Basis set expansion of the dirac operator without variational collapse

TLDR: In this article, a review of the possible ways to avoid the variational collapse is given, in terms of various criteria that one wants to fulfill, and the most promising approach consists of a freeparticle Foldy-Wouthuysen (FW) transformation on operator level and a back transformation on matrix level (approaches C2 and C3).

Abstract: The eigenstates of the matrix representation of the Dirac operator for c → ∞ do not approach their nonrelativistic counterparts in the same basis. This wrong “Schrodinger limit” is shown to be the main reason for the phenomenon known as “variational collapse.” After a short review of existing proposals to overcome the “variational collapse,” a systematic study of the possible ways to avoid it is given. All discussed approaches are analyzed in terms of various criteria that one wants to fulfill. The most promising approach consists of a free-particle Foldy–Wouthuysen (FW) transformation on operator level and a back transformation on matrix level (approaches C2 and C3). This implies a modification of the free-electron part of the matrix representation of the Dirac operator and leads to the correct Schrodinger limit (and if one wishes even the correct Pauli limit) in the same basis (and to the exact results for a complete basis). The potential energy is unchanged, which makes the application to n -electron systems straightforward. Projection of the Dirac operator to positive energy states does not remove the variational collapse unless this is done in a very special way.