Quote from Eugene Paul Wigner
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2 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.
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4 Selection Rule for IR Spectroscopy The molecular dipole moment may undergo a change during a molecular vibration If the vibration or normal mode has the same symmetry as x, y, or z, then it involves a change in the molecular dipole moment during the vibration, by symmetry, and otherwise not The criterion for absorption of infrared (IR) radiation by resonant excitation of a molecular vibration is that it must involve a change in molecular dipole moment This is because the interaction is between the oscillating electric dipole of the electromagnetic radiation and the changing molecular dipole arising from the electron distribution in the vibrating molecule The symmetry of the electric dipole operator is the same as x, y, and z as listed in the character tables
5 Selection Rule for IR Spectroscopy The molecular dipole moment may undergo a change during a molecular vibration If the vibration or normal mode has the same symmetry as x, y, or z, then it involves a change in the molecular dipole moment during the vibration, by symmetry, and otherwise not The criterion for absorption of infrared (IR) radiation by resonant excitation of a molecular vibration is that it must involve a change in molecular dipole moment This is because the interaction is between the oscillating electric dipole of the electromagnetic radiation and the changing molecular dipole arising from the electron distribution in the vibrating molecule The symmetry of the electric dipole operator is the same as x, y, and z as listed in the character tables
6 Selection Rule for IR Spectroscopy The molecular dipole moment may undergo a change during a molecular vibration If the vibration or normal mode has the same symmetry as x, y, or z, then it involves a change in the molecular dipole moment during the vibration, by symmetry, and otherwise not The criterion for absorption of infrared (IR) radiation by resonant excitation of a molecular vibration is that it must involve a change in molecular dipole moment This is because the interaction is between the oscillating electric dipole of the electromagnetic radiation and the changing molecular dipole arising from the electron distribution in the vibrating molecule The symmetry of the electric dipole operator is the same as x, y, and z as listed in the character tables
7 Selection Rule for IR Spectroscopy The molecular dipole moment may undergo a change during a molecular vibration If the vibration or normal mode has the same symmetry as x, y, or z, then it involves a change in the molecular dipole moment during the vibration, by symmetry, and otherwise not The criterion for absorption of infrared (IR) radiation by resonant excitation of a molecular vibration is that it must involve a change in molecular dipole moment This is because the interaction is between the oscillating electric dipole of the electromagnetic radiation and the changing molecular dipole arising from the electron distribution in the vibrating molecule The symmetry of the electric dipole operator is the same as x, y, and z as listed in the character tables
8 Selection Rule for IR Spectroscopy The molecular dipole moment may undergo a change during a molecular vibration If the vibration or normal mode has the same symmetry as x, y, or z, then it involves a change in the molecular dipole moment during the vibration, by symmetry, and otherwise not The criterion for absorption of infrared (IR) radiation by resonant excitation of a molecular vibration is that it must involve a change in molecular dipole moment This is because the interaction is between the oscillating electric dipole of the electromagnetic radiation and the changing molecular dipole arising from the electron distribution in the vibrating molecule The symmetry of the electric dipole operator is the same as x, y, and z as listed in the character tables
9 Water Normal Modes of Vibration
10 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
11 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
12 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
13 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
14 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
15 Procedure for IR Vibrational Symmetry Analysis Assign the structure to a point group, and refer to the character table For each class of symmetry operation, write down the number of atoms unshifted by carrying out the operation Note: only unshifted atoms can contribute to the reducible representation we are generating Multiply by the characters for x, y, and z, here representing the three degrees of freedom for each atom in the structure; this generates the reducible representation for all 3N degrees of freedom Subtract the characters for the rotations and translations and you are left with a reducible representation for the 3N 6 vibrational degrees of freedom (nonlinear molecule) Reduce this reducible representation to find the normal mode symmetries; apply the selection rule
16 Water Vibrational Symmetry Analysis in C2v 5.03 Inorganic Chemistry
17 Water Vibrational Symmetry Analysis in C 2v
18 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
19 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
20 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
21 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
22 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
23 Water Vibrational Symmetry Analysis in C 2v We have a mathematical expression for finding out how many times a particular irreducible representation occurs in a reducible one It is the same as taking the dot product of two vectors a i = 1 h χ(r)χ i (R) R h is the group order the procedure is to multiply together the characters of the reducible representation and the irreducible representation and take the sum, and then divide by the order of the group example: number of times B 2 appears in Γ red : (1 3) + ( 1 1) + ( 1 1) + (1 3) = 4, now divide by h = 4, so B 2 appears once
24 Water Normal Modes of Vibration The point group is maintained during the course of a totally symmetric vibration
25 IR Spectroscopy as a Structural Tool Different point group geometries for a given molecular formula may lead to the prediction of differing numbers of IR-Active bands Procedure: carry out vibrational symmetry analysis for various reasonable geometries and note the predictions Find out which structure(s) is/are consistent with the experimental data We will consider the SF 6 molecule in three different geometries
26 IR Spectroscopy as a Structural Tool Different point group geometries for a given molecular formula may lead to the prediction of differing numbers of IR-Active bands Procedure: carry out vibrational symmetry analysis for various reasonable geometries and note the predictions Find out which structure(s) is/are consistent with the experimental data We will consider the SF 6 molecule in three different geometries
27 IR Spectroscopy as a Structural Tool Different point group geometries for a given molecular formula may lead to the prediction of differing numbers of IR-Active bands Procedure: carry out vibrational symmetry analysis for various reasonable geometries and note the predictions Find out which structure(s) is/are consistent with the experimental data We will consider the SF 6 molecule in three different geometries
28 IR Spectroscopy as a Structural Tool Different point group geometries for a given molecular formula may lead to the prediction of differing numbers of IR-Active bands Procedure: carry out vibrational symmetry analysis for various reasonable geometries and note the predictions Find out which structure(s) is/are consistent with the experimental data We will consider the SF 6 molecule in three different geometries
29 SF6 Vibrational Analysis in D3h : 5 IR-Active Bands 5.03 Inorganic Chemistry
30 SF 6 Vibrational Analysis in D 3h : 5 IR-Active Bands
31 SF6 Vibrational Analysis in D6h : 3 IR-Active Bands 5.03 Inorganic Chemistry
32 SF 6 Vibrational Analysis in D 6h : 3 IR-Active Bands
33 SF6 Vibrational Analysis in Oh : 2 IR-Active Bands 5.03 Inorganic Chemistry
34 SF 6 Vibrational Analysis in O h : 2 IR-Active Bands
35 1 Sulfur hexafluoride INFRARED SPECTRUM Relative Transmittance Wavenumber (cm-1) NIST Chemistry WebBook (
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