Third-Order Electron Correlation Effective Operators

1 Third-Order Electron Correlation Effective Operators 

The two-particle nature of Coulomb interaction in equation (10.27) is the reason that among the third-order contributions to the transition amplitude, in addition to one particle effective operators (as in the standard J-O approach), two particle objects are also present. However, the numerical analysis based on ab initio calculations performed for all lanthanide ions, applying the radial integrals evaluated for complete radial basis sets (due to perturbed function approach), demonstrated that the contributions due to two-particle effective operators are relatively negligible [11,44-58]. This is why here they are not presented in an explicit tensorial form (see for example Chapter 17 in [13]). At the same time it should be pointed out that two-particle effective operators, as the only non-vanishing terms, play an important role in determining the amplitude of transitions that are forbidden by the selection rules of second- and the third-order approaches. This is the only possibility, at least within the non-relativistic model, to describe the so-called special transitions like, 0 — 0 in Eu +, for example, as discussed above. 

Finally, the one particle effective operators that are defined up to the third order have the following form (based on the static model, as the original J-O Theory), 

Rjoi ) represents the standard Judd-Ofelt theory (see 10.22), and the third-order radial terms  

Finally it should be concluded that one-particle parametrization scheme of the standard J-O Theory is preserved at the third order. The limitations caused by the original derivation based on the single configuration approximation are compensated in a perturbative way by the third-order electron correlation contfibutions analyzed here. 

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The extension of the relativistic model by the third-order contributions is rather straightforward. However, the expressions for such new terms are more complex than those of the non-relativistic approach, since the closure procedure has to be performed twice, for the spin and orbital parts of three inter-shell double tensor operators. When the electron correlation effects are taken into account, again at the third-order two particle effective operators are expected as originating from the Coulomb interaction. The third-order relativistic model of the Judd-Ofelt theory is discussed in detail in Chapter 18 of Wybourne and Smentek [13]. 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

L. Smentek-Mielczarek, Electron correlation third-order contributions to the electric dipole transition amplitudes of rare earth ions in crystals II. A general effective operator formulation based on static and dynamic models. Molecular Physics, 61,161-11A (1987). 

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