Perturbation theory spin-orbit interaction

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

Since the spin-orbit interaction energy is small, the solution of equations (7.43) to obtain E is most easily accomplished by means of perturbation theory, a technique which is presented in Chapter 9. The evaluation of E is left as a problem at the end of Chapter 9. 

Using first-order perturbation theory, show that the spin-orbit interaction energy for a hydrogen atom is given by... 

Robinson and Frosch<84,133> have developed a theory in which the molecular environment is considered to provide many energy levels which can be in near resonance with the excited molecules. The environment can also serve as a perturbation, coupling with the electronic system of the excited molecule and providing a means of energy dissipation. This perturbation can mix the excited states through spin-orbit interaction. Their expression for the intercombinational radiationless transition probability is... 

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... 

Hi H2 this is the so-called intermediate couphng. When the electrostatic and spin orbit interactions are of the same order of magnitude - and this is the case of the actinides - both should be included in first-order perturbation theory. 

Consider a dn configuration present in a crystal field that leaves the ground state nondegenerate except for spin. The ground state then consists of (25+ l)-spin states and the effect of the spin-orbit interaction plus the magnetic field can be computed using first- and second-order perturbation theory. If we take as the perturbation operator... 

We now turn our attention to the second-order contributions. In order to see how these are derived, let us consider in particular the contributions of the spin-orbit interaction, 3u o. Before we can use second-order perturbation theory to evaluate these contributions, we need to write down the general matrix elements of this operator. We can do this easily if we write the expression in equation (7.72) in the simplified form... 

The calculation of the ligand-field matrices within an Z electron configuration is made by standard tensorial methods. The independent perturbations are the interelectronic repulsion, the ligand-field, and the spin-orbit interaction, and they will be discussed in this order using, as mentioned above, the Russell-Saunders case of the theory of atomic spectra. 

The forbidden 3E — 1E+ transition has been discussed earlier [Section 6.3.2, Eqs. (6.3.43a) - (6.3.46b), Table 6.3]. It is useful to return to a specific example here. 3E — 1E+ transitions have been observed between two states belonging to the same it2 configuration (molecules with 6 valence electrons, NH, PH, etc. and heteronuclear molecules with 12 valence electrons, SO, NF, NCI, PF). Wayne and Colbourn (1977) have discussed the spin-orbit interaction between this pair of isoconfigurational states. Inserting the first-order perturbation theory definitions of the 0 and 7 mixing coefficients of Eq. (6.3.46a), an expression analogous to Eq. (6.4.15) is obtained,... 

Finally, Zaitsevskii et al [95] proposed a DGCF method differing mainly from the previous ones by the scalar effective Hamiltonian used and by the content of the spin-orbit interaction. In this work a dressed intermediate Hamiltonian is constructed using the spin-adapted many-body multipartitioning perturbation theory (MPPT) up to the second order. The MPPT theory is based on the simultaneous use of several quasi-one-electron zero-order Hamiltonians (see for... 

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