Introduction to Character Tables
The Character Table for C2v
The Character Table for C3v
Outline
1Introduction to Character Tables
2The Character Table for C2v
3The Character Table for C3v
5.03 Inorganic Chemistry
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Understanding Orbital Symmetry in Molecular Chemistry: A Study of C2v Point Group, Exams of Inorganic Chemistry
An in-depth exploration of orbital symmetry in molecular chemistry through the C2v point group. It covers the transformation properties of px, py, and pz orbitals, the concept of irreducible representations, and symmetry restrictions on molecular orbitals. The document also includes examples of water molecular orbitals and their symmetry.
What you will learn
- What is the significance of symmetry in molecular orbitals?
- How can you determine the symmetry of a molecular orbital?
- What is an irreducible representation in molecular chemistry?
- Which orbitals may mix in the formation of molecular orbitals for the C2v water molecule?
- What are the transformation properties of px, py, and pz orbitals in the C2v point group?
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2021/2022
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The Character Table for C 2 v The Character Table for C 3 v
Outline
1 Introduction to Character Tables
2 The Character Table for C 2 v
3 The Character Table for C 3 v
The Character Table for C 2 v The Character Table for C 3 v
Quote from Eugene Paul Wigner
See also: Current Science, vol. 69, no. 4, 25 August 1995, p. 375
From the preface to his book on group theory:
Wigner relates a conversation with von Laue on the use of group
theory as the natural tool with which to tackle problems in
quantum mechanics. “I like to recall his question as to which
results... I considered most important. My answer was that the
explanation of Laporte’s rule (the concept of parity) and the
quantum theory of the vector addition model appeared to me most
significant. Since that time, I have come to agree with his answer
that the recognition that almost all rules of spectroscopy follow
from the symmetry of the problem is the most remarkable result.”
The Character Table for C 2 v The Character Table for C 3 v
What Makes Up a Character Table
Character tables contain information about how functions transform in response to the
operations of the group
Five parts of a character table
1 At the upper left is the symbol for the point group
2 The top row shows the operations of the point group,
organized into classes
3 The left column gives the Mulliken symbols for each of the
irreducible representations
4 The rows at the center of the table give the characters of the
irreducible representations
5 Listed at right are certain functions, showing the irreducible
representation for which the function can serve as a basis
The Character Table for C 2 v The Character Table for C 3 v
What Makes Up a Character Table
Character tables contain information about how functions transform in response to the
operations of the group
Five parts of a character table
1 At the upper left is the symbol for the point group
2 The top row shows the operations of the point group,
organized into classes
3 The left column gives the Mulliken symbols for each of the
irreducible representations
4 The rows at the center of the table give the characters of the
irreducible representations
5 Listed at right are certain functions, showing the irreducible
representation for which the function can serve as a basis
The Character Table for C 2 v The Character Table for C 3 v
What Makes Up a Character Table
Character tables contain information about how functions transform in response to the
operations of the group
Five parts of a character table
1 At the upper left is the symbol for the point group
2 The top row shows the operations of the point group,
organized into classes
3 The left column gives the Mulliken symbols for each of the
irreducible representations
4 The rows at the center of the table give the characters of the
irreducible representations
5 Listed at right are certain functions, showing the irreducible
representation for which the function can serve as a basis
The Character Table for C 2 v The Character Table for C 3 v
The C 2 v Character Table
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
What happens when the E operation is applied?
The E operation is a rotation by 360
about an arbitrary axis
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
The E operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
What happens when the C 2 operation is applied?
The C 2 operation is a rotation by 180
about the z axis
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
What happens when the C 2 operation is applied?
The C 2 operation is a rotation by 180
about the z axis
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
The C 2 operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
What happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
The Character Table for C 2 v The Character Table for C 3 v
Transformation Properties of an s Orbital in C 2 v
The σv (xz) operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
The Character Table for C 2 v The Character Table for C 3 v