Selection and sum rules

Conditions of non-zero values of the submatrix elements of the electron transition operators define the selection rules for radiation. The latter coincide with those non-zero conditions for the quantity Q. On the other hand, the selection rules for Q are defined by the conditions of polygons for 3nj-coefficients, in terms of which they are expressed. The requirement (24.21) must also be kept in mind. The selection rules for transitions (25.8)-(25.17) are summarized in Table 25.1. In all cases the selection rules JJ k, hhk and l2 +12 + k is even number are valid. The table contains only those polygons which have the quantum numbers of both configurations, because only in such a case do these conditions serve as the selection rules for radiation. If a certain quantum number has no restrictions from this point of view, this means that it does not form a polygon with quantum numbers of the other configuration. Such quantities are placed in curly brackets. 

The most complicated selection rules - the quadrangle conditions -occur for transitions between configurations where the energy levels of 

We have in mind that the condition S(aiLiSu /iLiS[) is included in (25.18)-(25.21). It is possible to establish many other sum rules with respect to L, L, etc., however, they are of little importance. Analogous 

The submatrix element of non-relativistic fc-transitions between the levels of a shell of equivalent electrons is as follows  

Actually, only transitions described by one term in (25.23), take place. Formulas (25.22) and (25.23) are valid for the first and second forms of the Efc-transition operator (formulas (4.12) and (4.13)). The corresponding one-electron submatrix elements are given by (25.5) and (25.6). Analogous expressions for the third form of the /c-radiation operator are established in [77]. The appropriate selection and sum rules may be found in a similar way as was done for transitions between different configurations. It is interesting to mention that the non-zero conditions for submatrix elements for the operator Uk with regard to a seniority quantum number suggest new selection rules for the transitions in the shell of equivalent electrons v = v at odd and v = v, v 2 at even k values. 

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In conclusion of this section let us briefly discuss the selection and sum rules for Efc-transitions with the participation of core electrons. 

The Bethe Sum Rule and Basis Set Selection in the Calculation of Generalized Oscillator Strengths... 

Figure 3. Partial sum rule shown for underdoped (a) and (b), and overdoped (c) samples, for selected cutoff frequencies. Full symbols represent the spectral weight, integrated from 0+, hence without the superfluid contribution. Open symbols include (below Tc) the superfluid weight. Fig3-b and -c represent the intraband spectral weight, hence —Ek, as a function of temperature. The dotted lines are 12 best fits to the normal state data.

The work in Ref. [3] also includes sum rules for the energy levels and selection rules for the rotation spectra, as well as some properties of the most asymmetric case. The latter is reviewed in the following section. 

If the selected space group of the starting structure has high symmetry, the number of free parameters will be smaller than the number of constraints and it is not possible for all the predicted distances to be realized. If the deviations are small, the structure may be stable, but if they are large, the structure will relax or, in extreme cases, be so unstable that it cannot be prepared. Relaxation may involve only a small adjustment to the bond lengths so that the valence sum rule continues to be obeyed (at the expense of the equal valence rule), or it may involve a reduction in the symmetry as is found in the case of CaCrF5 described above. If the symmetry is reduced, the number of free variables is increased and the atomic coordinates may be under-determined. In this case, the constraints on the sizes of non-bonding distances become important. 

A precise theoretical and experimental determination of polarizability would provide an important probe of the electronic structure of clusters, as a is very sensitive to the presence of low-energy optical excitations. Accurate experimental data for a wide range of size-selected clusters are available only for sodium, potassium [104] and aluminum [105, 106]. Theoretical predictions based on DFT and realistic models do not cover even this limited sample of experimental data. The reason for this scarcity is that the evaluation of polarizability by the sum rule (46) requires the preliminary computation of S(co), which, with the exception of Ref. [101], is available only for idealized models. Two additional routes exist to the evaluation of a, in close analogy with the computation of vibrational properties static second-order perturbation theory and finite differences [107]. Again, the first approach has been used exclusively for the spherical jellium model. In this case, the equations to be solved are very similar to those introduced in Ref. [108] for the computation of atomic polarizabilities. Applications of this formalism to simple metal clusters are reported, for instance, in Ref. [109]. 

It is of course necessary to employ a correlation diagram in order to select the frequencies of the more symmetrical molecules over which the summation is to be extended. Such a diagram, appropriate for the benzene superposition indicated in (19), is given in Fig. 8-2. This shows that four separate sum rules apply, corresponding to symmetry coordinates of species A , B , A,c, and under the common group, 62 - In particular, the first sums must be extended over the A and Rf, species of C6H4D2, and over the Ai , A, and species (the latter being counted... 

W,j is the integral taken over the angular variables in (1.2) and summed over m, m N f e) is the partial density of band states. The weight factor W, i, determines the selection rules it is other than zero only when / = Taking into account the values of IV,we have (e = + e ") for X-spectrum... 

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