Spin-orbit coupling/interaction

Most transition metal ions have the orbital motion quenched to first order by the crystal field, but the presence of a large spin-orbit interaction coupled with Eq. (32) will cause second-order orbital terms in the hyperfine interaction of the magnitude... 

For the proton shieldings, the dominant term is the one-electron spin-orbit interaction coupled to the FC interaction. The two-electron SO term and the SD terms are less than 10% of the relativistic correction for HBr and HI. This also indicates that the neglect of the Breit interaction in the four component calculations does not give rise to large errors because it only modifies the... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

There is appreciable coupling between the resultant orbital and resultant spin momenta. This is referred to as LS coupling and is due to spin-orbit interaction. This interaction is caused by the positive charge Ze on the nucleus and is proportional to Z". The coupling between L and S gives the total angular momentum vector J. 

In the lowest optieally excited state of the molecule, we have one eleetron (ti ) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaetion and can either form a singlet 5 state (5 = 0), or a triplet T state (S = 1). Since the electric dipole matrix element for optical transitions — ep A)/(me) does not depend on spin, there is a strong spin seleetion rule (AS = 0) for optical electric dipole transitions. This strong spin seleetion rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the w eak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

The spin—orbit coupling constants f4f and fsd, which represent the relativistic spin—orbit interaction in the 4f and 5d shells, also determined by means of the radial wave functions Rrd of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23... 

Robinson and Frosch<84,133> have developed a theory in which the molecular environment is considered to provide many energy levels which can be in near resonance with the excited molecules. The environment can also serve as a perturbation, coupling with the electronic system of the excited molecule and providing a means of energy dissipation. This perturbation can mix the excited states through spin-orbit interaction. Their expression for the intercombinational radiationless transition probability is... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

Frosch(84,133) have explained the external heavy-atom effect in intersystem crossing by postulating that the singlet and triplet states of the solute, which cannot interact directly, couple with the solvent singlet and triplet states, which themselves are strongly coupled through spin-orbit interaction. Thus the transition integral becomes<134)... 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

This mode of hyperfine interaction will become important only when the impaired electron is able to partially occupy a low-lying excited state (AE small), and the ground state has orbital angular momentum (L 0). The adsorbed nitric oxide molecule and the superoxide ion with 170 are typical examples where hyperfine coupling via spin-orbit interaction may be observed. 

The spin-orbit interaction is also called spin-orbit effect or spin-orbit coupling, which is one cause of magnetocrystalline anisotropy. SOC, the intrinsic interaction between a particle spin and its motion, is responsible for various important phenomena, ranging from atomic fine structure to topological condensed matter physics. SOC plays a major role in many important condensed matter phenomena and applications, including spin and anomalous Hall effects, topological insulators, spintronics, spin quantum computation, and so on. 

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