Second-order spin-orbit splitting

Further, so may be expressed formally as = J2i where the s, are the usual one-electron spin operators and the first rank tensor operators denote the spatial part of so related to electron i. (In the BP Hamiltonian (Eq. [104]), for example, 2 corresponds to the terms in braces.) One then obtains 

The effective operator in Eq. [209] has exactly the same structure as the second term of the Breit-Pauli SSC operator in Eq. [162]. The fine-structure tensor of rank two has thus two major contributions 

For a comparison with experimentally determined parameters, the calculated total (SO and SS) is equated with 3) as used in the phenomenological electronic spin-spin operator, Eq. [164]. 

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains 

Usually, experimentalists fit spin-spin splittings in terms of these parameters D and E. For linear molecules, E = 0 by symmetry, since Dyy = Dxx. In the experimental literature, a different notation (Ass) is sometimes used in this 

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Spin-Spin Coupling Although we focus on spin-orbit coupling (SOC), we need to consider the tensorial structure of electronic spin-spin coupling Second-order SOC mimics perfectly first-order spin-spin coupling and vice versa, so that they cannot be told apart (see the later section on second-order spin-orbit splitting). 

If there is a second-order spin-orbit splitting, can we predict which of the triplet sublevels is lowered in energy due to spin-orbit coupling ... 

Kotochigova, S., Tiesinga, E., and Julienne, P.S., Relativistic ab initio treatment of the second-order spin-orbit splitting of the a S potential of rubidium and cesium dimers, Phys. Rev. A, 63, 012517, 2001. 

Gillies [237], They used the technique of laser ablation to form nozzle beams of the molecules, injected into a Fourier transform microwave spectrometer. In both cases the ground electronic state is 4X, an example of which we met in chapter 9, the excited a state of the CH radical. However in that example the coupling was close to case (b), whereas VO and NbO provide examples of a case (a) 4 XV state. The = 1/2 and 3/2 states are split by second-order spin-orbit coupling the splitting is... 

The effect of suUur participation on the orbital g -shifts in the EPR spectra, illustrated in Pig. 20, accounts for the qualitatively different spectra observed for tyrosyl phenoxyl and Tyr-Cys phenoxyl radicals (Gerfen et al., 1996). The rhombicity of the simple tyrosyl radical EPR spectrum is a consequence of the splitting between gx and gy principal g -values. These g -shifts deviate from the free electron g--value ge = 2.00023) as a result of orbital angular momentum contributions. While a nondegenerate electronic state (such as the A" ground state for ere) contains no hrst-order unquenched orbital momentum, second-order spin-orbit mixing between close-lying a and a" functions results... 

In Fe + ions, d-d transitions are both multiplicity- and Laporte-forbidden. Splitting of S is possible due to a second-order spin-orbit coupling of the S manifold with excited quartet spin states or due to higher order effects of the crystal field. ... 

For A > 0 states, second-order spin-orbit effects cause two types of J-independent level shifts proportional to AE (the form of the diagonal spin-orbit matrix element) or to 3E2 — S(S+1) (the form of the diagonal spin-spin matrix element). Second-order spin-orbit effects are unobservable for E and 2E states because they result in no new level splittings or changes in existing separations. For E states with S > 3, second-order spin-orbit effects are observable only as shifts similar to the form of the spin-spin interaction. See Levy (1973) for a demonstration of this fact. 

Figure 1. Experimental variations of 2p-core ionization energies (in eV) for atoms from A1(Z = 13) to Ba(Z = 56). Upper left. 2p /2 and 2p3/2 energies lower left, their weighted-average second-order discrete derivative, as functions of Z upper right spin-orbit splitting between the 2pi/2 and 2p3/2 levels lower right, its second derivative, as functions of Z. On the derivative diagrams shell effects appear about fully filled and half-filled shells and near filling irregularities of the transition elements.

Figure 8. Schematic diagram of the spin-orbit splitting of the r 5 bands. The intraband terms (first-order perturbation theory) and their interband counterpart (second order) due to interaction with Ffs are illustrated.

When the spin part is integrated off, then the integrals over the (spatial) molecular orbitals ) survive and the whole second-order contribution is split into a pure one-electron and a two-electron term... 

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