Parentage coefficients

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. 

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

As before [A] indicates orbital degeneracy Eq. (21) also involves W or Racah coefficients for spin and orbital coupling. Values for the spin coefficients may be calculated from formulae quoted by Brink and Satchler (17). Frequently, however, it may not be necessary to use this formula, since it may be clear from the selection rules that a given function (S5A5) of the ionised shell can only produce one allowed resultant state (S2 A 2), and in this case the intensity is entirely determined by the fractional parentage coefficient for the ionised shell ... 

Table 8 contains the values of ion-atom exchange interaction potentials for different atom fine states. Thus, the parentage coefficients are the same in this case as in the halogen case, and the fine structure parameter are identical to the case of positive oxygen atom. Because formula (22) contains the parameters of the atomic core, the exchange interaction potentials of Table 6 and 7 are different. 

If a reduced matrix element between many-electron states is to be expressed in terms of those of single-electron states, we have to introduce the concept of fractional parentage coefficients pioneered by Racah. Collecting the general parts of a reduced matrix element between states with n equivalent electrons, we are led to the definition of a Racah tensor and, in this case, the fractional parentage coefficients will appear explicitly only in this definition ... 

Fig. 14. Photoemission spectra of oxidized (YbjOj) and metallic Yb (after Johansson et al. 19801 The vertical bars represent fractional parentage coefficients of f and f ground states (see fig. 8e), the arrows mark the states recorded as a function of photon energy for obtaining the Fano profile shown in fig. 15.

Fig. 29. XPS spectrum of TmSe analyzed with the intensities of the f" and f final state multiplets (vertical bars) predicted by the fractional parentage coefficients (see fig. 8d,e) (after Campagna et al. 1976>...

We expanded the X operators into the primitive products of creation and annihilation operators to achieve this factorization. Instead of first expanding the x in terms of X, we can expand them directly in the primitive products of creation and annihilation operators, to obtain the sums of products of parentage coefficients. As an example, consider the second off-diagonal block, for which the two-particle density matrix factorizes naturally ... 

The coefficients in this expression are grandparentage coefficients, and could be stored instead of using the parentage coefficients, at a cost of 2n fM. The contribution... 

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