Matrix spin-orbit coupling

For the paramagnetic case the expre.ssion for the photo current in Eq. (2) can be simplified to a concentration weighted sum over the products of the K-resolved partial density of states (DOS) ri (F) and a corresponding matrix element that smoothly varies with energy [13]. This simple interpretation of the XPS-spectra essentially also holds for the more complex spin-resolved ca e in the presence of spin-orbit coupling as studied here. 

Mixing of LS-states by spin orbit coupling will be stronger with an increasing number of f-electrons. As a consequence, intermediate values of Lande g factor and reduced crystal field matrix elements must be used. 

Theoretical analyses (75-77) of the matrix-induced changes in the optical spectra of isolated, noble-metal atoms have also been made. The spectra were studied in Ar, Kr, and Xe, and showed a pronounced, reversible-energy shift of the peaks with temperature. The authors discussed the matrix influence in terms of level shift-differences, as well as spin-orbit coupling and crystal-field effects. They concluded that an increase in the matrix temperature enhances the electronic perturbation of the entrapped atom, in contrast to earlier prejudices that the temperature dilation of the surrounding cage moves the properties of the atomic guest towards those of the free atom. 

The only term which has non vanishing matrix elements with both A g and T2g terms is the arising from the electron configuration t gSg- If the spin-orbit coupling constant is denoted with the matrix elements are [126] ... 

The g-value of a free electron is a scalar, ge = 2.00232. In a radical species, g becomes a matrix because of the admixture of orbital angular momentum into S through spin-orbit coupling. The components of the g-matrix thus differ from ge to the extent that p-, d-, or f-orbital character has been incorporated, and they differ from one another, depending on which p-, d-, or f-orbitals are involved. 

The AO composition of the SOMO can often be deduced from the dipolar hyperfine matrix, particularly when the radical has enough symmetry to restrict possible hybridization. Thus an axial hyperfine matrix can usually be interpreted in terms of coupling to a SOMO composed of a single p- or d-orbital. A departure from axial symmetry may be due to spin orbit coupling effects, if (for example) /) Az and Ax AyxP(gx gy). If the departure from axial symmetry is larger, it is usually caused by d-orbital hybridization. The procedure is best illustrated by examples. 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

Characterization of Triplet States by Optical Spectroscopy Some of the largest spin-orbit coupling matrix elements are given in Table 4. 

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