Spin-orbit coupling definition

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

With only minor modifications, the spin-independent relativistic corrections can be put into this Schrodinger equation and so be absorbed in the definition of the basis orbitals ipi the spin orbit coupling potential, however, is treated explicitly in the effective Hamiltonian. 

Figure 5. CASSCF collinear evolution of spin-orbit couplings between (a) doublet-doublet, quartet-quartet and (b) doublet-quartet electronic states of CC>2+, where the other CO distance is kept fixed at the equilibrium geometry of the neutral molecule (i.e. 2.2 Bohr). See Table 1 for the definition of these terms. Strictly speaking, the g-u symmetry is only applicable for Rco = 2.2 Bohr.

Figure 3. Evolution of the spin-orbit couplings involving the S2 (4Ag) state along the internuclear distance. See Table 3 for the definition of these terms.

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. 

This scheme of adding angular momenta is called the j—j coupling scheme however, despite its fundamental validity it is rarely used in applications. The reason for this lies in the fact that the spin-orbitals j, mj) need to be determined by a two-component or four-component Dirac equation, which implies a complex algebra and much more computational effort. Definitely such a procedure is ultimate for heavy element atoms. The j—j coupling scheme assumes that the spin-orbit coupling dominates over the interelectron repulsion H ° > F 6. 

In order to settle a controversy over the effects of deuteriation on the radiative lifetimes of aromatic hydrocarbons, some systems have been examined, using two independent techniques. Table 7 shows the rr values obtained from phosphorescence lifetimes and quantum yields. Evidence for the intramolecular nature of the deuterium isotope effect on rr comes from the lack of sensitivity to solvent. Although no definitive evidence as to the origin of the isotope effect is put forward, it is suggested that there may be two important contributing mechanisms to the 7i -> S0 radiational process, an isotope-independent and an isotope-sensitive one. The isotope-insensitive mechanism is almost certainly first-order spin-orbit coupling. It therefore remains to determine the nature of the other process, which will only be important when spin-orbit coupling is inefficient.1886... 

Finally, some spectroscopic applications for pseudopotentials within SOCI methods are presented in section 3. We focus our attention on applications related to relativistic averaged and spin-orbit pseudopotentials (other effective core potentials applications are presented in chapters 6 and 7 in this book). Due to the large number of theoretical studies carried out so far, we have chosen to illustrate the different SOCI methods and discuss a few results, rather than to present an extensive review of the whole set of pseudopotential spectroscopic applications which would be less informative. Concerning the works not reported here, we refer to the exhaustive and up-to-date bibliography on relativistic molecular studies by Pyykko [21-24]. The choice of an application is made on the basis of its ability to illustrate the performances on both the pseudopotential and the SOCI methods. One has to keep in mind that it is not easy to compare objectively different pseudopotentials in use since this would require the same conditions in calculations (core definition, atomic basis set, SOCI method). The applications are separated into gas phase (section 3.1) and embedded (section 3.2) molecular applications. Even if the main purpose of this chapter is to deal with applications to molecular spectroscopy, it is of great interest to underline the importance of the spin-orbit coupling on the ground state reactivity of open-shell systems. A case study is presented in section 3.1.4. 

New (magnetic) interactions in the Hamiltonian operator due to electron spin. The spin-orbit coupling, for example, destroys the picture of an orbital having a definite spin. 

Fine structure splittings have been observed in the far-IR [1 to 3] and mid-IR [4] LMR spectra, in the IR emission [6] and absorption [7] spectra, and in the near-UV emission [8 to 12] and absorption [13] spectra of PH and PD. The fine-structure constants for the electronic ground state X i.e., X for the spin-spin coupling, y for the spin-rotation coupling, and Xq, Yd for their respective centrifugal distortions, and constants for the excited state A rij, i.e.. A, Ad for the spin-orbit coupling and p, q, Po or C (i = 0 to 3) for the A-type doubling, have been derived for definitions of the constants and relations with reference to earlier notations, see [8, table 3]. 

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