Electrostatic and Spin-Orbit Interaction

2] (ri)liSi are different for different states of the same configuration and partly remove 

The electrostatic interaction Hi within a configuration contains a direct and an exchange part. 

The spin-orbit interaction H2 for each electron i can be written in the form  

It is considered as an interaction between the spin and the orbital angular momenta of the electron i, and is represented by the coupling parameter ,  

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D is the zero-field splitting tensor, a traceless, rank-two tensorial quantity. The ZFS tensor is a property of a molecule or a paramagnetic complex, with its origin in the mixing of the electrostatic and spin-orbit interactions (80). In addition, the dipole dipole interaction between individual electron spins can contribute to the ZFS (81), but this contribution is believed to be unimportant... 

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. 

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. 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Electrostatic and spin-orbit interactions as first-order perturbations... 

For the electrostatic and spin-orbit interaction, steps 1 and 2 have already been carried out the appropriate expressions are given in formulae 1.21 and 1.3 respectively. 

In this section formulae are given for the algebraic matrix elements of the electrostatic and spin-orbit interactions between antisymmetric functions of various electronic configurations characteristic of the rare earth spectra. All these formulae were constructed in the ySLJM coupling scheme. ... 

Energy-level calculations conducted to first-order perturbation theory and based on the hamiltonian (1.2b), which comprises (only) the electrostatic and spin-orbit interactions, do indeed reproduce the energy level structure qualitatively however, the quantitative agreement obtained between calculated and observed energy levels and other related properties is rather poor, even when all the radial integrals F, G, / and ( are considered as adjustable parameters. 

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. 

Sugar (1965b) used the conventional hamiltonian (including only the electrostatic and spin-orbit interactions within the 4f configuration) and added to it... 

The electrostatic and spin-orbit interactions represent the major effects in the parametric free-ion Hamiltonian. Nevertheless the combined diagonalization of these matrices and fitting of the F and parameters sometimes results in calculated energy levels which are off by 100 cm or more from the experimental values. These deviations result from the neglect of configuration interaction. 

The parameters and Cf correspond to Slater-Condon electrostatic and spin-orbit integrals, respectively, and / and Aso represents matrix elements for the angular parts of these electrostatic and spin-orbit interactions. The integrals as used in the effective-operator approach are different from those used in the Hartree-Fock (HF) approach because they absorb some effects of configuration interaction that are not included in the... 

Energy parameter values have been determined for the three low even configurations (5d+6s) from 31 known levels, first by Childs et al. [19], and later by Traber [31]. Unpublished calculations are also due to Bauche [18] and a systematic study of (5d +6s) configurations in neutral atoms led Wyart [28] to N-expanslon of energy parameters. In all studies, assumptions have been made for reducing the number of free parameters. In [27], two magnetic parameters of crossed second-order effects of electrostatic and spin-orbit interactions are added to the usual ones and the fit between theory and experiment is significantly improved. The fitted parameters are collected in Table 2/16, p. 184. 

Approximate values for the radial integrals associated with the electrostatic and spin-orbit interactions can be obtained from both relativistic and non-... 

Although the electrostatic and spin-orbit interactions are by far the most important terms in the Hamiltonian, other smaller interactions have to be considered in order to get a good agreement between experimental and calculated energy levels. Diagonalization of the energy matrix which incorporates only the electrostatic and spin-orbit interaction, often results in discrepancies between experimental and calculated levels of several hundred cm (Wyboume 1965). 

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