Brown-Ravenhall Disease

Fortunately, if we use DPT, the Brown-Ravenhall disease does not show up at all, because we start from an n-electron state in the nrl, and there is no chance for unphysical positronic or ultrarelativistic components to mix in. This is easily seen in the quasidegenerate formalism, where we insist on relating the upper and lower components via the correct X-operator for electrons, which clearly takes care that we only move in the world of... 

There is no problem of variational collapse or of a Brown-Ravenhall disease. The need for a no-pair projection never arises. 

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) ... 

The use of the Coulomb (4.18) Breit (4.19) or Gaunt (4.21) interaction operators in combination with Dirac Hamiltonians causes that the approximate relativistic Hamiltonians does not have any bound states, and thus, becomes useless in the calculations of relativistic interactions energies. This feature is known under the name of the Brown-Ravenhall disease and is a consequence of the spectrum of the Dirac Hamiltonian [38],... 

DHF calculations on molecules using finite basis sets require considerably more computational effort than the corresponding nonrelativistic calculations and cause several problems due to the presence of the Dirac one-particle operator. It is therefore desirable to find (approximate) relativistic Hamiltonians for many-electron systems which are not plagued by unboundedness from below and therefore do not cause problems like the variational collapse at the self-consistent field level or the Brown-Ravenhall disease at the configuration interaction level. It is also desirable to find forms in which the quality of a matrix representation of the kinetic energy is more stable than for the Dirac Hamiltonian, i.e., forms which are not affected by the finite basis set disease . 

We have updated the material considering the latest developments in the field over the past five years. These developments comprise both computational and more fundamental advances such as exact two-component approaches and the study of explicitly correlated two-electron wave functions in the context of the Brown-Ravenhall disease, respectively. Other topics, such as relativistic density functional theory and its relation to nonrelativistic spin-... 

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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