Irreducible symmetries

This reducible representation (the occupancy of two e orbitals in the anion gives rise to more than one state, so the direct product e x e contains more than one symmetry component) can be decomposed into pure symmetry components (labels T are used to denote the irreducible symmetries) by using the decomposition formula given in Appendix E ... 

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... 

For all the calculator files on the CDROM, this display comprises the standard character table for the particular group and identifies the irreducible symmetries of translations and rotations about the origin of the coordinate system for molecular stmctures with this point symmetry. 

For all the spreadsheet files of the GT Calculator the convention applied is to mark the non-zero components of a direct sum decomposition of a permutation character by green background shading. For the regular character in the example of Figure 1.8, all the irreducible symmetries contribute and so all the direct sum cells exhibit green shadings. 

For example, the 5Hu command button on the keypad leads to the display. Figure 1.26, of the 5 copy of five polynomial functions, which, on the unit sphere, are mutually orthonormal to one another and to the 20 other polynomial functions of this irreducible symmetry forming the F , 2", and 4 sets of functions also of this symmetry, displayed when the other buttons on the keypad labelled with this symmetry are selected by a mouse click. The third function of 5hu irreducible symmetry in Figure 1.26 has leading polynomial terms... 

With reference to Figure 3.1, we have defined at each position, Pj, a local coordinate system j), e(i), cr( j) is the unit vector towards the central origin on the unit sphere while 7te ]) points south and Tt ]) points to the east. This local coordinate system provides for the construction of local functions of a, n and 5 orientations upon which group orbitals of these irreducible symmetries can be formed as linear combinations exhibiting angular momentum components k = 0, 1 and 2 about the radial vectors to each vertex of the structure orbit. 

Thus, for any central function, Yim(0, < ), the a group orbital of the corresponding irreducible symmetry in the F over a particular structure orbit of the point group G is the linear combination of the a-oriented local functions with the coefficients modulated by the magnitude of the central function at the vertices of the orbit, equations 3.20. 

Table 3.11 lists the real forms for the U and spherical X = 0) and vector (X = 1 ) harmonics for the 1 = 0 to 3 central harmonic functions. Group orbitals of the particular irreducible symmetries for which these central functions provide bases follow simply by making linear combinations of the u and vj at each vertex point modulated by the values of Uj and to form local resultants, which interconvert from vertex to vertex of the orbit under the actions of the symmetry operations of the point group. 

Figure 3.15 Projections identifying the group orbitals and their irreducible symmetries for local o -orbital decoration of the vertices of the Oi2v orbit, Table 3.15, of a molecular structure with point symmetry. The icons "A", and are applied as in Figure 3.8.

However, now the direct sum of the permutation character. Table 3.15, requires the identification of two sets of group orbitals of e and e" irreducible symmetries and we cannot expect that the simple superposition procedure sampling very few points on the unit sphere amplitude of the central function will return mutually orthogonal linear combinations. These... 

The listing for the Oi2h orbit permutation character in Table 3.15 presents a similar problem, since two sets of group orbitals of Eiu and E2g irreducible symmetries have to be identified. However, this example identifies an unusual case, because of the large number of orbit vertices in the plane. As can be seen in Figure 3.19, it is necessary to find central functions up to level... 

For the Oi2h orbit, the analysis of Figure 3.19 up to level f = 4 harmonics is not sufficient to identify a second distinct group orbital of Eiu symmetry. However, we know that there is a level 4 component of E2g irreducible symmetry and that level 5 components can be generated using the products of the form z(x iy)i. Since z transforms as A2u, level 5 harmonics of the required symmetry [2eiu] are likely to be found and this is the case for the extended superposition analysis of Figure 3.19. 

Solution of the ag block of the Hiickel determinant for Cso leads to two linear combinations of ag irreducible symmetry. The first of these is the linear combination resulting from the projection of the lag polynomial of Table 3.20 on the 80 vertices, so that each coefficient is of equal value, while the second linear combination is identical to the one found by projecting the level 6 polynomial of the table onto the 80 vertices and rendering the result orthogonal to the first. 

Table 3.20 Central polynomials of Ag irreducible symmetry for tbe Ih point group.

The deficiency is rectified in Figure 3.24 by taking suitable sums and differences of the motions of Figure 3.23. For molecular motions these transformations correspond to the orthog-onalization of the like symmetry linear combinations, which occur as repetitions. Note that this particular example is an especially favourable one, as it is a single-orbit problem and is also one where the vibrational character contains not more than one copy of any irreducible symmetry, so the forms of all the vibrational modes are entirely determined by symmetry. In a more general case, symmetry considerations provide a basis for the normal modes rather than the modes themselves. 

Figure A2.5 displays the results of carrying out the similarity transform of the final five orthonormal functions of hu irreducible symmetry on the Htickel Hamiltonian for the decorated regular orbit cage. Given the accuracy level applied in the calculations, with no precautions to ensure that notoriously unreliable functions are being calculated appropriately, the 3 figure accuracy in the overall diagonalization is acceptable.

SALCs are nonnalized to unity, neglecting overlap between the sites. The matrix T transforms the locaUzed orbitals on the sites to delocalized molecular orbitals with irreducible symmetry characteristics. The inverse matrix fiiUills the opposite role and localizes the molecular orbital set back on the atomic sites. 

The resulting RHF MOs may belong (and most often do) to the irreducible symmetry representations (Appendix C in p. 903) of the Hamiltonian. But this is not necessarily the case. 

In many cases the A-B system is centrosymmetric, i.e. there is at least one symmetry operation that transforms A into B and vice versa. Then 0i and 02 belong to different irreducible symmetry representations, and consequently the wave func-... 

In the second part (Sections 6.4-6.7) we will consider what the vibrations belonging to a given irreducible representation look like . This involves the construction of linear combinations of the basis of atomic movements that are consistent with the characters of the irreducible representation. These combinations are known as symmetry adapted linear combinations (SALCs). SALCs are a general way to visualize the molecular properties that correspond to the objects with the irreducible symmetries identified by the reduction formula. The method we shall use to obtain these SALCs is the projection operator, which is introduced in Section 6.6 and will also be employed in Chapter 7 to find molecular orbitals. 

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