Spin-orbit coupling perturbation theory

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

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. 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

Second-order perturbation theory gives rise to two additional mechanisms involving an intermediate state that is vibrationally coupled to one and spin-orbit coupled to the other manifold (Fig. 11). 

In the absence of an external magnetic field, orbitally nondegenerate levels with spin multiplicity greater than 2 split due to direct electron spin—spin coupling (in first order) and spin—orbit coupling (in second and higher orders of perturbation theory). This phenomenon is called zero-field splitting (ZFS). The SH that describes this phenomenon can be formulated in... 

The limitations of the SH formalism can be overcome when we omit the perturbation theory and involve the spin-orbit coupling in a variational principle. The straightforward way is to diagonalize the (complex) interaction... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

Perturbation theory, including relativistic effects without the contribution of spin-orbit coupling from Reference 32. 

Next, we consider the spin-orbit coupling, hi WZ structure, one may apply a unitary transformation, which diagonalises at the T point, to the k.p Hamiltonian and then use a perturbation theory for the states close enough to the T point, as described by Bir and Pikus [3]. This leads to the following closed expressions for the hole masses ... 

Lo et a/,102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanis103 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin orbit coupling... 

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