Efficient spin-orbit coupling

We should note that if g = ge, the contact shift is isotropic (independent of orientation). If g is different from ge and anisotropic (see Section 1.4), then the contact shift is also anisotropic. The anisotropy of the shift is due to the fact that (1) the energy spreading of the Zeeman levels is different for each orientation (see Fig. 1.16), and therefore the value of (Sz) will be orientation dependent and (2) the values of (5, A/s Sz S, Ms) of Eq. (1.31) are orientation dependent as the result of efficient spin-orbit coupling. On the contrary, the contact coupling constant A is a constant whose value does not depend on the molecular orientation. 

An efficient spin-orbit coupling calculation will take advantage of symmetry and various computational techniques to improve the efficiency. This paper begins with a consideration of the applicable symmetry rules. This will be followed by a discussion of the computational aspects of the spin-orbit coupling matrix elements. The symmetry rules elucidated in this work are applicable to the recently developed relativistically transformed spin-orbit coupling operators, provided that the rotational properties of the transformed operators are unchanged, which is the case for transformations explicitly dependent upon momentum p (15,16,17). 

The mixing coefficients a and b in (4.10) depend upon the efficiency of the spin-orbit coupling process, parameterized by the so-called spin-orbit coupling coefficient A (or for a single electron). As A O, so also do a or b. Spin-orbit coupling effects, especially for the first period transition elements, are rather small compared with either Coulomb or crystal-field effects, so the mixing coefficients a ox b are small. However, insofar that they are non-zero, we might write a transition moment as in Eq. (4.11). 

Intersystem crossing (i.e. crossing from the first singlet excited state Si to the first triplet state Tj) is possible thanks to spin-orbit coupling. The efficiency of this coupling varies with the fourth power of the atomic number, which explains why intersystem crossing is favored by the presence of a heavy atom. Fluorescence quenching by internal heavy atom effect (see Chapter 3) or external heavy atom effect (see Chapter 4) can be explained in this way. 

In general, the presence of heavy atoms as substituents of aromatic molecules (e.g. Br, I) results in fluorescence quenching (internal heavy atom effect) because of the increased probability of intersystem crossing. In fact, intersystem crossing is favored by spin-orbit coupling whose efficiency has a Z4 dependence (Z is the atomic number). Table 3.3 exemplifies this effect. 

Pure n-n phosphorescences tend to be too long-lived for efficient emission. Either spin orbit coupling or mixing with more allowed CT states must exist to increase the allowedness of the n-n phosphorescences so that it can compete with radiationless decay. 

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