Irreducible Components in

Theorem 5.10. There is exactly one in — 1)-dimensional irreducible component in... 

To find the symmetry species of the molecular orbitals, we shall construct the representation generated by the six valence orbitals, and reduce it into irreducible components. In the following table, each entry gives the result of applying the operation heading the column, to the orbital at the left of the row. 

These facts suggest that the excess absorption is due to irreducible, ternary dipole components. We mention that classical estimates of the long-range, irreducible component in hydrogen (where quadrupolar induction prevails) have suggested a weak reduction of the absorption [402]. The observed excess absorption is, therefore, believed to arise from a different, non-classical mechanism. Overlap induction seems to be the most... 

Reduction of (Eq. 3.1) yields an u, and a b2 as the irreducible components. In order for atomic orbitals on nitrogen and oxygen to be suitable for linear combination into... 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

Assume moreover that the closed subschemes V(a ) defined by a have no irreducible component in common and are reduced in their maximal points. Then... 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

The product representation D XI/ is in general reducible. It can be decomposed into its irreducible components... 

The coupling tensor Rlm in the laboratory frame is time dependent due to the motions of spin-bearing molecules. It can be expressed in terms of the rotational transformation of the corresponding irreducible components pln in the principal axis system (PAS) to the laboratory frame by... 

For Z may be zero, but need not be. It follows that we can generate from j > at most c vectors transforming like the fc th basis vector of fM. When the operations of are applied to these, each may separately generate the representation I M, but it cannot be generated more times than this. Thus, the representation fW appears in T at most the same number of times that y appears in the subduced representation of r( ), and its decomposition into irreducible components is... 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

Examples for non-totally-symmetric components in the decomposition of density matrix into irreducible tensor components are the one-particle spin density matrices ... 

In 2 we determine how the fibres of the components of Ag,d over the finite primes split up into irreducible components. It turns out that this question is equivalent to a question in the theory of finite abelian groups (Proposition 2.3). 

In 3 we continue the study of the stratification by p-rank of Ag,d 0FP which was started in [NO]. The locus Z = Vg-i Vg 2 C A5,d Fp corresponding to abelian varieties of p-rank exactly g 1 is decomposed into irreducible components. Finally we determine which components of Z are contained in a fixed component of Ag, Fp again this turns out to be equivalent to a question in the theory of abelian groups (Proposition 3.4). 

We have seen above that the pairwise-additive parts of the ternary induced dipole component are well known for a number of systems. The irreducible, ternary components, on the other hand, are poorly known, for any system. Even less is known about the N-body irreducible components of the induced dipole for N > 3. Attempts exist to model three- and four-body dipole components on the basis of classical relationships, but the evidence indicates that overlap effects are important the known many-body dipole models do in general not account for quantum effects. 

In order to estimate the strength of the ternary, irreducible dipole components, Guillot et al. used the one effective electron model [171, 173]. For H2-He-He at liquid state densities, the (admittedly crude) model suggests significant enhancement of the absorption due to overlap-induced, irreducible dipole components in molecular dynamics studies, in qualitative agreement with the above conclusion. 

We have neglected the (weak) dipole component arising from three identical atoms, i.e., paaa and pBBB 0 and the irreducible components of order greater than 3. The dynamic functions G play the same role as the analogous static functions in the expressions for the spectral moments. 

At this point, the analogy with the static case is complete and one may take advantage of the results obtained for the static case. Interesting quantal expressions for the dynamic pair and triplet distribution functions have been communicated [297] the latter may be separated into a pairwise additive and an irreducible component, as in the static case. 

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