Coefficients of fractional parentag

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... 

When in the valence configuration of the atomic system there are N equivalent electrons expressions (9) and (17) should be multiplied by N and by the appropriate Coefficient of Fractional Parentage (CFP) [10,12], In all the present transitions N has been taken equal to 5, as it is one of the five 3p electrons the one that experiences the transition. The CFP varies with the initial and final states. All the CFP values have been taken from Ref. 12. 

Both these techniques (graphs with arrows and heavy lines) in fact are equivalent and they were developed to sum up 3nj- and ym-coefficients. There have been attempts to generalize them to cover the cases of summing up 3ny-coefficients and coefficients of fractional parentage or even to calculate matrix elements, but without great success [8, 13, 17, 88]. Some refinements of graphical technique, particularly efficient for large n values of 3ny-coefficients, are presented in [89]. 

The efficient way of constructing the wave function of the states of equivalent electrons permitted by the Pauli exclusion principle is by utilization of the methods of the coefficients of fractional parentage (CFP). The antisymmetric wave function xp(lNolLSMlMs) of a shell nlN is constructed in a recurrent way starting with the antisymmetric wave function of N— 1 electrons xp(lN lociLiSiMLlMsl). Let us construct the following wave function of coupled momenta ... 

The quantities ( ) are called the coefficients of fractional parentage (CFP) with one detached electron. They ensure the antisymmetry of the wave function and the occurrence only of states permitted by the Pauli principle. 

Coefficients of fractional parentage play a fundamental role in the theory of many-electron atoms. There are algebraic formulas for them (see Chapter 16), however, they are not very convenient for practical utilization, and normally tables of their numerical values are used. They can be generated in the recursive way, starting with the formula... 

As was already mentioned, due to the Pauli exclusion principle, which states that no two electrons can have the same wave functions, a wave function of an atom must be antisymmetric upon interchange of any two electron coordinates. For a shell of equivalent electrons this requirement is satisfied with the help of the usual coefficients of fractional parentage. However, for non-equivalent electrons the antisymmetrization procedure is different. If we have N non-equivalent electrons, then a wave function that is antisymmetric upon interchange of any two electron coordinates can be formed by taking the following linear combination of products of one-electron functions [16] ... 

Whether the quantum numbers L,S in (10.11) are coupled or not, this function is not antisymmetric with respect to exchange of pairs of electron coordinates between shells, e.g. the coordinates and r +j. The necessary additional antisymmetrization can be accomplished through use of the generalized coefficients of fractional parentage [14] or of modification of the coordinate permutation scheme employed with one-electron functions in (10.8). 

Submatrix elements of creation and annihilation operators. Coefficients of fractional parentage... 

These submatrix elements are exceedingly important for atomic theory, since they are proportional to the usual coefficients of fractional parentage ( (aiLiSi H/ aLS). In order to establish the relation between the coefficients of fractional parentage and the submatrix elements of creation operators we shall consider the irreducible tensorial product... 

A similar treatment is possible, if we take into account (16.30), for the coefficients of fractional parentage with two detached electrons. Specifically, for an odd L2 we have... 

CFP (9.11) also have a simple algebraic form. In the previous paragraph we discussed the behaviour of coefficients of fractional parentage in quasispin space and their symmetry under transposition of spin and quasispin quantum numbers. The use of these properties allows one, from a single CFP, to find pertinent quantities in the interval of occupation numbers for a given shell for which a given state exists [92]. 

The term LjS can be chosen in an arbitrary manner, and the normalization factor is found from the normalization"condition for coefficients of fractional parentage at fixed momenta L SJ and L,S. Equation (16.66) holds for repeating terms that are uniquely classified by the seniority quantum number v, but for non-repeating terms (when 5(L2S2,LS) = 0) that equation becomes the conventional Redmond formula [109]. 

Let us consider approximate selection rules for the case of LS coupling. Similar examples for other coupling schemes will be presented in the next two sections together with the corresponding expressions for the submatrix elements of the respective transition operators in terms of the coefficients of fractional parentage, 3 j-coefficients and one-electron submatrix elements. The dependence of each submatrix element of Ek-... 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

Table 48 Coefficients of fractional parentage for dn configurations G lLlSl [35,40,41]...

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