Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 13, Problem 13.43E
Interpretation Introduction

Interpretation:

The 12×12 matrices for all three planes of symmetry σ, σ', and σ" are to be stated. Also the similarities and differences between the three matrices are to be stated.

Concept introduction:

The group of symmetry operations of which atleast one point is kept fixed is called point group. The symmetry operations can be identity, rotation, reflection, inversion and improper rotation.

Expert Solution & Answer
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Answer to Problem 13.43E

The 12×12 matrices that specify the change in atomic coordinates of NH3 upon operation of σ, σ', and σ" symmetry operations are shown below.

σ=[ 100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

σ'=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

σ"=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

The coordinates for hydrogen in NH3 for σ', and σ" are present off diagonal while in case of σ, the coordinates are present on the diagonal. In all the matrices, there are four submatrices which follow the same pattern.

Explanation of Solution

The structure of ammonia molecule is,

Physical Chemistry, Chapter 13, Problem 13.43E , additional homework tip  1

Figure 1

The total coordinates in the given system is 3N, where N is the number of atoms. Therefore, total coordinates are 12.

The ammonia molecule has non-linear structure and C3 as principal axis. In this molecule 3σv is present, each passes through one of the hydrogen atom. Therefore, the point group is C3v.

On applying the σ(yz) or σ symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of x as shown below.

Physical Chemistry, Chapter 13, Problem 13.43E , additional homework tip  2

Figure 2

The 12×12 matrix that specifies the change in atomic coordinates of NH3 upon operation of the σ(yz) symmetry operation is shown below.

σ=[ 100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

Similarly on applying the σ(xz) or σ' symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of y as shown below. The y coordinate of H1 moves to the y coordinate of H2 which implies 0 on the diagonal position and 1 on the off diagonal position.

The 12×12 matrix that specifies the change in atomic coordinates of NH3 upon operation of the σ(xz) symmetry operation is shown below.

σ'=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

Similarly on applying the σ(xy) or σ" symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of z as shown below. The z coordinate of H1 moves to the z coordinate of H2 which implies 0 on the diagonal position and 1 on the off diagonal position.

The 12×12 matrix that specifies the change in atomic coordinates of NH3 upon operation of the σ(xy) symmetry operation is shown below.

σ"=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

The coordinates for hydrogen for σ', and σ" are on off diagonal while in case of σ, the coordinates are on the diagonal. In all the matrices there is submatrix which follows the same pattern.

Conclusion

The 12×12 matrices that specify the change in atomic coordinates of NH3 upon operation of σ, σ', and σ" symmetry operations are shown below.

σ=[ 100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

σ'=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

σ"=[ 000000000100000000000010000000000001000100000000000010000000000001000000001000000000000100000000000010000000100000000000010000000000001000000000 ][ xH1yH1zH1xNyNzNxH2yH2zH2xH3yH3zH3 ]

The coordinates for hydrogen in NH3 for σ', and σ" are present on the off diagonal while in case of σ, the coordinates are present on the diagonal. In all the matrices, there are four submatrices which follow the same pattern.

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Chapter 13 Solutions

Physical Chemistry

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