User:Lshou14/1.5: Character Tables

1.5: Character Tables

A character table (in inorganic chemistry) is a collection of the characters of the matrices that make up the irreducible representations of a particular point group. Character tables are used heavily in the study of molecular symmetry and the application of group theory to molecular orbitals.

Layout
Character tables typically appear in the format shown below. To the right is an example for the C3v point group.





Representations
A representation is generally defined as a set of matrices that correspond to the individual operations of a group and combine in the same manner as those operations. There are theoretically an infinite number of possible representations for a group-however, only irreducible representations are listed in character tables as individual rows. Here, irreducible representations may be defined as those which cannot be reduced into any other combination of representations that still fulfills the rules of the group itself (see irreducible representation for a more precise, mathematical definition and great orthogonality theorem for a detailed description of reducing representations). However, rather than listing out individual matrices that correspond to symmetry operations, only their characters are shown. For square matrices, the only types that symmetry operations deal with, the character is defined as the sum of the matrix diagonal elements aii (the diagonal running from the top left to the bottom right). The region of the table encompassing these characters as well as the symmetry classes is known as region I.

Irreducible Representation Rules
Irreducible representations will always follow certain rules in relation to point group character tables. These include the following:


 * The number of irreducible representations in a group will always equal the number of classes for a group.
 * As discussed in the section below, one symmetry class corresponds to one column of a character table. Since each irreducible representation is depicted using one row in the table, this rule simplifies down to the following statement: the number of rows must equal the number of columns in a character table.


 * The characters of all symmetry operations belonging to a symmetry class will always be the same in any representation, reducible or irreducible.
 * This rule is a natural extension of the definition of a symmetry class. From below, "all elements [in this case symmetry operations and their associated matrices] of a class must behave identically in the properties of the character table." This includes their effects on any basis vectors or binary products (discussed below) that form the representation.


 * The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group.
 * The dimension of a representation may be defined as its character under identity. Using the C3v group as an example, the sum of the squares of the dimensions is calculated as follows:

$$\sum\nolimits_{}d^2 = 1^2 + 1^2 + 2^2 = 6$$
 * The order of a group is defined as the number of symmetry operations (not classes) in the group. This can be found by adding the coefficients present in the symmetry classes of a character table. For C3v,

$$\mathrm{Order} = 1 + 2 + 3 = 6$$


 * The sum of the squares of the characters ($$\chi$$) in any irreducible representation is equal to the order of the group.
 * Using the E (see below for a discussion of Mulliken symbols) representation of C3v,

$$\sum\nolimits_{}\chi^2 = (1)2^2 + 2(-1)^2 + (3)0^2 = 6$$
 * It should be noted here that each term is the product of two components-the squares of the characters from the representation listed in the character table and the symmetry class coefficient (i.e. the number of operations present in each class). This is because the rule above refers to the "unabbreviated" irreducible representation where each operation has its own character, regardless of if it belongs in a class with others or not. Following this logic, the "unabbreviated" representation for E is as follows:

$$E = (2, -1, -1, 0, 0, 0)$$
 * Multiplying each set of characters by the symmetry class coefficient yields the same result as if the expansion above was used in the calculation.


 * The vectors made using the components of two irreducible representations in a group will always be orthogonal.
 * Orthogonality of two vectors is proved if the dot product of the two vectors is zero. For the A2 and E representations of the C3v group,

$$A_2 \cdot E = 1(1)(2) + 2(1)(-1) + 3(-1)(0) = 0$$
 * Here, each term is the product of three components-the characters from the two representations and the coefficient of the symmetry class. Similar to the above rule, this is because the irreducible representation "vector" should contain all the symmetry operations of the group.

Symmetry Classes
Symmetry classes are represented by individual character table columns. This leads to the conclusion that all elements of a class must behave identically in the properties of the character table-that is, each produces the same character in each irreducible representation (otherwise, the character table is incorrect and more columns should be added). In this case, this means that applying any symmetry operation of a class to any basis vector present in the character table must yield an identical result. All symmetry operations in a class must therefore be equivalent. More formally, operations belong in a class when one can be replaced by the other in a new coordinate system accessible by a symmetry operation and are known as equivalent.

A matrix-based approach uses similarity transforms. Members of a class must be conjugates of each other. In equation form, $$X$$ and $$Y$$ are conjugates if there exists a matrix $$Z$$ such that $$X = ZYZ^{-1}$$ where $$X, Y,$$ and $$Z$$ are members of the group.

One of the more typical examples of this can be found using proper rotation axes. If one $$C_n$$ proper rotation is present in a molecule, there must be $$nC_n$$ proper rotations present that are usually denoted as $$C_n^1, C_n^2, ... C_n^n$$ that correspond to successive rotations by $$360\over n$$ degrees (for more on this, see here and here). The character of a proper rotation is defined as $$2cos(\theta) - 1$$ where $$\theta$$ is the number of degrees of the rotation. However, the cosine function is even-that is, $$cos(\theta) = cos(-\theta)$$. Therefore, the direction of rotation does not matter when calculating the character of the operation. This means that the rotations $$C_n^m and\ C_n^{n - m}$$ (one clockwise and one counterclockwise) are equivalent and belong to a single class. Consequently, the identity operation (equivalent to $$C_n^n$$) is always in its own class, since $$C_n^0$$ does not exist.

Mulliken Symbols
Mulliken symbols are shorthand designations for irreducible representations. They obey the following rules where a representation is symmetric with respect to an operation if the corresponding character is 1 and is not symmetric with respect to an operation if the corresponding character is -1. These rules (with the exception of 1 and 2) only apply to one-dimensional representations.


 * 1) If the character under identity (also known as the representation dimension) is 1, the representation is assigned a letter of A or B. If the character is 2, the letter is E. If the character is 3, the letter is T.
 * 2) To differentiate between A and B, look at the character of the highest order proper rotation. If the representation is symmetric with respect to Cn, use the letter A. If the representation is not symmetric with respect to Cn, use the letter B.
 * 3) A subscript of 1 is used if the representation is symmetric with respect to the C2 axis perpendicular to the principal Cn rotation axis. A subscript of 2 is used if the representation is not symmetric with respect to the perpendicular C2 axis. If there is no perpendicular C2 axis, subscript numbering is determined using symmetry with respect to the $$\sigma_v$$ mirror plane (if the representation is symmetric the subscript is 1 and vice versa).
 * 4) The subscript g (gerade) is used if the representation is symmetric with respect to inversion. The subscript u (ungerade) is used if the representation is not symmetric with respect to inversion.
 * 5) If the representation is symmetric with respect to the $$\sigma_h$$ mirror plane, one prime is added. If the representation is not symmetric with respect to the $$\sigma_h$$ mirror plane, two primes are added.



Each Mulliken symbol will, at minimum, have a letter. Each additional element (in order of the rules above) will only be added if two representations cannot be differentiated without adding that element. For instance, in the C2h character table (shown to the right), the first representation is denoted using the symbol Ag. Since there are no perpendicular C2 axes or $$\sigma_v$$ mirror planes, the gerade/ungerade designations are used. Per usual, representations Ag and Bg are symmetric ($$\chi = 1$$) with respect to inversion and representations Au and Bu are not ($$\chi = -1$$). Theoretically, since there is a remaining $$\sigma_h$$ mirror plane, primes may also be added to the Mulliken symbols (i.e. the first representation could become Ag', the second Bg"). However, since there are only four different representations and already four unique symbols, there is no need to.

There are two possible subscripts for a Mulliken symbol-1/2 and g/u. Traditionally, the number will come before the letter, so a representation that is symmetric with respect to the principal rotation axis Cn, C2 axes perpendicular to the Cn, and inversion will have a Mulliken symbol of A1g.

The region of the table with these symbols is known as region II.

Basis Vectors
The basis vectors present in a character table indicate the vector(s) that are used to create the irreducible representation. Each representation is derived by determining how the basis vector is transformed by the symmetry operations listed in the character table. Typical vectors shown here are the x, y, and z unit vectors (corresponding to p orbitals) as well as rotational unit vectors Rx, Ry, and Rz. Individual basis vectors are separated by commas.

If there are parentheses present around a set of two or more vectors, it means that they are coupled. Coupled vectors form a basis together because the symmetry operations in the character table transform them into each other. Therefore, they cannot be separated. This can be seen in the C3v character table, where the E representation can have a basis of either the coupled (x, y) or (Rx, Ry) vectors (here, the "mixing" happens in the C3 rotation).

The region of the table containing these vectors is known as region III.

Binary Products
Binary products usually point to how d-orbitals transform in the point group. They are obtained simply by finding the representation that results from multiplying together each corresponding character of its components ($$\chi_x\chi_y = \chi_{xy}$$). To find the binary product of xz in C2h:

$$\Gamma_x\Gamma_z = A_uB_u = \begin{matrix}1(1) & 1(-1) & (-1)(-1) & (-1)(1)\end{matrix} = 1 -1\quad 1 -1 = B_g$$

The region of the table with these products is known as region IV.