Group, mathematical characters

An introduction to the mathematics of group theory for the non-mathematician. If you want to learn formal group theory but are uncomfortable with much of the mathematical literature, this book deserves your consideration. It does not treat matrix representations of groups or character tables in any significant detail, however. 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

The relative importance of a and r contributions to the overall bonding is unclear, but several different combinations of relative strengths lead to limiting case models. When there are 2 electrons in the forward (T-bond and 2 electrons in the ir-backbond, there are 2 bonding electrons for each metal-carbon bond. This is mathematically equivalent to 2tr-bonds and a metallocyclopropane structure (72). This model does not necessitate strict sp3 hybridization at the carbon atoms. Molecular orbital calculations for cyclopropane (15) indicate that the C—C bonds have higher carbon atom p character than do the C—H bonds. Thus, the metallocyclopropane model allows it interactions with substituent groups on the olefin (68). 

In the next chapter, we will present various chemical applications of group theory, including molecular orbital and hybridization theories, spectroscopic selection rules, and molecular vibrations. Before proceeding to these topics, we first need to introduce the character tables of symmetry groups. It should be emphasized that the following treatment is in no way mathematically rigorous. Rather, the presentation is example- and application-oriented. 

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

The sum of the diagonal elements of a matrix is called the character (%) of the matrix. Hereafter, we use the term character rather than the representation since there is a one-to-one correspondence between them and since mathematical manipulation with x is simpler than with the representation. The characters of the reducible representations for the E, and o operations are 3, 0 and 1, respectively. The characters for C3 (counterclockwise rotation by 120°) is the same as that of C, and those for cr2 and cr3 are the same as that of crj. By grouping symmetry operations of the same character ( class ), we obtain... 

This partially oxidized material contains 4.2-5.4 mmol g of carboxylic acid functional groups, which account in part for its generally acidic character, its charge distribution as a function of pH, its complexation of metal ions, and its interaction with mineral surfaces in the environment. Phenolic hydroxyl groups are present at concentration of —1.5-1.8 mmol g in FAs, HAs, and NOM. The ratio of carboxyl-to-phenolic groups is thus —3 1, which is significantly greater than the ratio of 2 1 that is commonly used in some mathematical models of the acidity of these materials. 

Counting the isomers arising by addition to, or substitution in, a basic framework is a mathematical problem with many practical applications in chemistry. In classical organic chemistry, for example, the number of derivatives of a compound was often cited as proof or disproof of structure. Point group theory that uses concepts familiar to most chemists and is easy to apply when the number of addends/substituents is small provides a unified method for deciding, for example, the number of dihydrides C70H2 of fullerene C70, or the number of trihalo-derivatives C2oHi7FClBr of dodecahedrane. All that is needed to determine such matters is the availability of the permutation character. Ter, of the atoms in the parent molecule. 

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