Symmetry and Group Theory

Chapter 4

Symmetry and Group Theory

• The symmetry properties of molecules can be useful in predicting infrared spectra, describing the types of orbitals used in bonding, predicting optical activity, and interpreting electronic spectra (to name a few).

• The materials in this chapter will be used extensively throughout the remaining semester.

Symmetry and Group Theory

• Symmetry element – a geometric entity with respect to which a symmetry operation is performed.

• Symmetry operation – a rearrangement of a body after which it appears unchanged.

Several objects for examples– Cup, snowflake in the book, your body, and a key

(other objects). For each name the operations and the elements.

Types of Molecular Operations and Elements

• Identity operation (E) – causes no change in the molecules.– Every molecule possesses this symmetry.

• Rotation operation or proper rotation (Cn) – rotation through 360/n about a rotation axis.– CHCl3 possesses a C3 (clockwise) and a C3

2 (counterclockwise) rotation angle

– C4H4 (planar) and C6H6 (benzene) – identify the rotation angles.

Types of Molecular Operations and Elements

• Rotation operation or proper rotation (Cn)

– Principal rotation axis – the Cn axis that has the highest value of n of multiple rotation axes exist.

• Examine CH3Cl, C4H4, and C6H6. Identify other rotation axes if present.

• C2 passes through several atoms and C2 passes between the C2 axes and the atoms.

Note: The principal axes is usually chosen as the z-axis.

Types of Molecular Operations and Elements

• Reflection operation () – contains a mirror plane.– CH3Cl contains multiple mirror planes that contain the

principal axis. These mirror planes are v or d.

• If applicable, the v plane usually intersects several atoms while d goes between them.

– C4H4 and C6H6 also contain a horizontal plane perpendicular to the principal axis of rotation. This plane is called h.

Types of Molecular Operations and Elements

• Inversion (i) – each point moves through the center of the molecule to a position opposite the original position and as far from the central point as when it started. The environment at the new point is the same as the environment at the old point.– Invert the molecule. If the inversion creates a molecule that

appears identical, the molecule possesses a center of inversion.

– CH3Cl, C4H4, and CH4 – Determine if the molecules have inversion symmetry.

Types of Molecular Operations and Elements

• Improper rotation or rotation-reflection (Sn) – requires rotation of 360/n followed by reflection through a plane perpendicular to the axis of rotation.– C4H4 and H3C-CH3 (ethane) Name and identify the

Sn operations performed on ethane.

S2 i (preferred)

S1 (preferred)

Identify the Symmetry Elements

• C4H4

• CH3Cl

• C2H6

• CO

• CO2

It will help to build these molecules with your model kits (especially in the beginning).

Point Groups

• The set of symmetry elements for an object/molecule define a point group. The properties of a particular group allow the use of group theory. Group theory can be used to determine the molecular orbitals, vibrations, and other properties of a molecule.

Website for software:

http://www.emory.edu/CHEMISTRY/pointgrp/index.html

Examine Figure 4-7.

Finding the Point Group

• Determine whether the molecule belongs to one of the special cases of low or high symmetry.– Low symmetry

• C1 (only E), Cs (E and h), and Ci (E and i)

– High symmetry• Linear with inversion will be Dh; without will be Cv.

• Other point groups; Td, Oh, and Ih

• Find the rotation axis with the highest n.– This will be the principal axis.

Finding the Point Group

• Does the molecule have any C2 axes to the Cn axis?– If so, the molecule is in the D set of groups.– If not, the molecule is in the C or S set.

• Does the molecule have a mirror plane (h).– If so, the molecule is Cnh or Dnh.– If not, continue with other mirror planes.

• Does the molecule contain any mirror planes that contain the Cn axis?– If so, the molecule is Cnv or Dnd.– If not and in the D group, the molecule is Dn. – If not and in the C group, continue to next.

Finding the Point Group

• Is there any S2n axis collinear with the Cn axis?– If so, the molecule is S2n.– If not, the molecule is Cn.

• This assignment is very rare.

Vertical planes contain the highest order Cn axis. In the Dnd case, the planes are dihedral because they are between the C2 axes.

Purely rotation groups of Ih, Oh, and Td are I, O, and T, respectively (only other symmetry operation is E). These are rare.

The Th point group is derived by adding inversion symmetry to the T point group. These are rare.

Determining Point Groups

• HCl

• CO2

• PF5

• H3CCH3

• NH3

• CH4

• CHFClBr

• H2C=CClBr

• HClBrC-CHClBr

• SF6

• H2O2

• 1,5-dibromonaphthalne• 1,3,5,7-

tetrafluorocyclooctatetraene

• B12H122-

Properties and Representations of Groups

• Properties of a group– Each group must have an identity operation.– Each group must have an inverse.– The product of any two group operations must

also be a member of the group.– The associative property holds.

Understand each property.

Matrices

• Information about the symmetry aspects of point groups are summarized in character tables. Character tables can be thought of as shorthand versions of matrices that are used to describe symmetry aspects of molecules.

• A matrix is an ordered array of numbers represented in columns and rows.– Illustrate an example.

Multiplying Matrices• The number of vertical columns of the first matrix must

be equal to the number of horizontal rows of the second matrix.

• The product is found, term by term, by summing the products of each row of the first matrix by each column of the second.

• The product matrix is the resulting sum with the row determined by the row of the first matrix and the column determined by the column of the second matrix.

Let’s do a few matrix multiplications. kjikij BAC

Construction of Character Tables• Construction of the x, y, and z axes follows the

right-hand rule.– The principal rotation axis is usually collinear with

the z-axis.

• A symmetry operation can be expressed as a transformation matrix.– [new coordinates]=[transformation matrix][old coordinates]

Let’s examine the symmetry operations of a C2v point group (e.g. H2O). All the symmetry operations of this point group can be represented by transformation matrices.

Construction of Character Tables

• This set of matrices satisfies the properties of a mathematical group. This is a matrix representation of the C2v point group. Each matrix corresponds to an operation in the group. A set of matrices can describe the symmetry operations of any group and satisfy the properties of a group specified in Table 4-6.

Construction of Character Tables

• The character is the traces of matrix or the sum along the diagonal (show).– The set of characters also forms a representation. This is

called a reducible representation since it is a combination of irreducible representations (later).

• The matrices for the symmetry operations are “block diagonalized”.– Can be broken down into smaller matrices along the

diagonal with all other elements equal to zero. Illustrate this form the symmetry operations in the C2v point group.

Construction of Character Tables• The x, y, and z axes are also block diagonalized and, as a

consequence, are independent of each other.– Each character set forms a row in the character table and is an

irreducible representation (i.e. cannot be simplified further). • Illustrate this in the character table.

• The three IRs or set of characters can be added together to produce the reducible representation, (illlustrate).– Same result produced from combining the matrices. The character

format is a shorthand version of matrix representation.Note: The row under each symmetry operation corresponds to

the result of the operation on that particular dimension.

Character Tables• A complete set of irreducible representations for a point group

is called the character table for that group.

Explanation of labels on page 97.Go over properties of character of IRs in point groups on page 98 (with

relation to the C2v point group).

Where did the A2 representation come from? Property 3. Using property 6 of orthogonality the characters of this representation can be determined.

C2v E C2 v(xz)

v(yz)

A1 1 1 1 1 z x2,y2, z2 A2 1 1 -1 -1 Rz xy B1 1 -1 1 -1 x, Ry xz B2 1 -1 -1 1 y, Rx yz

The Character Table for the C3v Point Group

• The matrices cannot be block diagonalized into 11 matrices. It can, however, be block diagonalized into 22 matrices,– x and y are not independent of each other. In this case, they

form a doubly degenerate representation.

• E and A1 representations can be found by the matrices and the A2 matrix can be found by the properties of a group.Go over Table 4-7 with this point group.

Additional Features of the Character Tables

• C32 and C3 combine to form 2C3.

• C2 axes to the principal axis are designated with primes.– C2 passes through several atoms.– C2 passes between the atoms.

• Mirror plane to the principal axis is designated as h. v and d planes (explain)

• Expressions on the right indicated the symmetry of mathematical functions of the coordinates x, y, and z. These can be used to find the orbitals that match the representations (discuss).

Additional Features of the Character Tables

• Labeling IRs (do this with C2v)– The characters of the IRs

• Symmetric with respect to the operation is 1

• Antisymmetric with respect to the operation is –1

– Letter assignments and dimension (degeneracy)• The letter indicates the dimension/degeneracy of the IR. It also

indicates if the representation is symmetric to the principal rotation operation.

– The subscripts 1 or 2 on the letter indicates a representation symmetric or antisymmetry, respectively, to a C2 rotation to the principal axis.

Additional Features of the Character Tables

• Labeling IRs (do this with C2v)

– If no C2 axes exist, 1 designates a representation symmetric to a vertical plan and 2 designates a representation antisymmetric.

• Show with C2v and D4h point groups.

– Subscript g (gerade) designates symmetric to inversion, and subscript u (ungerade) designates antisymmetric (D4h).

– Single primes are symmetric to h and double primes are antisymmetric (C3h, C5h, and D3h (look at pz)).

Chirality

• Molecules that are not superimposable on their mirror images are labeled as chiral or disymmetric.– CBrClI (the nonsuperimposable mirror images are called

enantionmers).

• In general a molecule is chiral if it has no symmetry operations (E) or if it has only a proper rotation axis.

• A chiral molecule will rotate the plane of polarized light.– One enantiomer will rotate the plane in a clockwise direction

and the other in an anticlockwise direction. Termed as optical activity.