Doubly reduced matrix element

The last term on the right in Equation 1.22 represents a doubly reduced matrix element, which can be calculated by recursive formula in terms of the coefficients of fractional parentage [4, 14], tabulated in the work of Nielson and Koster [27]. Finally, Equation 1.18 is rewritten as... 

The 3j and 6j symbols as well as the doubly reduced matrix elements needed in the calculation of crystalline field energies have been programmed for us by Gelio Alves and Chamindra Williams at Florida Atlantic University. 

The doubly reduced matrix elements of U K] are tabulated by Nielson and Koster [318], The necessary matrix elements needed here are given in Table 8.40. 

Table 13 Doubly reduced matrix elements used in the calculations of the dipole strengths for...

The doubly reduced matrix elements (/ t5Z / r SZ ) have been tabulated by... 

Sjj are the Kronecker delta symbols, for which Sjj = 0 for i j and Sjj = 1 for i = j. a stands for all additional quantum numbers which describe the initial and hnal states of T. The doubly reduced matrix elements (raLS E Z a L 5 ), containing the spin-orbit operator are tabulated [34]. The term between curly brackets is the six-j symbol, which describes the coupling of three momenta, in this case L, S and J. Online calculators are available to determine these, or they are tabulated [35]. Erom the 6-j symbol selection rules arise, as it is only non-zero when ... 

The doubly reduced matrix elements(i j 11/ y/fsc) are specific to a given lanthanide ion. It was shown that the crystal field does not affect them substantially. Therefore, instead of ealeulating them for each system, the values tabulated by Nielson and Koster can be used [34]. 

Luminescence of Lanthanide Ions In Coordination Compounds and Nanomaterials Table 1.9 Selected doubly reduced matrix elements yfrsrWu Wv rs L ) for the f Pr ion [34 ... 

Not only does the frozen core approximation reduce the number of configurations, but it also reduces the computational effort required to evaluate matrix elements between the configurations which remain. Assuming that all frozen core orbitals are doubly occupied and orthogonal to all other molecular orbitals, it can be shown126 that... 

The problem of determining the sign before 5 E is reduced to one of finding the signs of the eigenvalues of a certain Hermitian matrix A [32] whose elements are determined from the known formulas for matrix elements of the HF Hamiltonian between the singly excited states as well as between the ground and the doubly excited states [33, 34]. 

ICP-AES was validated for the simultaneous determination of Al, B, Ba, Be, Cd, Co, Cr, Cu, Fe, Li, Mn, Ni, Pb, Se, Sr and Zn in human serum in a clinical laboratory. The samples underwent digestion and yttrium was used as an internal standard. The LOD were as follows 0.002-0.003 (xM for Ba, Cd, Mn and Sr 0.014-0.07 (xM for Be, Co, Cr, Cu, Fe, Li, Ni, Pb and Zn and 0.2-0.9 (xM for Al, B and Se. The concentrations of Al, Be and Co in human serum were found to be above the LOD, while those of Cd, Cr, Ni and Pb were below the LOQ however, in case of acute intoxication with the latter elements the method is valid . Matrix effects were evaluated for ICP-AES analysis using solution nebulization and laser ablation (LA) techniques. The main matrix-related interferences stem from elements with a low second ionization potential however, these are drastically reduced when pure He is used as carrier gas. This points to Ar (the usual carrier) participation in the interference mechanism, probably by interacting with doubly charged species. ... 

---