Irreducible full representation

Most texts on the use of group representation theory in physical science list the character tables for the commonly occurring point groups euid the better ones will list the full standard irreducible representation matrices. If one has the full representation matrices it is possible to use these to effect a complete solution of the problem of the formation of symmetry-adapted basis from a given set of basis functions. 

The dimension of a space-group full representation (degeneracy of energy levels in a crystal) for a given k is equal to the product of the number of rays in the star k and the dimension of the point group irreducible representation (ordinary or projective). In particular, for the space group under consideration at the X point the dimensions of full representations are 6 and at the W point - 12. As to each of the... 

As the starting geometries for iterative calculation, we take all the possible structures in which bond lengths are distorted so that the set of displacement vectors may form a basis of an irreducible representation of the full symmetry group of a molecule. For example, with pentalene (I), there are 3, 2, 2 and 2 distinct bond distortions belonging respectively to a, b2 and representations of point group D21,. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

What has been mentioned up to now allows us to infer that the relevant information needed for a representation is given by the characters of its matrices. In fact, the full information for a given group is given by its character table. This table contains the character files of a particular set of representations the irreducible representations. Table 7.2 shows the character table of the Oh point group. A character table, such as Table 7.2, contains the irreducible representations (10 for the Oh group) and their characters, the classes (also 10 for the Oh group), and the set of basis functions. 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

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. 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

The simplest molecules are atoms, which belong to point group %h (often called the full rotation-reflection group). The character table (which we omit) contains irreducible representations of dimensions 1,3,5,... these representations correspond to energy levels with electronic orbital angular-momentum quantum number /=0,1,2,... we have the (2/+1)-fold degeneracy associated with different values of the quantum number... 

Each of these configurations, except d1 and d9, will give rise to terms under the action of HER as a perturbation. The terms, of course, bear the labels of the cubic group, here Oh. The terms which arise are determined qualitatively by the decomposition of the group theoretical direct product of the electrons (or holes) involved into irreducible representations of the cubic groups. A t2s or eR orbital set more than half-full is treated as the equivalent number of holes, and a filled one is ignored. Then, for instance... 

This implies that the fields B(V), B 2, and B 2 are also operators of the full rotation group, and are therefore irreducible representations of the full rotation group. Specifically... 

Usually, an individual VB structure assembled from the localized bonding components does not share the point group symmetry of the molecule anymore. However, the overall VB wavefunction, PVB, should retain the same symmetry properties as the MO wavefunction (in the sense of full Cl, they are in fact identical). Therefore, TVs can be classified by an irreducible representation associated with a given point group. In order to sort vFra by symmetry, a project operator can be introduced as follows ... 

We next seek the irreducible representations of the full rotation group, formed by the infinite number of finite rotations R(ait). Because all such rotations can be expressed in terms of the infinitesimal rotation operators Jx, Jy and Jz (or equivalently J+, J and Jz), we start from these. 

The functions, here occurring in standard order, are our standard basis functions for the real irreducible representations of the full three-dimensional rotation-reflection group, Rg x I, and for its subgroups, Aoft. and Coo . All functions are normalized to 4nj(2l- -1), where / is the azimuthal quantum number. 

The standard real three-dimensional, DW (R()), an)), irreducible representations of the full three-dimensional rotation group 7 3. The real basis functions are listed in Table 1. 

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