# Character Table

Below is an example of a character table.

File:D4h.png
D4h Character Table

The higher the symmetry of the point group, the more complicated the character table is. As the result, sometimes, the complexity and symmetry of the point group can be reduced and approximated by a lower symmetry point group. For example, in D4h, the character table is shown above. However, to simplify the table, the point group can be approximated by C4v point group. As the one below.

File:C4v.png
C4v Character Table

A character table is made up of 5 parts.

1. The symbol of the point group is given at the upper left hand corner of the character table.

2. The operations that belonged to the point group is given at the top row, organized into classes.

3. Mulliken symbols for irreducible representations are given at the left column.

4. The characters of the irreducible representations are given at the center of the table. These are the outcome of the basis function in response to the operations of the group

5. Some certain functions are listed on the right. These functions show the irreducible representations for which the function can be served as basis.

Mulliken symbols

A – singlely degenerate, meaning only one orbital has that particular symmetry and level of energy. Symmetric with respect to the primary rotational axis.

B – singlely degenerate, meaning only one orbital has that particular symmetry and level of energy. Anti-symmetric with respect to the primary rotational axis.

E – doubly degenerate, meaning that two orbitals have the same symmetry and the same level of energy. These orbitals transform together.

T – triply degenerate, meaning that three orbitals have the same symmetry and the same level of energy. These orbitals transform together.

Subscript g – symmetric with respect to inversion center.

Subscript u – anti-symmetric with respect to inversion center.

Subscript 1 – symmetric with respect to perpendicular C2 axis

Subscript 2 – anti-symmetric with respect to perpendicular C2 axis.

‘(prime) – designates symmetric with respect to horizontal reflection plane

“(double prime) – designates anti-symmetric with respect to horizontal reflection plane

# How to use a character table

An orbital of a central atom is placed at the origin. The symmetry operations are perform on the orbital.

If the sign of the orbital does not change, a positive 1 is returned for the character.

If the sign inverts, a negative 1 is returned for the character.

# Example

S orbital at the origin

Above is an S orbital of the central atom sitting at the origin. Remember, it is very critical to always place the orbital of the central atom at the origin. Refer to Figure VIIII, there are a total of five operations in C4v point group.

1. If an E operation is performed on this S orbital, a positive 1 is returned for the character since the sign of the S orbital remains the same.

2. If a C4 operation is performed on this S orbital, a positive 1 is returned for the character (for convention, the primary rotation axis is always placed along the z axis).

3. If A C2 operation is performed on this S orbital, a positive 1 is returned for the character.

4. If a mirror plane reflection through the bonds is performed on this S orbital, a positive 1 is returned for the character.

5. If a mirror plane reflection between the bonds is performed on this S orbital, a positive 1 is returned for the character.

Px orbital at the origin

Above is a Px orbital of a central atom sitting at the origin. Refer to C2v character table, there are a total of 4 symmetry operations in this point group.

1. If an E operation is performed on this Px orbital, a positive 1 is returned for the character.

2. If a C2 operation along the z axis is performed on this Px orbital, a negative 1 is returned because the sign inverts

3. If a vertical mirror reflection on xz plane is performed on this Px orbital, a positive 1 is returned.

4. If a vertical mirror reflection on yz plane is performed on this Px orbital, a negative 1 is returned.

Since 1, -1, 1, -1 are returned for the character, the corresponding Mulliken symbol is B1.

Since 1, 1, 1, 1, 1 is returned for all the character, matching the Mulliken symbol on the left of the table, A1 is the irreducible representation of this S orbital. S orbital is a perfect sphere, so it is immediately obvious that it is a perfectly symmetrical orbital.

# Reference

Figueroa, Joshua. "Intro to Symmetry and Symmetry Element." Inorganic Chemistry. University of California, San Diego, La Jolla. Oct. 2012. Lecture.

Figueroa, Joshua. "Symmetry operation and character table." Inorganic Chemistry. University of California, San Diego, La Jolla. Oct. 2012. Lecture.

Figueroa, Joshua. "Character tables, irreducible representations of central atom." Inorganic Chemistry. University of California, San Diego, La Jolla. Oct. 2012. Lecture.

Figueroa, Joshua. "Symmetry elements and point groups." Inorganic Chemistry. University of California, San Diego, La Jolla. Oct. 2012. Lecture.

Figueroa, Joshua. "SALCS, molecular orbital diagrams and high symmetry point groups." Inorganic Chemistry. University of California, San Diego, La Jolla. Oct. 2012. Lecture.

"Symmetry Resources." Otterbein University. N.p., n.d. Web. 20 Nov. 2012. <http://symmetry.otterbein.edu/>.

"Point Group Symmetry Character Tables - Chemistry Online Education." Point Group Symmetry Character Tables - Chemistry Online Education. N.p., n.d. Web. 20 Nov. 2012. <http://www.webqc.org/symmetry.php>.

Miessler, Gary L., and Donald A. Tarr. Inorganic Chemistry. Upper Saddle River, NJ: Pearson Prentice Hall, 2011. Print.