Group Theory and Chemistry

Outline:

• Raman and infra-red spectroscopy • Symmetry operations • Point Groups and Schoenflies symbols • Function space and matrix representation • Reducible and irreducible representation • Normal modes and normal coordinates • Selection rules • CH4 and CH3D example • Some notes on real spectra • References

Infra-red Spectroscopy I

en.wikipedia.org/wiki/File:Infrared_spectrum.gif

Far infra-red: Wavelength λ: 25–1000 μm Wavenumber 𝑣 =1 λ :400–10 cm −1

rotational spectroscopy Mid infra-red: λ: 2.5–25 μm 𝑣 : 4000–400 cm-1 fundamental vibrations and associated rotational-vibrational structure. Structure analysis region Near infra-red: λ: 0.8–2.5 μm 𝑣 : 14000–4000 cm−1

excite overtone or harmonic vibrations Finger print region

Infra-red Spectroscopy II

http://www.ir-spektroskopie.de/

Absorbed frequency resonant frequency frequency of the bond or group that vibrates In order for a vibrational mode in a molecule to be "IR active," it must be associated with changes in the dipole.

𝜇 = 𝛿 ∙ 𝑟 1D = 3.336 ∙ 10−30Cm 𝛿: partial charge

𝑟 : distance vector between two partial charges D: Debye

Dipole moment:

Infra-red Spectroscopy II

http://www.ir-spektroskopie.de/

Absorbed frequency resonant frequency frequency of the bond or group that vibrates In order for a vibrational mode in a molecule to be "IR active," it must be associated with changes in the dipole.

𝜇 = 𝛿 ∙ 𝑟 1D = 3.336 ∙ 10−30Cm 𝛿: partial charge

𝑟 : distance vector between two partial charges D: Debye

Dipole moment:

Raman Spectroscopy I

http://en.wikipedia.org/wiki/Raman_spectroscopy

The Raman effect corresponds, in perturbation theory, to the absorption and subsequent emission of a photon via an intermediate vibrational state, having a virtual energy level

Raman Spectroscopy II

Rayleigh Anti-Stokes Stokes

• Source: visible, monochromatic light => laser • One measures the scattered intensity • About 1 1000 of the intensity is scattered as Rayleigh radiation • An even smaller fraction is shifted

If a molecule is placed in an electric field a dipole moment is induced:

α: polarizability; ν0: vibrational frequency

Raman Spectroscopy II

Rayleigh Anti-Stokes Stokes

• Source: visible, monochromatic light => laser • One measures the scattered intensity • About 1 1000 of the intensity is scattered as Rayleigh radiation • An even smaller fraction is shifted

If a molecule is placed in an electric field a dipole moment is induced:

α: polarizability; ν0: vibrational frequency

Symmetry operations I Symmetry operations for a symmetric tripod

like the ammonia molecule NH3

David M. Bishop; Group Theory and Chemistry

Symmetry operations II Symmetry operations for a symmetric tripod

David M. Bishop; Group Theory and Chemistry

Symmetry operations III

David M. Bishop; Group Theory and Chemistry

1. Identity (𝑬): doing nothing operation 2. Rotation (𝑪𝒏): operation of rotation a molecule clockwise about an

axis by an angle 2𝜋

𝑛. The axis with the highest n or with the smallest

angle to produce coincidence is called principle axis. 3. Reflection (𝝈): the reflection of a molecule can be distinguished by its

orientation to the principle axis: • If the reflection plane is perpendicular to the principle axis: 𝝈𝒉 • If the plane contains the principle axis: 𝝈𝒗 • If the plane contains the principle axis and bisects the angle

between two 2 fold axes (𝐶2) perpendicular to the principle axis: 𝝈𝒅

4. Rotation-reflection (𝑆𝑛): a combined operation of a clockwise rotation

by 2𝜋

𝑛 followed by reflection in a plane perpendicular to this axis (or

vice versa). 5. Inversion (𝒊): operation of inverting all points about the same center.

Symmetry operations III

David M. Bishop; Group Theory and Chemistry

1. Identity (𝑬): doing nothing operation 2. Rotation (𝑪𝒏): operation of rotation a molecule clockwise about an

axis by an angle 2𝜋

𝑛. The axis with the highest n or with the smallest

angle to produce coincidence is called principle axis. 3. Reflection (𝝈): the reflection of a molecule can be distinguished by its

orientation to the principle axis: • If the reflection plane is perpendicular to the principle axis: 𝝈𝒉 • If the plane contains the principle axis: 𝝈𝒗 • If the plane contains the principle axis and bisects the angle

between two 2 fold axes (𝐶2) perpendicular to the principle axis: 𝝈𝒅

4. Rotation-reflection (𝑆𝑛): a combined operation of a clockwise rotation

by 2𝜋

𝑛 followed by reflection in a plane perpendicular to this axis (or

vice versa). 5. Inversion (𝒊): operation of inverting all points about the same center.

Symmetry operations III

David M. Bishop; Group Theory and Chemistry

1. Identity (𝑬): doing nothing operation 2. Rotation (𝑪𝒏): operation of rotation a molecule clockwise about an

axis by an angle 2𝜋

𝑛. The axis with the highest n or with the smallest

angle to produce coincidence is called principle axis. 3. Reflection (𝝈): the reflection of a molecule can be distinguished by its

orientation to the principle axis: • If the reflection plane is perpendicular to the principle axis: 𝝈𝒉 • If the plane contains the principle axis: 𝝈𝒗 • If the plane contains the principle axis and bisects the angle

between two 2 fold axes (𝐶2) perpendicular to the principle axis: 𝝈𝒅

4. Rotation-reflection (𝑆𝑛): a combined operation of a clockwise rotation

by 2𝜋

𝑛 followed by reflection in a plane perpendicular to this axis (or

vice versa). 5. Inversion (𝒊): operation of inverting all points about the same center.

Symmetry operations III

David M. Bishop; Group Theory and Chemistry

1. Identity (𝑬): doing nothing operation 2. Rotation (𝑪𝒏): operation of rotation a molecule clockwise about an

axis by an angle 2𝜋

𝑛. The axis with the highest n or with the smallest

angle to produce coincidence is called principle axis. 3. Reflection (𝝈): the reflection of a molecule can be distinguished by its

orientation to the principle axis: • If the reflection plane is perpendicular to the principle axis: 𝝈𝒉 • If the plane contains the principle axis: 𝝈𝒗 • If the plane contains the principle axis and bisects the angle

between two 2 fold axes (𝐶2) perpendicular to the principle axis: 𝝈𝒅

4. Rotation-reflection (𝑆𝑛): a combined operation of a clockwise rotation

by 2𝜋

𝑛 followed by reflection in a plane perpendicular to this axis (or

vice versa). 5. Inversion (𝒊): operation of inverting all points about the same center.

Symmetry operations III

David M. Bishop; Group Theory and Chemistry

1. Identity (𝑬): doing nothing operation 2. Rotation (𝑪𝒏): operation of rotation a molecule clockwise about an

axis by an angle 2𝜋

𝑛. The axis with the highest n or with the smallest

angle to produce coincidence is called principle axis. 3. Reflection (𝝈): the reflection of a molecule can be distinguished by its

orientation to the principle axis: • If the reflection plane is perpendicular to the principle axis: 𝝈𝒉 • If the plane contains the principle axis: 𝝈𝒗 • If the plane contains the principle axis and bisects the angle

between two 2 fold axes (𝐶2) perpendicular to the principle axis: 𝝈𝒅

4. Rotation-reflection (𝑆𝑛): a combined operation of a clockwise rotation

by 2𝜋

𝑛 followed by reflection in a plane perpendicular to this axis (or

vice versa). 5. Inversion (𝒊): operation of inverting all points about the same center.

Point Groups I

David M. Bishop; Group Theory and Chemistry

Grouprequirements: 1. The combination of two elements must yield another element of

the same group. 2. There must exist a neutral element, which leaves the elements of

the group unchanged and commute with all other elements. 3. There must exists an inverse element which be combination with

the element itself yields the neutral element. 4. The associative law holds.

Point Groups I

David M. Bishop; Group Theory and Chemistry

Grouprequirements: 1. The combination of two elements must yield another element of

the same group. 2. There must exist a neutral element, which leaves the elements of

the group unchanged and commute with all other elements. 3. There must exists an inverse element which be combination with

the element itself yields the neutral element. 4. The associative law holds.

Point Groups I

David M. Bishop; Group Theory and Chemistry

Grouprequirements: 1. The combination of two elements must yield another element of

the same group. 2. There must exist a neutral element, which leaves the elements of

the group unchanged and commute with all other elements. 3. There must exists an inverse element which be combination with

the element itself yields the neutral element. 4. The associative law holds.

Point Groups I

David M. Bishop; Group Theory and Chemistry

Grouprequirements: 1. The combination of two elements must yield another element of

the same group. 2. There must exist a neutral element, which leaves the elements of

the group unchanged and commute with all other elements. 3. There must exists an inverse element which be combination with

the element itself yields the neutral element. 4. The associative law holds.

first

second

Point Groups I

David M. Bishop; Group Theory and Chemistry

Grouprequirements: 1. The combination of two elements must yield another element of

the same group. 2. There must exist a neutral element, which leaves the elements of

the group unchanged and commute with all other elements. 3. There must exists an inverse element which be combination with

the element itself yields the neutral element. 4. The associative law holds.

first

second

Point Groups II Further properties: • Order of the group (𝑔): 𝑔 = 6 • 𝑃 and 𝑅 are conjugate to each other if: 𝑃 = 𝑄−1𝑅𝑄

the elements which are conjugate to each other form a class. 𝑔𝑖 denotes the number of elements in the 𝑖th class. • 𝐸 is conjugate with itself: 𝑔1 = 1 • 𝜎𝑣′, 𝜎𝑣′′, 𝜎𝑣′′′ form a class: 𝑔2 = 3 • 𝐶3

−1 and 𝐶3 form a class: 𝑔3 = 2

Point Groups III In general: • 𝐸, 𝑖, 𝜎ℎ form a class on their own

• 𝐶𝑛𝑘 and 𝐶𝑛

1−𝑘 belong to the same class, if there exists a mirror plane

which contains the 𝐶𝑛𝑘 axis or a 𝐶2 axis perpendicular to the

𝐶𝑛1−𝑘 axis. The same holds for 𝑆𝑛

1 and 𝑆𝑛1−𝑘.

• Two reflection operations 𝜎′ and 𝜎′′ belong to the same class if there exists a symmetry operation which transforms all points of the 𝜎′ plane into the 𝜎′′ plane.

Point Groups IV Classification of the Point Groups by Schoenflies symbols:

Point Groups V Determination of a particular Point Group by Schoenflies symbols and a flow chart:

Point Groups V Example: Dichloromethane

𝐶2𝑣

Point Groups V Example: Methane

𝑇𝑑

Function space and matrix representation I

The five different 𝑑𝑖(𝑥1, 𝑥2, 𝑥3) orbitals can be presented in Cartesian coordinates by following equations:

Function space and matrix representation II For the 𝐶3𝑣 point group one finds the following matrix representation for the six symmetry operations for the five 𝑑𝑖(𝑥1, 𝑥2, 𝑥3)

Reducible and Irreducible Representation The set of 5 × 5 matrices (reducible representation) can in general be transformed in block diagonal form by symmetry transformation. No further block diagonalization possible => irreducible representation

Character Character of a symmetry opperation:

𝜒 𝑅 = tr M = 𝑀𝑖𝑖

𝑖

For the 𝐶3𝑣 point group one obtains for the irreducible representations:

Character Tables I

Construction rules: • The number of classes is equal to the number of irreducible representation. • The sum of the squares of the dimension of the irreducible representation 𝑛𝜇 is

equal to the order of the group.

Since the identity operation is always represented by the unit matrix, the first ee column is 𝜒𝜇 𝐸 = 𝑛𝜇 and the order of the group is also given by:

• The rows must fulfill: • The columns have to satisfy:

Character Tables II 𝐶3𝑣 point group example: • Three irreducible representation • First rows only ones • First column must be 𝑛

• Applying second point:

• Applying third point: =>

Character Tables III Nomenclatrue of the irreducible representation:

• 1D Irreps are labeled 𝐴 or 𝐵, depending on if the character of a 2𝜋

𝑛 rotation is +1

or -1. • 2D Irreps are labeled 𝐸. • 3D Irreps are lebeled 𝑇. • If a group contains 𝑖, 𝑔 or 𝑢 is added as an index depending on if the character of

𝑖 is +1 or -1. • If a group contains 𝜎ℎ but no 𝑖 the symbol gets primed or double primed

depending on if the character of 𝜎ℎ is positive or negative. • If there remain ambiguities after the rules 1-5 the symbols are given consective

numbers 1, 2, 3, … as indices.

Examples: 𝑪𝒔 𝑬 𝝈𝒉

𝑨′ 1 1

𝑨′′ 1 1

𝑪𝒊 𝑬 𝒊

𝑨𝒈 1 1

𝑨𝒖 1 -1

Character Tables III Nomenclatrue of the irreducible representation:

• 1D Irreps are labeled 𝐴 or 𝐵, depending on if the character of a 2𝜋

𝑛 rotation is +1

or -1. • 2D Irreps are labeled 𝐸. • 3D Irreps are lebeled 𝑇. • If a group contains 𝑖, 𝑔 or 𝑢 is added as an index depending on if the character of

𝑖 is +1 or -1. • If a group contains 𝜎ℎ but no 𝑖 the symbol gets primed or double primed

depending on if the character of 𝜎ℎ is positive or negative. • If there remain ambiguities after the rules 1-5 the symbols are given consective

numbers 1, 2, 3, … as indices.

Examples: 𝑪𝒔 𝑬 𝝈𝒉

𝑨′ 1 1

𝑨′′ 1 1

𝑪𝒊 𝑬 𝒊

𝑨𝒈 1 1

𝑨𝒖 1 -1

Normal modes and coordinates I Normal coordinates: • Molecule with N nuclei in the groundstate

=> mass weighted displacement coorinates: 𝑞𝑥(1), 𝑞𝑦

(1), 𝑞𝑧(1) , … 𝑞𝑧

𝑁

kinetic energy of the moving nuclei: 𝑇 =1

2 𝛿𝑖𝑗𝑞 𝑖

3𝑁𝑗=1

3𝑁𝑖=1 𝑞 𝑗

the potential energy relative to the equilibrium position:

𝑉 = 𝜕𝑉

𝜕𝑞𝑖 𝑜

𝑞𝑖 +

3𝑁

𝑖=1

1

2

𝜕²𝑉

𝜕𝑞𝑖𝜕𝑞𝑗 𝑜

3𝑁

𝑗=1

𝑞𝑖𝑞𝑗

3𝑁

𝑖=1

classical equation of motion:

𝛿𝑖𝑗𝑑2𝑞𝑗

𝑑𝑡2+

𝜕²𝑉

𝜕𝑞𝑖𝜕𝑞𝑗 𝑜

3𝑁

𝑗=1

𝑞𝑗; for 𝑖 = 1,2 …N

3𝑁

𝑗=1

𝑄 = ℎ𝑗𝑞𝑗3𝑁

𝑗=1

• 3N − 6(5) degrees of vibritional freedom • To each normal coordinate is a motion called normal mode associated • Each normal coordinate belongs to one of the irreducible representations of the

point group of the molecule.

Normal modes and coordinates II

New example: H2O / 𝐶2𝑣

Normal modes and coordinates III How does the 𝐸 operation work on the H2O molecule?

Normal modes and coordinates IV How does the 𝐶2 operation work on the H2O molecule?

Normal modes and coordinates V How to obtain the representation Γ0 for the 3𝑁 degress of freedom? • Determine the number of atoms that do not move • Multiply for each symmetry operation the number of fixed atoms by the character

of the representation Γ𝑡 from the character table

Normal modes and coordinates V How to obtain the representation Γ0 for the 3𝑁 degress of freedom? • Determine the number of atoms that do not move • Multiply for each symmetry operation the number of fixed atoms by the character

of the representation Γ𝑡 from the character table

Normal modes and coordinates VI The representation Γ0 for the 3𝑁 degress of freedom is reducible. One obtains: Searching for the representation Γ𝑣 for 3𝑁 − 6 degress of vibrational freedom => substracting the representations of rotation Γ𝑟 and translation Γ𝑡

Normal modes and coordinates VI The representation Γ0 for the 3𝑁 degress of freedom is reducible. One obtains: Searching for the representation Γ𝑣 for 3𝑁 − 6 degress of vibrational freedom => substracting the representations of rotation Γ𝑟 and translation Γ𝑡

Selection rules I

• Selection rules give information which modes can be observed in a IR or Raman spectrum

• The selection rules make no reference to the intensities, they only state wheter a mode is allowed or forbidden

• In reality also forbidden modes might be observed due to deviations from the harmonic approximation

Infra-red: A mode is IR active if the dipole moment changes during a vibration The transition probability form from the vibrational ground Ψ0

𝑣 state to an excited

state Ψ𝑚 (fundamental state) depends on the integral:

Selection rules II

Raman: A mode is Raman active if the polarizability changes during a vibration The transition probability form from the vibrational ground Ψ0

𝑣 state to an excited

state Ψ𝑚𝜌 (𝜈𝜌 is the fundamental frequency) depends on the integral:

Define the polarizability tensor and transform as

Selection rules III IR:

A mode 𝜈𝛿 is IR active if its representation Γ𝛿 is contained in the representation Γ𝜇. Raman:

A mode 𝜈𝛿 is Raman active if its representation Γ𝛿 is contained in the representation Γ𝛼.

Steps to predict modes in an IR or Raman spectrum

• Determine the representation Γ0 for the 3𝑁 degrees of freedom • Determine the representation Γ𝑣 by substracting Γ𝑡 and Γ𝑟 • Compare Γ𝑣 with Γ𝜇 (IR) and Γ𝛼 (Raman)

CH4 and CH3D example CH4: 𝑇𝑑 point group

CH4 and CH3D example CH4: 𝑇𝑑 point group

CH4 and CH3D example Monodeuteromethane CH3D : 𝐶3𝑣 point group (like ammonia NH3)

CH4 and CH3D example Monodeuteromethane CH3D : 𝐶3𝑣 point group (like ammonia NH3)

CH4 and CH3D example

Methane CH4 Monodeuteromethane CH3D

Number of fundamental frequencies which appear in the infra-red and Raman spectra are the same

Number of fundamental frequencies which appear in the infra-red and Raman spectra are different

Sufficient information to distinguish these two molecules

Some notes on real spectra

• Many IR spectra show more lines than predicted by symmetry arguments

• There exist other transition besides the fundamental normal modes which are generally less intence:

1. Overtones: They occur when a mode is exited beyond the fundamental state: 𝜓1(0)𝜓2(0) 𝜓3(0) → 𝜓1(0)𝜓2(3) 𝜓3(0)

2. Combination bands: A combination band is observed when more than on vibration is excited by one photon: 𝜓1(0)𝜓2(0) 𝜓3(0) → 𝜓1(1)𝜓2(1) 𝜓3 0 symmetry of such a mode can be calculated by the direct product of the Irreps of the normal modes: Γ(𝜓1)⨂Γ(𝜓2)

3. Hot bands: A hot band is observed when an already excited vibration is further excited: 𝜓1(0)𝜓2(1) 𝜓3(0) → 𝜓1(0)𝜓2(2) 𝜓3(0)

Some notes on real spectra

• Many IR spectra show more lines than predicted by symmetry arguments

• There exist other transition besides the fundamental normal modes which are generally less intence:

1. Overtones: They occur when a mode is exited beyond the fundamental state: 𝜓1(0)𝜓2(0) 𝜓3(0) → 𝜓1(0)𝜓2(3) 𝜓3(0)

2. Combination bands: A combination band is observed when more than on vibration is excited by one photon: 𝜓1(0)𝜓2(0) 𝜓3(0) → 𝜓1(1)𝜓2(1) 𝜓3 0 symmetry of such a mode can be calculated by the direct product of the Irreps of the normal modes: Γ(𝜓1)⨂Γ(𝜓2)

3. Hot bands: A hot band is observed when an already excited vibration is further excited: 𝜓1(0)𝜓2(1) 𝜓3(0) → 𝜓1(0)𝜓2(2) 𝜓3(0)

References

1. David M. Bishop; Group Theory and Chemistry; Dover Publication

2. http://www.raman.de/

3. http://www.ir-spektroskopie.de/

Thank you for your

attention!