Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 27 Molecular Symmetry

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Objectives Discussion of molecular symmetry Understanding of spectroscopic selection rules Identifying the normal modes of vibration for a molecule Determining if a molecular vibration is infrared active and/or Raman active

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Outline 1.Symmetry Elements, Symmetry Operations, and Point Groups 2.Assigning Molecules to Point Groups 3.The H 2 O Molecule and the C 2v Point Group 4.Representations of Symmetry Operators, Bases for Representations, and the Character Table 5.The Dimension of a Representation

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Outline 6.Using the C 2v Representations to Construct Molecular Orbitals for H 2 O 7.The Symmetries of the Normal Modes of Vibration of Molecules 8.Selection Rules and Infrared versus Raman Activity

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.1 Symmetry Elements, Symmetry Operations, and Point Groups Individual molecule has an inherent symmetry based on the spatial arrangement of its atoms. Symmetry of a molecule determines its properties.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.1 Symmetry Elements, Symmetry Operations, and Point Groups Symmetry elements are geometric entities with respect to which operations can be carried out. Symmetry operations are actions with respect to the symmetry elements that leave the molecule in a new configuration. A set of symmetry elements forms a group.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.2 Assigning Molecules to Point Groups The assignment is made using the logic diagram. It is useful to first identify the major symmetry elements.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.2 Assigning Molecules to Point Groups A number of point groups are applicable to small molecules.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.3 The H 2 O Molecule and the C 2v Point Group We would express the symmetry operators mathematically and show that the symmetry elements form a group. Convince yourself that the 2 mirror planes belong to different classes and have different symbols.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.3 The H 2 O Molecule and the C 2v Point Group Elements that belong to the same class can be transformed into one another by other symmetry operations of the group. For example, the operators belong to the same class.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Example 27.1 a.Are the three mirror planes for the NF 3 molecule in the same or in different classes? b. Are the two mirror planes for H 2 O in the same or in different classes?

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution a. NF3 belongs to the C3v group, which contains the rotation operators and the vertical mirror planes. These operations and elements are illustrated by this figure:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution We see that converts, and converts. Therefore, all three mirror planes belong to the same class. b. The figure shows that neither the nor the operation converts. Therefore, these two mirror planes are in different classes.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.3 The H 2 O Molecule and the C 2v Point Group The effect of the four operators can also be deduced from the following:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.3 The H 2 O Molecule and the C 2v Point Group The operators can be described by the following 3 × 3 matrices:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Example 27.3 Evaluate. What operation is equivalent to the two sequential operations? Solution:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table The matrices derived are called representations where the multiplication table of the group can be reproduced with the matrices. Each operation can be represented by +1 or -1 (represented by ) and multiplication table still satisfied.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table The irreducible representations are the matrices of smallest dimension that obey the multiplication table of the group. A group has as many irreducible representations as it has classes of symmetry elements. AOs forms a basis for one of the representations.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table The information on representations can be assembled in a character table. The representation in which all entries are +1 is called the totally symmetric representation.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table Columns 2 through 5 has an entry for each operation of the group in each representation called characters. Columns 6 through 8 shows many possible bases for each representation, called basis functions.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.5 The Dimension of a Representation The dimension of a representation is defined as the size of the matrix used to represent the symmetry operations. 3×3 matrix operations can be reduced to three 1×1 matrix operations, which consist of the numbers +1 and -1.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.5 The Dimension of a Representation The dimension of the different irreducible representations, d j, and the order of the group, h, defined as the number of symmetry elements in the group, are related by the equation This sum is over each class of symmetry elements in the group, rather than over the elements.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Example 27.4 The C 3v group has the elements and three σv mirror planes. How many different irreducible representations does this group have, and what is the dimensionality of each irreducible representation?

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution The order of the group is the number of elements, so h=6. The number of representations is the number of classes. belong to one class, and the same is true of the three σv reflections. Although the group has six elements, it has only three classes. Therefore, the group has three irreducible representations. The equation is solved to find the dimension of the representations, and one of the values must be 1. The only possible solution is. C 3v group contains one two-dimensional representation and two one-dimensional representations.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.5 The Dimension of a Representation To transform x–y coordinate system by a mirror plane,σ

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.5 The Dimension of a Representation The character for an operator in a representation of dimension higher than one is given by the sum of the diagonal elements of the matrix. If the set of characters associated with a representation of the group is viewed as a vector,, with one component for each element of the group, the following condition holds:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.5 The Dimension of a Representation Equivalently,. The sum is over all elements of the group.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Example 27.5 Determine the unknown coefficients a, b, and c for the preceding partially completed character table and assign the appropriate symbol to the irreducible representation.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution We know that the unknown representation is one dimensional and for different values of the index i are orthogonal. Therefore, We can take the sum over classes and multiplied each term by the number of elements in the class, because all elements in a class have the same character.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution We also know that a=1 because it is the character of the identity operator. Solving the equations gives the results of b=1 and c=-1. Because the character of C 3 is +1, and the character of σv is -1, the unknown representation is designated A 2.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.6 Using the C 2v Representations to Construct Molecular Orbitals for H 2 O The character for an operator of the direct product of two representations is given by Overlap integral between two combinations of AOs is nonzero only if the combinations belong to the same representation.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Example 27.6 Which of the oxygen AOs will participate in forming symmetry-adapted water MOs with the antisymmetric combination of hydrogen AOs defined by ?

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry Solution The antisymmetric combination of the H AOs is given by shown in the margin. By considering the C 2v operations, convince yourself that the characters for the different operations are. Therefore, belongs to the B 2 representation. Of the valence oxygen AOs, only the 2 py orbital belongs to the B 2 representation. Therefore, the only MOs formed from and the 2s and 2p orbitals that have the symmetry of the H2O molecule and also have a nonzero overlap among the AOs are the MOs denoted 1b 2 and 2b 2.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.7 The Symmetries of the Normal Modes of Vibration of Molecules All normal modes are independent in the harmonic approximation. Random motion of the atoms in a molecule can be expressed as a linear combination of the normal modes of that molecule. For harmonic approximation,

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.7 The Symmetries of the Normal Modes of Vibration of Molecules The general method for decomposing a reducible representation into its irreducible representations utilizes the vector properties of the representations Take the scalar product between the reducible representation and each of the irreducible representations in turn, and divide by the order of the group.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.7 The Symmetries of the Normal Modes of Vibration of Molecules The result of this procedure is a positive integer ni that is the number of times each representation appears in the irreducible representation. This statement is expressed by the equation

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 27: Molecular Symmetry 27.8 The Symmetries of the Normal Modes of Vibration of Molecules Normal modes of a molecule are Raman active if the bases of the representation to which the normal mode belongs are the x 2, y 2, z 2, xy, yz, or xz functions. All normal modes that belong to a particular representation have the same frequency.