Symmetry operations reduction

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

The internal coordinates for the water molecule are chosen as changes in the structural parameters defined in Fig. 3. The effect of each symmetry operation of the symmetry group ( 2 on these internal coordinates is specified in Table 2. Clearly, the internal coordinate Ace is totally symmetric, as the characters xy(Aa) correspond to those given for the irreducible representation (IR) Ai. On die other hand, the characters x/(Ar), as shown, can not be identified with a specific IR. By inspection of Table 2, however, it is apparent that the direct sum Ai B2 corresponds to the correct symmetry of these coordinates. In more complicated cases the magic formula can always be employed to achieve the correct reduction of the representation in question. 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

It is apparent that Dn(Ct) is in block form and since the same block form appears for all the other symmetry operations of the point groupf the T° representation has been reduced by the change to the normal coordinate basis. That such a reduction will always occur, is a point taken up in the next section. Needless to say T0 and Tn have identical characters i.e. °(/l) f°r... 

The order of the group (h, the number of symmetry operations) is 16 (first line of Table 6.18). The reduction formula (6.5) enables us to decompose the four-dimensional representation F into a sum of irreducible representations, The only non-zero values of at are ... 

It is important to note that no symmetry operation exchanges an axial ligands with an equatorial one. As these two types of ligands are therefore non-equivalent, both chemically and according to group theory, they can he considered separately. The characters of the representations Per (eq) and P (ax) are given in Table 6.22. From the reduction formula (6.5), we find ... 

The entry under ojyz) is of differem sign to the corresponding one in 4.13 since this symmetry operation sends each Fp orbital to -p. Reduction of 4.17 either by inspection or using equation 4.15 gives us the result that these two py. orbitals transform as 2 +. ... 

Consider two sets of orbitals, transforming as the irreps Fa and Fi, respectively, each occupied by one electron. A two-electron wavefunction with electron 1 in the Ya component of the first set, and electron 2 in the yj, component of the second set is written as a simple product function FaYai l)) rbYb 2)). Clearly, since the one-electron function spaces are invariants of the group, their product space is invariant, too. Now the question is to determine the symmetry of this new space. The recipe to find this symmetry can safely be based on the character theorem first determine the character string for the product basis, and then carry out the reduction according to the character theorem. Symmetry operators are all-electron operators affecting all particles together hence, the effect of a symmetry operation on a ket product is to transform both kets simultaneously. 

Further reductions may arise from the remaining symmetry operations, such as inversion, which provide good fermion quantum numbers. 

This leaves us with 21 independent components, just as for the quantities C 3 A further reduction of the number of independent components is possible if the invariance of the elastic energy density w [given by (3.123)] against the symmetry operations of the crystal is used. In the case of cubic crystals there are only three independent elastic constants [3.1,19, 21]. 

In this equation, / is the character of the symmetry operation C , and a the rotation angle. For the identity operation /( ) = 27+1, which corresponds to the degeneracy of the term. For the operations a and i the characters are /(cr) = —xiCf) and x i) = -xiE), respectively. Using these formulas it is possible to determine the reducible representation T associated with each term and, upon reduction into its irreducible components, utilising the reduction formula (Equation 1.58), the symmetries of the individual crystal field split terms. 

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