Effective interactions electrostatic-spin-orbit

In recent calculations, the agreement between theory and experiment was further improved by including in the calculations additional interactions, namely, the spin-dependent spin-spin, spin-other-orbit and effective electrostatic-spin-orbit (effective EL-SO) interactions. 

Fig. 18.1 The effects of the electrostatic, spin-orbit, and crystal field interactions on the/ configuration.

In a single-configurational non-relativistic approach, the integrals of electrostatic interactions and the constant of spin-orbit interactions compose the minimal set of semi-empirical parameters. Then for pN and dN shells we have two and three parameters, respectively. However, calculations show that such numbers of parameters are insufficient to achieve high accuracy of the theoretical energy levels. Therefore, we have to look for extra parameters, which would be in charge of the relativistic and correlation effects not yet described. 

For pN shells the effective Hamiltonian Heff contains two parameters F2 and 4>i, as well as the constant of spin-orbit interaction. The term with k = 0 causes a general shift of all levels, which is usually taken from experimental data in semi-empirical calculations, and can therefore be neglected. The coefficient at 01 is proportional to L(L + 1). Therefore, to find the matrix elements of the effective Hamiltonian it is enough to add the term aL(L + 1) to the matrix elements of the energy of electrostatic and spin-orbit interactions. Here a stands for the extra semi-empirical parameter. 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

M k = 0, 2, and 4. They also considered the effect of additional configurations on the spin-orbit interaction to produce the electrostatically correlated spin-orbit interaction. 

As fits to the crystal spectra became more detailed, a lack of balance in the theory appeared. The Coulomb interactions within the 4f shell and the effects of configuration interaction to second order can be taken into account by means of the four Slater integrals F (4f, 4f), the three Trees parameters a, J5, y, and the six three-electron parameters T . In contrast to these 13 electrostatic parameters, the spin-orbit interaction, until 1968, was represented by the single parameter This scheme overlooks the terms that arise from the Breit interaction, which was developed on relativistic grounds to account for the fine structures of the multiplets of Hel Isnp (see Bethe and Salpeter 1957). In the non-relativisitc limit parts of the Breit interaction, such as the retardation of the Coulomb interaction and the magnetic interactions that exist between the electrons in virtue of their orbital motions, can be represented by adjustments to the electrostatic parameters. Two terms cannot be absorbed in that way the spin-spin interaction and the spin-other-orbit interaction Marvin (1947) showed that, for the configurations I", ... 

On remembering that the two-dominant interactions for many-electron atoms are the electrostatic interaction = e lnj and the spin-orbit interaction Hi = 2/ CiUi Si), the effective interactions have the form... 

In this section we summarize the results of investigations of several types of configurations characteristic of the lanthanide spectra, which, in the first approximation, may be considered as isolated and therefore treated separately. In this case, the electrostatic interaction with other configurations is introduced only to second order perturbation theory, namely, by including in the hamil-tonian of the investigated configuration effective electrostatic interactions in addition to the real electrostatic and spin-orbit interactions. It is shown that further improvement between calculated and observed levels can be obtained through the inclusion of the spin-dependent interactions (SDI), i.e. ss, soo and effective EI SO. 

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