Breit-Pauli relativistic coupling

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

The analysis of the relativistic Renner coupling has been extended to 11 states, including the two-electron part of the Breit-Pauli operator, thus generalizing previous result of Hougen [45,46]. Other extensions of the theory are S — 11 coupling in the doublet manifold [47] and SO coupling in a half-filled n shell, as found, for example, in carbenes [48]. 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

First attempts to calculate molecular parity violating potentials within a two-component framework have been undertaken by Kikuchi and coworkers [168,169]. They have added the Breit-Pauli spin-orbit coupling operator Hso to the usual non-relativistic Hamiltonian Hq... 

In the case of a relativistic system, as a first (and useful) approximation, the zero-order spectrum can be taken as the nonrelativistic one, with Hq defined explicifly as fhe Coulomb Hamiltonian. Then, the perturbation V is also written explicitly as the relativistic Breit-Pauli operators, and it is this perturbation that turns the initially discrete state into a resonance. For example, this type of advanced calculation, with multichannel coupling included, has been shown to explain quantitatively the positions and lifetimes of the relativistic levels of mefastable states in negative ions [90]. However, if the more accurate four-component relativistic Dirac treatment for each electron is invoked for cases of high effective nuclear charge, then the stability against autoionization implies not only the exclusion of components representing decay to... 

For atoms from the first row of the periodic table and molecules containing them, the Breit-Pauli approximation appears to afford an accurate description of relativistic effects. The most important of these effects is usually the spin-orbit coupling... 

Sadlej and co-workers (Sadlej and Urban 1991, Kello and Sadlej 1991a, b) have had considerable success in using the Breit Pauli Hamiltonian in all-electron relativistic calculations which included electron-correlation effects. Techniques which include electron-correlation effects such as the coupled cluster method or Moller-Plesset (MP) perturbation method have been adapted to include relativistic effects. The authors have carried out such computations on molecules such as AgH, AuH, etc. 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

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