Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf ·...
Transcript of Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf ·...
![Page 1: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/1.jpg)
Introduction to Character TablesThe Character Table for C2v
Character Tables
Christopher C. Cummins
Massachusetts Institute of Technology
5.03 Lecture 3 Character Tables
![Page 2: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/2.jpg)
Introduction to Character TablesThe Character Table for C2v
Outline
1 Introduction to Character Tables
2 The Character Table for C2v
5.03 Lecture 3 Character Tables
![Page 3: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/3.jpg)
Introduction to Character TablesThe Character Table for C2v
What Makes Up a Character TableCharacter tables contain information about how functions transform in response to theoperations 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 theirreducible representations
4 The rows at the center of the table give the characters of theirreducible representations
5 Listed at right are certain functions, showing the irreduciblerepresentation for which the function can serve as a basis
5.03 Lecture 3 Character Tables
![Page 4: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/4.jpg)
Introduction to Character TablesThe Character Table for C2v
What Makes Up a Character TableCharacter tables contain information about how functions transform in response to theoperations 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 theirreducible representations
4 The rows at the center of the table give the characters of theirreducible representations
5 Listed at right are certain functions, showing the irreduciblerepresentation for which the function can serve as a basis
5.03 Lecture 3 Character Tables
![Page 5: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/5.jpg)
Introduction to Character TablesThe Character Table for C2v
What Makes Up a Character TableCharacter tables contain information about how functions transform in response to theoperations 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 theirreducible representations
4 The rows at the center of the table give the characters of theirreducible representations
5 Listed at right are certain functions, showing the irreduciblerepresentation for which the function can serve as a basis
5.03 Lecture 3 Character Tables
![Page 6: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/6.jpg)
Introduction to Character TablesThe Character Table for C2v
What Makes Up a Character TableCharacter tables contain information about how functions transform in response to theoperations 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 theirreducible representations
4 The rows at the center of the table give the characters of theirreducible representations
5 Listed at right are certain functions, showing the irreduciblerepresentation for which the function can serve as a basis
5.03 Lecture 3 Character Tables
![Page 7: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/7.jpg)
Introduction to Character TablesThe Character Table for C2v
What Makes Up a Character TableCharacter tables contain information about how functions transform in response to theoperations 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 theirreducible representations
4 The rows at the center of the table give the characters of theirreducible representations
5 Listed at right are certain functions, showing the irreduciblerepresentation for which the function can serve as a basis
5.03 Lecture 3 Character Tables
![Page 8: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/8.jpg)
Introduction to Character TablesThe Character Table for C2v
The C2v Character Table
5.03 Lecture 3 Character Tables
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Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 10: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/10.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 11: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/11.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe E operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 12: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/12.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe E operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 13: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/13.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 14: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/14.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 15: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/15.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe C2 operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 16: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/16.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe C2 operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 17: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/17.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 18: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/18.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 19: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/19.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe σv (xz) operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 20: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/20.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe σv (xz) operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 21: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/21.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 22: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/22.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 23: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/23.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe σ′
v (yz) operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 24: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/24.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s Orbital in C2vThe σ′
v (yz) operation returns the original configuration of the s orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 25: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/25.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 26: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/26.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 27: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/27.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 28: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/28.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 29: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/29.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 30: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/30.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of an s OrbitalThese observations pertain to any central-atom s orbital in any point group
Consider an s orbital located on a central atom
An example of a central atom is O in the case of water, or Nin the case of ammonia
Carrying out any operation on a central atom s orbital returnsthe s orbital in its original configuration
The central-atom s orbital “belongs to” or “serves as a basisfor” the totally symmetric (A1) irreducible representation
All the characters of the totally symmetric irreduciblerepresentation are 1
The totally symmetric irreducible representation is alwayssingly degenerate
5.03 Lecture 3 Character Tables
![Page 31: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/31.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 32: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/32.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 33: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/33.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe E operation returns the original configuration of the px orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 34: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/34.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe E operation returns the original configuration of the px orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 35: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/35.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 36: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/36.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 37: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/37.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe C2 operation inverts the phase of the px orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 38: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/38.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe C2 operation inverts the phase of the px orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 39: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/39.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 40: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/40.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 41: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/41.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe σv (xz) operation does nothing to the phase of the px orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 42: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/42.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe σv (xz) operation does nothing to the phase of the px orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 43: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/43.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 44: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/44.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 45: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/45.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe σ′
v (yz) operation inverts the phase of the px orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 46: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/46.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a px Orbital in C2vThe σ′
v (yz) operation inverts the phase of the px orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 47: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/47.jpg)
Introduction to Character TablesThe Character Table for C2v
A px Orbital has B1 Symmetry in C2v
We carried out the operations of C2v on a central-atom px
orbital
This generated the following row of characters: 1,−1, 1,−1
This row of characters in the C2v character table is labeled B1
Any orbital having these transformation properties in C2v issaid to have B1 symmetry
5.03 Lecture 3 Character Tables
![Page 48: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/48.jpg)
Introduction to Character TablesThe Character Table for C2v
A px Orbital has B1 Symmetry in C2v
We carried out the operations of C2v on a central-atom px
orbital
This generated the following row of characters: 1,−1, 1,−1
This row of characters in the C2v character table is labeled B1
Any orbital having these transformation properties in C2v issaid to have B1 symmetry
5.03 Lecture 3 Character Tables
![Page 49: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/49.jpg)
Introduction to Character TablesThe Character Table for C2v
A px Orbital has B1 Symmetry in C2v
We carried out the operations of C2v on a central-atom px
orbital
This generated the following row of characters: 1,−1, 1,−1
This row of characters in the C2v character table is labeled B1
Any orbital having these transformation properties in C2v issaid to have B1 symmetry
5.03 Lecture 3 Character Tables
![Page 50: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/50.jpg)
Introduction to Character TablesThe Character Table for C2v
A px Orbital has B1 Symmetry in C2v
We carried out the operations of C2v on a central-atom px
orbital
This generated the following row of characters: 1,−1, 1,−1
This row of characters in the C2v character table is labeled B1
Any orbital having these transformation properties in C2v issaid to have B1 symmetry
5.03 Lecture 3 Character Tables
![Page 51: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/51.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 52: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/52.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 53: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/53.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe E operation returns the original configuration of the py orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 54: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/54.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe E operation returns the original configuration of the py orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 55: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/55.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 56: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/56.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 57: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/57.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe C2 operation inverts the phase of the py orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 58: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/58.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe C2 operation inverts the phase of the py orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 59: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/59.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 60: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/60.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 61: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/61.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe σv (xz) operation inverts the phase of the py orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 62: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/62.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe σv (xz) operation inverts the phase of the py orbital
The result of this corresponds to a character of −1
5.03 Lecture 3 Character Tables
![Page 63: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/63.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 64: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/64.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 65: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/65.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe σ′
v (yz) operation does nothing to the phase of the py orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 66: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/66.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a py Orbital in C2vThe σ′
v (yz) operation does nothing to the phase of the py orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 67: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/67.jpg)
Introduction to Character TablesThe Character Table for C2v
A py Orbital has B2 Symmetry in C2v
We carried out the operations of C2v on a central-atom py
orbital
This generated the following row of characters: 1,−1,−1, 1
This row of characters in the C2v character table is labeled B2
Any orbital having these transformation properties in C2v issaid to have B2 symmetry
5.03 Lecture 3 Character Tables
![Page 68: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/68.jpg)
Introduction to Character TablesThe Character Table for C2v
A py Orbital has B2 Symmetry in C2v
We carried out the operations of C2v on a central-atom py
orbital
This generated the following row of characters: 1,−1,−1, 1
This row of characters in the C2v character table is labeled B2
Any orbital having these transformation properties in C2v issaid to have B2 symmetry
5.03 Lecture 3 Character Tables
![Page 69: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/69.jpg)
Introduction to Character TablesThe Character Table for C2v
A py Orbital has B2 Symmetry in C2v
We carried out the operations of C2v on a central-atom py
orbital
This generated the following row of characters: 1,−1,−1, 1
This row of characters in the C2v character table is labeled B2
Any orbital having these transformation properties in C2v issaid to have B2 symmetry
5.03 Lecture 3 Character Tables
![Page 70: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/70.jpg)
Introduction to Character TablesThe Character Table for C2v
A py Orbital has B2 Symmetry in C2v
We carried out the operations of C2v on a central-atom py
orbital
This generated the following row of characters: 1,−1,−1, 1
This row of characters in the C2v character table is labeled B2
Any orbital having these transformation properties in C2v issaid to have B2 symmetry
5.03 Lecture 3 Character Tables
![Page 71: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/71.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 72: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/72.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the E operation is applied?
The E operation is a rotation by 360◦ about an arbitrary axis
5.03 Lecture 3 Character Tables
![Page 73: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/73.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe E operation returns the original configuration of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 74: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/74.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe E operation returns the original configuration of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 75: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/75.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 76: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/76.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the C2 operation is applied?
The C2 operation is a rotation by 180◦ about the z axis
5.03 Lecture 3 Character Tables
![Page 77: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/77.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe C2 operation does nothing to the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 78: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/78.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe C2 operation does nothing to the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 79: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/79.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 80: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/80.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the σv (xz) operation is applied?
The σv (xz) operation is a reflection through the xz plane
5.03 Lecture 3 Character Tables
![Page 81: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/81.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe σv (xz) operation inverts the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 82: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/82.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe σv (xz) operation inverts the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 83: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/83.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 84: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/84.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vWhat happens when the σ′
v (yz) operation is applied?
The σ′v (yz) operation is a reflection through the yz plane
5.03 Lecture 3 Character Tables
![Page 85: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/85.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe σ′
v (yz) operation does nothing to the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 86: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/86.jpg)
Introduction to Character TablesThe Character Table for C2v
Transformation Properties of a pz Orbital in C2vThe σ′
v (yz) operation does nothing to the phase of the pz orbital
The result of this corresponds to a character of 1
5.03 Lecture 3 Character Tables
![Page 87: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/87.jpg)
Introduction to Character TablesThe Character Table for C2v
A pz Orbital has A1 Symmetry in C2v
We carried out the operations of C2v on a central-atom pz
orbital
This generated the following row of characters: 1, 1, 1, 1
This row of characters in the C2v character table is labeled A1
Any orbital having these transformation properties in C2v issaid to have A1 symmetry
5.03 Lecture 3 Character Tables
![Page 88: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/88.jpg)
Introduction to Character TablesThe Character Table for C2v
A pz Orbital has A1 Symmetry in C2v
We carried out the operations of C2v on a central-atom pz
orbital
This generated the following row of characters: 1, 1, 1, 1
This row of characters in the C2v character table is labeled A1
Any orbital having these transformation properties in C2v issaid to have A1 symmetry
5.03 Lecture 3 Character Tables
![Page 89: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/89.jpg)
Introduction to Character TablesThe Character Table for C2v
A pz Orbital has A1 Symmetry in C2v
We carried out the operations of C2v on a central-atom pz
orbital
This generated the following row of characters: 1, 1, 1, 1
This row of characters in the C2v character table is labeled A1
Any orbital having these transformation properties in C2v issaid to have A1 symmetry
5.03 Lecture 3 Character Tables
![Page 90: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/90.jpg)
Introduction to Character TablesThe Character Table for C2v
A pz Orbital has A1 Symmetry in C2v
We carried out the operations of C2v on a central-atom pz
orbital
This generated the following row of characters: 1, 1, 1, 1
This row of characters in the C2v character table is labeled A1
Any orbital having these transformation properties in C2v issaid to have A1 symmetry
5.03 Lecture 3 Character Tables
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Introduction to Character TablesThe Character Table for C2v
Symmetry Restrictions on Molecular Orbitals (MOs)
Only orbitals of the same symmetry may mix
“Orbitals of the same symmetry” belong to the sameirreducible representation
For the C2v water molecule, the oxygen s and pz atomicorbitals may contribute to any molecular orbital of A1
symmetry, but px and py may not
Any valid molecular orbital must transform according to oneof the irreducible representations of the molecular point group
5.03 Lecture 3 Character Tables
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Introduction to Character TablesThe Character Table for C2v
Symmetry Restrictions on Molecular Orbitals (MOs)
Only orbitals of the same symmetry may mix
“Orbitals of the same symmetry” belong to the sameirreducible representation
For the C2v water molecule, the oxygen s and pz atomicorbitals may contribute to any molecular orbital of A1
symmetry, but px and py may not
Any valid molecular orbital must transform according to oneof the irreducible representations of the molecular point group
5.03 Lecture 3 Character Tables
![Page 93: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/93.jpg)
Introduction to Character TablesThe Character Table for C2v
Symmetry Restrictions on Molecular Orbitals (MOs)
Only orbitals of the same symmetry may mix
“Orbitals of the same symmetry” belong to the sameirreducible representation
For the C2v water molecule, the oxygen s and pz atomicorbitals may contribute to any molecular orbital of A1
symmetry, but px and py may not
Any valid molecular orbital must transform according to oneof the irreducible representations of the molecular point group
5.03 Lecture 3 Character Tables
![Page 94: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/94.jpg)
Introduction to Character TablesThe Character Table for C2v
Symmetry Restrictions on Molecular Orbitals (MOs)
Only orbitals of the same symmetry may mix
“Orbitals of the same symmetry” belong to the sameirreducible representation
For the C2v water molecule, the oxygen s and pz atomicorbitals may contribute to any molecular orbital of A1
symmetry, but px and py may not
Any valid molecular orbital must transform according to oneof the irreducible representations of the molecular point group
5.03 Lecture 3 Character Tables
![Page 95: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/95.jpg)
Introduction to Character TablesThe Character Table for C2v
The C2v Character Table
5.03 Lecture 3 Character Tables
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Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables
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Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables
![Page 98: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/98.jpg)
Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables
![Page 99: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/99.jpg)
Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables
![Page 100: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/100.jpg)
Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables
![Page 101: Character Tables - MITweb.mit.edu/5.03/www/readings/character_tables/character_tables.pdf · Introduction to Character Tables The Character Table for C2v What Makes Up a Character](https://reader033.fdocuments.net/reader033/viewer/2022050902/5ab619667f8b9ab7638d6c22/html5/thumbnails/101.jpg)
Introduction to Character TablesThe Character Table for C2v
The Molecular Orbitals of Water
Notice that the water HOMO is a pure oxygen px orbital ofB1 symmetry
The hydrogen atoms with their 1s valence orbitals lie in thenodal plane of the oxygen px orbital
The two hydrogen 1s orbitals give rise to linear combinationsof A1 and B2 symmetry
The O-H bonding molecular orbitals must likewise be of A1
and B2 symmetry
Given that all the irreducible representations of C2v are singlydegenerate, so must be all the MOs of the water molecule
Click on Link to Water MOs
5.03 Lecture 3 Character Tables