Full One- and Two-Electron Spin-Orbit Operators

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

The most renowned one- and two-electron spin-orbit Hamiltonian is given by 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as 

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

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