Operators spin-other-orbit, Breit-Pauli

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

In general, they can all be properly dealt with in the framework of perturbation (response) theory. According to the discussion in section 5.4, we may add external electromagnetic fields acting on individual electrons to the one-electron terms in the Hamiltonian of Eq. (8.66). Fields produced by other electrons, so that contributions to the one- and two-electron interaction operators in Eq. (8.66) arise, are not of this kind as they are considered to be internal and are properly accounted for in the Breit (section 8.1) or Breit-Pauli Hamiltonians (section 13.2). Although the extemal-field-free Breit-Pauli Hamiltonian comprises all internal interactions, such as spin-spin and spin-other-orbit terms, they may nevertheless also be considered as a perturbation in molecular property calculations. While our derivation of the Breit-Pauli Hamiltonian did not include additional external fields (such as the magnetic field applied in magnetic resonance spectroscopies), we now need to consider these fields as well. 

The consequence is that we must treat the spin-orbit and the spin-other-orbit interactions separately we cannot combine them as in the Breit-Pauli Hamiltonian. The reason is that the functions on which the momentum operators operate are derived from the small component, and only in the nonrelativistic limit where the large and small components are related by kinetic balance can we rewrite the spatial part of the spin-other-orbit interaction in the same form as the spin-orbit interaction. The reader... 

The Breit-Pauli SOC Hamiltonian contains a one-electron and two-electron parts. The one-electron part describes an interaction of an electron spin with a potential produced by nuclei. The two-electron part has the SSO contribution and the SOO contribution. The SSO contribution describes an interaction of an electron spin with an orbital momentum of the same electron. The SOO contribution describes an interaction of an electron spin with the orbital momentum of other electrons. However, due to a complicated two-electron part, the evaluation of the Breit-Pauli SOC operator takes considerable time. A mean field approximation was suggested by Hess et al. [102] This approximation allows converting the complicated two-electron Breit-Pauli Hamiltonian to an effective one-electron spin-orbit mean-field form... 

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. 

The matrix elements v,y and w,y may be derived from Breit-Pauli or Douglas-Kroll spin-orbit operators, or other operators that conveniently lend themselves to this type of expression. 

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