Fractional parentage

Cox PA (1975) Fractional Parentage Methods for Ionisation of Open shells of d and f Electrons. 24 59-81... 

In Nature, however, we always have a contiiinous distribution of particles. This means that we have all sizes, even those of fractional parentage, i.e.-18.56n, 18.57p, 18.58 p, etc. (supposing that we can measure 0.01 p differences). The reason for this is that the mecheuiisms for particle formation, i.e.- precipitation, embryo and nucleation growth, Ostwald ripening, and sintering, are random processes. Thus, while we may speak of the "statistical variation of diameters", and while we use whole numbers for the particle diameters, the actuality is that the diameters are fractional in nature. Very few particle-size" specialists seem to recognize this fact. Since the processes are random in nature, we can use statistics to describe the... 

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. 

If the open-shell is ionized, the relative probabilities of producing different ionic states will reflect the fractional parentage coefficients (44, 155, 156), which may, but will not in general, be proportional to spin-orbital degeneracies. 

If a molecule contains two or more open-shells, it is necessary to consider the coupling that already exists between the different open-shells in the molecule. The probability of ionization is usually expressed in terms of Racah coefficients and fractional parentage coefficients. 

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... 

Table 9.3. Numerical values of fractional parentage coefficients with one detached electron for j <1/2...

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... 

We refer to the last factor in this expression as the reduced coefficient (or subcoefficient) of fractional parentage (SCFP) [91]. It follows from (16.16) that these coefficients can be expressed fairly simply in terms of CFP for the term with N = v that occur for the first time ... 

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]. 

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