Spin-independent term

The total energy per site of polyacetylene has two contributions, i.e. the expectation value of the spin independent term, as given by Eq. (5), which only depends on the interatomic distances,... 

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. 

Generally speaking, the density p ri —ta)-, which appears in the nuclear spin-dependent terms, does not coincide with the nucleon density relevant for the nuclear spin-independent term (see for instance also equation 26 in ref. [90]). The situation is reminiscent of the magnetic moment distribution... 

Finally, the spin-dependent contribution is in principle separable from the spin-independent term because it gives an optical rotation which varies in a distinctive way from one hfs component to another. 

Although the static quarkonium potential is flavour independent to a good approximation, nevertheless, the potential does contain flavour dependence in its non-static parts. These include both spin-dependent and spin-independent terms. We shall focus mainly on the spin-dependent corrections, as these are directly testable by measvirements of the fine and hyperfine splittings in quarkoniimi. 

Fig. 4. Comparison of the Schrddinger equivalent potentials of the RIA model (solid curves) with the NRIA potentials (dashed curves) for 500 MeV p + °Ca (from ref. [Cl 83b]). The central, spin independent term and the spin-orbit term are shown in tte upper and lower half of the figure, respectively.

There are electron spin-independent and spin-dependent paramagnetic terms (linear in the vector potential), and a diamagnetic spin-independent term that (quadratic in the vector potential). By substituting (12.12a) or (12.12b) in the spin-free linear terms, one obtains the ZORA form of the Orbital Zeeman (OZ) operator... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

Although the same nuclear spin interactions are present in solid-state as in solution-state NMR, the manifestations of these effects are different because, in the solid, the anisotropic contribution to the spin interactions contributes large time-independent terms to the Hamiltonian that are absent in the liquid phase. Therefore, the experimental methods employed in solids differ from the ones in the liquid state. The spin Hamiltonian for organic or biological solids can be described in the usual rotating frame as the sum of the following interactions ... 

The further assumption that 3M is degenerate with the correlating molecular triplet state 3M provides an estimate of the energy (3M ) of this state in the region (XM ) > (3M ) > E(3M ) which may be spectroscopically inaccessible. Double intersystem crossing to different molecular triplet states of naphthalene87 is also apparently exhibited by the excimer of 1,6-dimethylnaphthalene40 in which the nonradiative process is characterized by a rate constant kf which is the sum of temperature-dependent and temperature-independent terms. The value of the latter is also consistent with a spin-prohibited process (Table XVI). 

Equation (28) can be simplified if we recognize that since the electric dipole operators are independent of spin any term will be zero unless M=M =M". Furthermore, as noted earlier, the axis of spin quantization is the direction of the applied magnetic field and terms will also be zero unless y=y. Therefore,... 

The six elements can be used to generate the n-electron ligand field energies, to which interelectronic repulsion and spin-orbit coupling, which are already defined on the dxy, dX2, dyz, dx2 y2, dz2 basis, are added. Alternatively, the five eigenvalues can be used, but an additional term is necessary to define the correct linear combinations of the dx2 y2 and dz2 orbitals on which to evaluate interelectronic repulsion and spin-orbit coupling. Either way, there are still six total, or five spectroscopically independent, terms in the ligand field potential matrix. 

Here A0 is the spin-independent part, Aq = —56.4 cm-1. H represents the one-center spin terms, which may be called exchange-induced zero-field splitting of the r5 state... 

Here, every bonded electron pair, in atomic or hybrid orbitals, contributes every pair of bonds (or bonds involving nonbonded atoms) with spin-independent electrons contributes — every electron pair with parallel spins contributes — Jy-. This expression tells us that bonding lowers the energy, and nonbonded atoms or nonpaired electrons raise the energy. Increased stability will therefore be found when the negative terms in (9) approach zero, as they must when the relevant atoms or electrons move apart. 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

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