The Rayleigh-Ritz method for Dirac Hamiltonians

At the beginning of this section we presented a non-rigorous version of the Rayleigh-Ritz method for Dirac Hamiltonians. It is now time to examine the conditions under which this can be justified rigorously as a foundation for practical calculations with the DHF(B) equations. Our presentation here is based on [76,86,87] which give much more detail. 

We start with the standard textbook problem of approximating eigenvalues and eigenvectors of a self-adioint operator T that is bounded below on some 

Evidently, there is a problem with Dirac Hamiltonians of the sort we have been discussing because the spectrum goes from — to -l- there is no global lower bound. It is conventional to assume that this is the end of the matter, and that variational methods cannot be applied to Dirac Hamiltonians. This is false. The bound state spectrum of an atom is indeed bounded below, more or less where one wants it to be, so that provided due care is taken with the choice of trial functions, we can proceed exactly as in nonrelativistic quantum mechanics. We can then extend this in the usual way to molecules and solids. Here, we shall merely summarize the argument leading to this conclusion. 

Provided that, as is always the case in practice [87, Table II], that = - infyg —IrmP-, this demonstrates that a lower bound exists in the 

A similar argument for y/ G shows that these eigenvalues also are lowered 

---