Symmetry and Group Theory

1
Symmetry and Group Theory

• Chapter 4

2
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.

3
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.

4
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 C32
  (counterclockwise) rotation angle
• C4H4 (planar) and C6H6 (benzene) identify the
  rotation angles.

5
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.

6
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.

7
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.

8
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)

9
Identify the Symmetry Elements

• C4H4
• CH3Cl
• C2H6
• CO
• CO2
• It will help to build these molecules with your
  model kits (especially in the beginning).

10
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.

11
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 D?h without will
  be C?v.
• Other point groups Td, Oh, and Ih
• Find the rotation axis with the highest n.
• This will be the principal axis.

12
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.

13
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.

14
Determining Point Groups

• HCl
• CO2
• PF5
• H3CCH3
• NH3
• CH4
• CHFClBr
• H2CCClBr

• HClBrC-CHClBr
• SF6
• H2O2
• 1,5-dibromonaphthalne
• 1,3,5,7-tetrafluorocyclooctatetraene
• B12H122-

15
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.

16
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.

17
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.
• Lets do a few matrix multiplications.

18
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 coordinatestransformation matrixold
  coordinates
• Lets 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.

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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.

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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.

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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.

22
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.

23
The Character Table for the C3v Point Group

• The matrices cannot be block diagonalized into
  1?1 matrices. It can, however, be block
  diagonalized into 2?2 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.

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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).

25
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.

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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)).

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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.