Spin-orbit interaction energy parameters

Compute the spin-orbit interaction energy in terms of the parameter y in Equation 10.24 for the all states of an atom with occupancy Is 2s 2p 3s 3d. ... 

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

The Slater integrals of the energy of electrostatic interaction and the spin-orbit interaction constant are the main parameters searched. It is obvious that the number of known levels must be equal to or exceed the number of unknown parameters. 

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. 

Starting with the method described above, extensive tables of the numerical values of mean energies, integrals of electrostatic and constant of spin-orbit interactions are presented in [137] for the ground and a large number of excited configurations, for atoms of boron up to nobelium and their positive ions. They are obtained by approximation of the corresponding Hartree-Fock values by polynomials (21.20) and (21.22). Such data can be directly utilized for the calculation of spectral characteristics of the above-mentioned elements or they can serve as the initial parameters for semi-empirical calculations [138]. 

The rank k can take values 0, 1 and 2 by the triangle rule. Of these, the scalar term with k = 0 has no A dependence and hence does not affect the relative positions of the ro-vibrational energy levels. It just makes a small contribution to the electronic energy of the state r], A). The first-rank term produces a second-order contribution to the spin orbit interaction because it is directly proportional to the quantum number A from the 3-j symbol in the first line of (7.119). The contribution to the spin-orbit parameter A(R) which arises in this way is given (in cm-1) by... 

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