Nuclear spin-orbit operator

The nuclear spin-orbit operator or orbital hyperfine operator describes the coupling of the nuclear magnetic moments to the orbital of the electrons... 

The H e operator is the one-electron part of the spin-orbit interaction, while the H e and He°° operators in eq. (8.36) define the two-electron part. The one-electron term dominates and the two-electron contribution is often neglected or accounted for approximately by introducing an effective nuclear charge in H ° (corresponding to a screening of the nucleus by the electrons). The effect of the spin-orbit operators is to mix states having different total spin, as for example singlet and triplet states. 

The V2AA term in eq. (10.68) gives a Diamagnetic Spin-Orbit operator, which is an operator of order c. Although we otherwise only consider terms up to order c, the nuclear spin-spin coupling constant only contains terms of order c, which is why we need to include... 

If one starts from a formally nonrelativistic Hamiltonian, third-order perturbation theory has to be used, as the spin-orbit operator has to be included in addition to the perturbations due to the nuclear magnetic moments and to the external magnetic field. As the spin-orbit operator permits spin polarization, a Fermi contact (FC) term and a spin-dipolar (SD) term also appear in the perturbed Hamiltonian and couple nuclear magnetic moment with electronic spin. 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

The vibronic coupling operates upon the nuclear and electronic functions of the electronic states involved in the internal conversion (Equation 6.74). Mk is the effective mass associated with the /cth vibrational mode. Because of the different spin labels of the states involved in the intersystem crossing, the spin-orbit coupling gives a finite value to the electronic coupling in Equation 6.74. 

From the three operators obtained by differentiating the Hamiltonian with respect to the nuclear magnetic moments (Equation (2.12)) only the singlet paramagnetic spin-orbital (PSO) operator... 

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. 

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