Spin-orbit interaction perturbation

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... 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

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. 

As described in Sections II.A. 1 and II.A.4, the numerical procedures required to calculate term parameters and C terms induced by the perturbation of the transition moment by spin-orbit coupling are nearly identical. Almost all of the comments concerning the calculation of terms made in Section III.A.1 apply equally well to the spin-orbit-induced C terms. The major difference between the two types of calculation is that the spin-orbit interaction is often significantly larger than the influence of a magnetic field. 

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... 

Spin-orbit interactions mix states with the same J but different L 8 by second (or higher) order perturbations, such perturbations become important when the separation between the levels is small. The spin-orbit coupling constants ( 4/) increase more rapidly through the rare earth series with increasing number of /-electrons than do the F s. This results in the breakdown of L 8 coupling even more near the middle of the rare earth series, because of the greater population of the upper... 

In this expression 2S + 1 is called the multiplicity because if the spin-orbit interaction of the electrons is taken into account in a perturbative approach, the pure LS state splits in energy for the different couplings of L with S leading to J (the LSJ-coupling case), and the number of term splittings is given for L > S by 2S + 1. It is therefore common to say singlet , doublet , triplet and so on for... 

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