Avoiding Variational Collapse The Minimax Principle

The unboundedness of the one-electron Dirac operator prohibits the application of the variational principle and we are in desperate need of a solution to this problem. In the 1980s, many authors discussed the issue of how a basis set expansion of the one-electron spinors affects the variational stability (we come back to this particular issue in chapter 10) and how this is related to the need to choose projection operators as discussed in section 8.2.3.2. 

A formal solution is the so-called minimax principle [354], which states that the problem of variational collapse is avoided by determining the minimum of the electronic energy with respect to the large component of the spinor, while guaranteeing a maximum of the energy with respect to the small component. How such a saddle point may look has been shown by Schwarz and Wechsel-Trakowski [217]. The minimax principle has also been discussed in great detail for the complicated two-electron problem [355] (see also Refs. [356,357]). 

For basis-set expansion techniques it turned out to be decisive to fulfill the kinetic-balance condition for the basis functions (see again chapter 10 for details and references), whereas fully numerical four-component calculations had already been carried out around 1970 without encountering variational collapse. In numerical approaches it is possible to search for optimized spinors in the vicinity of the nonrelativistic solution with a given number of nodes and associated orbital energy as we shall see in chapter 9. 

After numerous numerical (and formal) studies of these issues in the 1980s, rigorous results on the minimax principle were aimed at in the late 1990s and beginning of the new millermium [358-366]. Other studies attempted rewriting the Dirac operator such that the variational principle can be applied without precautions [367-370]. We will describe in chapters 9 and 10 numerically stable approaches that have been well established in atomic and molecular 

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