Local symmetry operations

The number of multipole parameters is reduced by the requirements of symmetry. As discussed in chapter 3, the only allowed multipolar functions are those having the symmetry of the site, which are invariant under the local symmetry operations. For example, only / = even multipoles can have nonzero populations on a centrosymmetric site, while for sites with axial symmetry the dipoles must be oriented along the symmetry axis. For a highly symmetric site having 6 mm symmetry, the lowest allowed / 0 is d66+ all lower multipoles being forbidden by the symmetry. The index-picking rules listed in appendix D give the information required for selection of the allowed parameters. 

At first order, it can be shown that only the symmetrized part of the interaction tensor contributes to the frequency shift. The majority of second-order contributions arise from large EFG, the EFG tensor being symmetric by definition. Thus, only symmetric second-rank tensors T can be considered, which can be decomposed into two contributions T = isol3 + AT with D3 the identity matrix. The first term is the isotropic part so = l/3Tr(T) that is invariant by any local symmetry operation. The second term is the anisotropic contribution AT, a symmetric second-rank traceless tensor, which depends then on five parameters the anisotropy 8 and the asymmetry parameter rj that measures the deviation from axial symmetry, and three angles to orient the principal axes system (PAS) in the crystal frame. The most common convention orders the eigenvalues of AT such that IAzzL and defines 8 — Xzz and rj — fzvv i )... 

These a-p operations are coincidence operations (one-way movement) and not local symmetry operations. Symbols for the homo-octahedral polytypes homomorphic to the meso-octahedral polytypes listed in this table. 

We can invent such local symmetry operations by using the fact that perms may have zeroes originally introduced to circumvent the problem of linear combinations. In the case of H2O the original perms array may be augmented to include the local symmetries of the oxygen atom. Since real basis functions are being used, this means C4 rotations, not Coo- Here is the augmented array, with appropriate zeroes. 

Additional symmetry conditions may require some 5 to cancel since Hqf must be invariant under local symmetry operators. For instance, if the local symmetry contains a fourfold axis, then the only non-zero values are 5g, B, B, B and B for =3. The crystal field depends therefore on only five parameters. 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

In essence, jSel is the interaction between the electronic wave functions of A and B. Obviously there must be some spatial overlap between the two in order to give rise to a finite value. Furthermore, if 0A. and />B fall in the same symmetry class or, in complex molecules, have similar local symmetry, the value of j8 may be relatively large. This will depend upon whether or not the perturbation operators, H, have preferred symmetry properties. The Woodward-Hoffman rules suggest that these operators can... 

Bent AH2 Molecules.—A bent AH molecule belongs to the symmetry class C2r. The definitions of the symbols appropriate to the non-localized orbitals of such a molecule are given below. The z axis bisects the HAH angle and lies in the molecular plane. The y axis also lies in the molecular plane and is parallel with the H H line. C2(z) means a rotation by 180° about the z axis. wave function does not or does change sign when one of the symmetry- operations C2(z) or av(y) is carried out. 

Figure 11.8 reproduces the important molecular orbitals and classifies them according to their symmetry with respect to the C2 axis, the element that defines the local symmetry during the conrotatory process. Orbital nlr antisymmetric under C2, must change continuously into an orbital of the product in such a way as to remain at all stages antisymmetric under the C2 operation. 

To find the symmetry of the normal modes we study the transformation of the atomic displacements xL y, z,, i 0,1,2,3, by setting up a local basis set e,i ci2 el3 on each of the four atoms. A sufficient number of these basis vectors are shown in Figure 9.1. The point group of this molecule is D3h and the character table for D3h is in Appendix A3. In Table 9.1 we give the classes of D3h a particular member R of each class the number of atoms NR that are invariant under any symmetry operator in that class the 3x3 sub-matrix r, (R) for the basis (e,i el2 cl31 (which is a 3 x 3 block of the complete matrix representative for the basis (eoi. .. e331) the characters for the representation T, and the characters for... 

As for the case of the two-leg ladder, we select the Local Configurations (LC) which are the much important for the ground state wave function. This selection of the relevant LC is made by combining energetic and symmetry considerations. For that purpose, we use the electron-hole symmetry operator, J, to classify the LC. On the Fock space of a single site n, the action of this operator are summarized as follows [40]... 

In Table 4 we summarize some global properties of the obtained JT minima for all charge and spin states. In these multi-mode JT systems, the local symmetry of an optimal distortion is described in terms of the subgroup GiOCai of symmetry operations which leave that minimum invariant. We remind that the minima in the... 

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