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 CH 4 and CH 3 D example Some notes on real spectra References

Infra-red Spectroscopy I 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 en.wikipedia.org/wiki/file:infrared_spectrum.gif 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 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. Dipole moment: μ = δ r 1D = 3.336 10 30 Cm δ: partial charge r: distance vector between two partial charges D: Debye http://www.ir-spektroskopie.de/

Raman Spectroscopy I 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 http://en.wikipedia.org/wiki/raman_spectroscopy

Raman Spectroscopy II 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; ν 0 : vibrational frequency Rayleigh Anti-Stokes Stokes

Symmetry operations for a symmetric tripod Symmetry operations I like the ammonia molecule NH 3 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 1. Identity (E): doing nothing operation 2. Rotation (C n ): operation of rotation a molecule clockwise about an axis by an angle 2π. The axis with the highest n or with the smallest n 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: σ h If the plane contains the principle axis: σ v If the plane contains the principle axis and bisects the angle between two 2 fold axes (C 2 ) perpendicular to the principle axis: σ d 4. Rotation-reflection (S n ): 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. David M. Bishop; Group Theory and Chemistry

Point Groups I 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. David M. Bishop; Group Theory and Chemistry

Point Groups I 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 David M. Bishop; Group Theory and Chemistry

Point Groups II Further properties: Order of the group (g): g = 6 P and R are conjugate to each other if: P = Q 1 RQ the elements which are conjugate to each other form a class. g i denotes the number of elements in the ith class. E is conjugate with itself: g 1 = 1 σ v, σ v, σ v form a class: g 2 = 3 C 3 1 and C 3 form a class: g 3 = 2

Point Groups III In general: E, i, σ h form a class on their own C n k and C n 1 k belong to the same class, if there exists a mirror plane which contains the C n k axis or a C 2 axis perpendicular to the C n 1 k axis. The same holds for S n 1 and S n 1 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 C 2v

Point Groups V Example: Methane T d

Function space and matrix representation I The five different d i (x 1, x 2, x 3 ) orbitals can be presented in Cartesian coordinates by following equations:

Function space and matrix representation II For the C 3v point group one finds the following matrix representation for the six symmetry operations for the five d i (x 1, x 2, x 3 )

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 = M ii i For the C 3v 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 n μ is equal to the order of the group. Since the identity operation is always represented by the unit matrix, the first ee column is χ μ E = n μ and the order of the group is also given by: The rows must fulfill: The columns have to satisfy:

C 3v point group example: Three irreducible representation First rows only ones First column must be n Character Tables II 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 rotation is +1 n 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 σ h but no i the symbol gets primed or double primed depending on if the character of σ h 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: C s E σ h A 1 1 A 1 1 C i E i A g 1 1 A u 1-1

Normal modes and coordinates I Normal coordinates: Molecule with N nuclei in the groundstate => mass weighted displacement coorinates: q x (1), q y (1), q z (1), q z N 3N i=1 3N j=1 kinetic energy of the moving nuclei: T = 1 δ 2 ij q i the potential energy relative to the equilibrium position: V = 3N i=1 V q i classical equation of motion: 3N j=1 o q i + 3N 1 2 3N 3N i=1 j=1 d 2 q j δ ij dt 2 + ²V q i q j j=1 o ²V q i q j o q j q i q j q j ; for i = 1,2 N Q = h j q j 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. 3N

Normal modes and coordinates II New example: H 2 O / C 2v

Normal modes and coordinates III How does the E operation work on the H 2 O molecule?

Normal modes and coordinates IV How does the C 2 operation work on the H 2 O molecule?

Normal modes and coordinates V How to obtain the representation Γ 0 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 Γ t from the character table

Normal modes and coordinates VI The representation Γ 0 for the 3N degress of freedom is reducible. One obtains: Searching for the representation Γ v for 3N 6 degress of vibrational freedom => substracting the representations of rotation Γ r and translation Γ t

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 Ψ 0 v state to an excited state Ψ m (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 Ψ 0 v state to an excited state Ψ m ρ (ν ρ is the fundamental frequency) depends on the integral: Define the polarizability tensor and transform as

Selection rules III IR: A mode ν δ is IR active if its representation Γ δ is contained in the representation Γ μ. Raman: A mode ν δ is Raman active if its representation Γ δ is contained in the representation Γ α. Steps to predict modes in an IR or Raman spectrum Determine the representation Γ 0 for the 3N degrees of freedom Determine the representation Γ v by substracting Γ t and Γ r Compare Γ v with Γ μ (IR) and Γ α (Raman)

CH 4 : T d point group CH 4 and CH 3 D example

CH 4 and CH 3 D example Monodeuteromethane CH 3 D : C 3v point group (like ammonia NH 3 )

CH 4 and CH 3 D example Methane CH 4 Monodeuteromethane CH 3 D Number of fundamental frequencies which appear in the infra-red and Raman spectra are different Number of fundamental frequencies which appear in the infra-red and Raman spectra are the same 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: ψ 1 (0)ψ 2 (0) ψ 3 (0) ψ 1 (0)ψ 2 (3) ψ 3 (0) 2. Combination bands: A combination band is observed when more than on vibration is excited by one photon: ψ 1 (0)ψ 2 (0) ψ 3 (0) ψ 1 (1)ψ 2 (1) ψ 3 0 symmetry of such a mode can be calculated by the direct product of the Irreps of the normal modes: Γ(ψ 1 ) Γ(ψ 2 ) 3. Hot bands: A hot band is observed when an already excited vibration is further excited: ψ 1 (0)ψ 2 (1) ψ 3 (0) ψ 1 (0)ψ 2 (2) ψ 3 (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!