead to the so-called continuum dissolution or Brown-Ravenhall disease. Both problems are avoided if the Hamiltonian is, at least formally, projected onto the electronic states by means of suitable operators (no-pair Hamiltonian) ... 

Figure 13. The continuum dissolution problem. An electron making a transition to an unoccupied negative-energy state imparts the resulting energy to the other electron.

In order to analyze the consequences of basis set expansion of the relativistic wave equation for electrons, it is sufficient to consider the one-electron Dirac equation. The only other fundamental complication we foresee when going to a many-electron model is continuum dissolution (Brown-Ravenhall disease), which we have already dealt with in chapter 5 and so need not consider in this context. 

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