Character representations

Products between the irreducible representation characters within a group will produce representations which are often reducible. A simple calculation can decompose this product to a sum of the irreducible representation characters, as is demonstrated in Table V for two representations from the S3-DP-S2 group. 

Proposition 6.11 has many applications. One is the fact that a character completely determines a representation. Compared to representations, characters are relatively simple objects — complex-valued functions on the group. Yet they carry all the information about the representation T. 

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

Table of symmetry elements, irreducible representations, characters of the individual irreducible representations and assignments of vector and tensor properties for a given point group. 

Now let us turn to consider the special case of the characteristic phenomenal yellow. Is it possible to identify phenomenal yellow with some other characteristic, say, C If we are to do so, there must be a way of explaining how it is possible to grasp phenomenal yellow experientially without appreciating its identity with C. This means that we must invoke an appearance/reality distinction of some sort. But folk psychology does not recognize a distinction between appearance and reality in this case. It fails to register the representational character of our awareness of phenomenal yellow, and by the same token, it fails to support any ambitions that we might have to identify phenomenal yellow with another characteristic. 

SOME PROPERTIES OF GROUP REPRESENTATIONS CHARACTER TABLES AND CORRELATION TABLES... 

Some Properties of Group Representations Character Tables and Coi... 

The classes of all factor systems and the corresponding projective representations characters can be found in [27] for all 32 crystaUographic point groups. Ten groups Cl, S2,Cs,C2, Se,Ce,Csh) are cyclic, only one factor system Kq belongs... 

For the Csv group, the characters for some direct products of bases for irreducible representations are shown in Table 13-25. The direct product has as characters the product of characters of E times itself. These characters (4,0,1) do not agree with any of the irreducible representation character sets, and so E E is reducible. We can tell, in fact, that E E is four-dimensional from the leading character. To resolve EiS>E,we employ the formula (13-7), which gives E E = Ai A2 E, and fits the observation that E E is four-dimensional. The other direct products listed... 

The C2V group multiplication table was constructed from the group operations in Problem 2.1 and the result is reproduced in Table 4.4a. The same table using the Bi representation characters is given in Table 4.4b. In this representation version, the operation symbols for the row and column headings are replaced by the corresponding Bi characters. The result of a product of operations then becomes a simple numerical multiplication of... 

Table 4.10 Subtraction of the irreducible representation characters from F for the y, z basis on the central Ni atom of the complex as [Ni(CN)4p. ...

Reduce the representation. The sums formed in the application of the reduction formula are given in Table 6.6. In this table the symmetry classes that have reducible representation characters of 0 are omitted as they will not contribute to the totals in any irreducible representation. The order of the Da point group h = 8, so that the reduction process yields... 

These functions which contmn the position of the atomic centre of mass as a parameter, yield a correct description of the states far a very large distance between tile two atoms, and the number of possibilities appearing for A already yields an exhaustive view of the totality of the reaction possibilities that can occur. Although for increasing dynamical coupling the structure of the atoms is perturbed as well, the representation character of the eigenfunction remains unaltered. 

Less complete information than the representations of non-Abelian groups are the characters, which are the traces of the representation matrices. (For Abelian groups, with their 1x1 matrices, the characters are the same as the representations.) Characters can be used in the projection formula but only to distinguish a result of zero from a nonzero result, that is, to distinguish whether the function has any part that transforms as the representation or not. Character tables provide this concise information. 

Structurally benzene is the simplest stable compound having aromatic character, but a satisfactory graphical representation of its formula proved to be a perplexing problem for chemists. Kekule is usually credited with description of two resonating structures which. 

The traces of the representation matrices are called the characters of the representation, and (equation Al.4.36) shows that all equivalent representations have the same characters. Thus, the characters serve to distingiush inequivalent representations. 

The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

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... 

We can therefore calculate the character, under a synnnetry operation R, in the representation generated by the of two sets of fiinctions, by multiplying together the characters under R in the representations generated by eac... 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The ROSDAL (Representation of Organic Structures Description Arranged Linearly) syntax was developed by S. Welford, J. Barnard, and M.F. Lynch in 1985 for the Beilstein Institute. This line notation was intended to transmit structural information between the user and the Beilstein DIALOG system (Beilstein-Ohlme) during database retrieval queries and structure displays. This exchange of structure information by the ROSDAL ASCII character string is very fast. 

A year later, a novel method of encoding chemical structures via typewriter input (punched paper tape) was described by Feldmann [42]. The constructed typewriter had a special character set and recorded on the paper tape the character struck and the position (coordinates) of the character on the page. These input data made it possible to produce tabular representations of the structure. 

Stereoisomerism at double bonds is indicated in SMILES by / and . The characters specify the relative direction of the connected atoms at a double bond and act as a frame. The characters frame the atoms of a double bond in a parallel or an opposite direction. It is therefore only reasonable to use them on both sides Figure 2-78). There are other valid representations of cis/trans isomers, because the characters can be written in different ways. Further details are listed in Section 2,3.3, in the Handbook or in Ref, [22]. 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

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