Vibrational states of molecules Diatomic molecules Polyatomic molecules
Diatomic molecules V v 1 v 0 Re Q
Birge-Sponer plot
The solution of the Schrödinger equation can be solved analytically for the Morse potential. One can show that the eigenvalues can be written as: E/hc=ω e (v+1/2)-ω e x e (v+1/2) 2 +B e [J(J+1)]-D e [J(J+1)] 2 -α e (v+1/2)[j(j+1)] D e =4B 3 e (1/ω2 e )--------Kratzer relationship α e = [6(ω e x e B 3 e )1/2-6B 2 e ](1/ω e )---------Pekeris Relationship More general form is given by Dunham (Dunham potential and application of Wentzel-Kramers-Brillouin ) theory
Selection rules in vibrational spectrum of diatomic molecules
μ(r) can be determined from stark effect measurements, from intensities of infrared transitions or from ab initio calculations
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Vibration-rotation transition of diatomic molecules V=1 [G 1 (v)+f 1 (J)] - [G 0 (v)+f 0 (J)] V=0 0
Vibration-rotation transition of diatomic molecules In general, molecules vibrate and rotate. The selection rules are obtained by combining the pure rotational selection rules ΔJ=+/-1 apply to molecules with no net spin and orbital momentum. For other molecules weak Q branches are possible (ΔJ=0) Transitions are organized into branches: vr(v,j +1 v,j)--------r-branch vp(v,j -1)<- v,j)--------p-branch The P and Q branch can be combined into the single expression: v=v 0 +(B +B )m+(b -B )m 2 m=j+1-----------r branch m=j-1-------------p branch band origin
Example of the spectra There is a gap at the band origin where a Q branch would be present if ΔJ=0 were allowed. This band gap between R(0) and P(1) of the two branches Is approximately 4B
Combination differences: allow the rotational constants of the upper and lower states to be determined independently A transition depends on both upper and lower state constants. The differences of line positions depends only on upper or lower state spectroscopic constants. They can be arrange as alower state combination or upper state combination difference: Δ 2 F (J)=v[R(J-1)-v[P(J+1)] =B (J+1)(J+2)-B (J-1)=4B (J+1/2) Δ 2 F (J)= =B =4B (J+1/2)
Th intensity distribution in vib-rot spectrum The intensity distribution in the rotational transitions in a vibration-rotation band is governed by the Boltzman population distribution among the initial states of the transitions N N q J ' (2 '' 1)exp[ ] rot = 1 = q rot kt hcb '' J + hcb'' J''( J'' + 1) kt rotation partition function
The Raman effect: a light scatterring phenomenon The processes which cause molecular Raman scattering involve electronic, vibrational or rotational transitions accompanying the scattering process. Monochromatic radiation E induces in molecule electric dipole μ μ=α E The polarizability α is a measure of the degree to which the electrons in the molecule can be displaced relative to the nuclei
Classical model When an electric field is incident on an molecule, the electrons and nuclei respond by moving in opposite directions in accordance with coulomb s law. If some internal motion of the molecule modulates this induced oscillating dipole moment, then additional frequencies can appear. Polarizability has α 0 static term a0 and sinusoidal oscillating term α 1 α=α 0 +α 1 cosωt Internal frequency: rotational mode or vibrational mode e.g., for vibrational mode it means that: α = α 1 Q = 0Qi Qi i If α Q i 0= 0 then there is no vibrational Raman effect
Classical predictions of the intensity change: μ ind = αe = ( α 0 + α cosωt) E 1 0 cosω inc t cosθ cosφ = μ ind 1 [cos( θ φ ) + cos( θ + φ )] 2 = α E cosω t 0 0 inc α E + 1 0 [cos( ω 2 inc ω) t + cos( ω inc + ω) t] For a classical oscillator the scattering (Rayleigh, Raman) is proportional to the fourth power of the frequency, and introducing the Boltzman distribution of vibrational populations one can obtain the intensities of the bands: Anti Stokes Stokes = ( ν + ν i ( ν ν i vib vib ) ) 4 4 e hν kt vib
Quantum Model For highly symmetric molecules, diatomics, or CH 4 the induced dipole moment is always in the same direction as the applied electric field. For less symmetric molecules, they will generally point in different directions. μ X ( μ ) = μ Y Z α ( α α XX XY XZ α α α XY YY YZ α α α XZ YZ ZZ E Χ )( EY ) E Z Polarizabilty tensor The intensity of the Raman effects depends, in general, on the square of integrals of the type: ψ 1 αij ψ 0dτ i, j = X, Y, Z
For the vibrational Raman effect the intensity depends on the square of integrals, and the integral is evaluated in the molecular coordinate system. Group theory analysis is neccesary to evaluate a direct product: * Γ( ψ 1 ) Γ( αij ) Γ( ψ 0) It must contain the A 1 irrepreducible representation in order for the corresponding integral to be nonzero and give a Raman transition Note: If molecule has a ceneter of symmetry, then both ψ0 for fundamentals and αij have g symmetry and ψ1 must be of g symmetry. All Raman active fundamental transitions have g symmetry if the molecule has a center of symmetry. Correspondingly, all infrared fundamentals must have u symmetry since μ has a u symmetry. The rule of mutual exclussion: no fundamental mode of a molecule with center of symmetry can be both infrared and Raman active
Selection rules in Raman effect Rotational Raman effect (derived by evaluating the integral): ΔJ=+/-2 (there are two photons involved in Raman transition!)
The rotational Raman is less restictive than is pure rotational spectroscopy because symmetric linear molecules without dipole moments, Cl 2, CO 2 etc have pure rotationa spectra. Notice: spherical top will not have observable rotationa spectrum (no an anisotropic polarizability is present)
Selection rules for vibration-rotation Raman Spectroscopy Δv=+/-1 ΔJ=0,+/-2 O-branch For 1 Σ + states S-branch Q-branch
Polyatomic molecules **
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Notation: Herzberg order 1.Symmetry consideration eg., A 1, A 2, B 1 using C 2v 2. For given symmetry type the frequencies of the modes are arranged in descending order For linear 3-atomics v2 is always the bending mode!
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Selection rules for vibrational transitions in polyatomics M = ψ μ = μ + f * μψ dτ = 3N 6 μ ( 0 ) 0 k= 1 Qk M i Q k ψ f * ( Q) μ( Q) ψ ( Q) dq 3N 6 μ = μ ψ f * ψ idq + ( ) 0 Q 0 ψ * f k= 1 k 0 the vibrational functions are orthogonal i Q ψ dq k i Δυ = ±1 Properties of Hermite polynomials selection rule follows: -harmonic oscyllator wf -truncation of the expansion of the dipole moment at terms linear in Q The intensity of electric dipole allowed vibrational transition is given ψ f * μψ i r Γ ψ *) Γ( μ) Γ( ψ ) ( f i I f i = 2 μ 2 ( ) 0 φ f * ( ξ j ) Q jφi ( ξ j ) dq j Q j must be totally symmetric, must contain the A 1 irreducible representation
Vibration-rotation transition of linear molecules Similar to diatomics; the molecular symmetry:d ~h C ~v ; 3N-5 modes of vibration The fundamental vibration transitions are: parallel type Σ Σ perpendicular Π Σ for bending modes Transition dipole moment is parallel to the molecular axes μ z Transition dipole moment is parallel to the molecular axes μ x, y
Examples R(0) u P(1) Antisymmteric stretching fundamental v 3 (σ+ u ) g u g
Other possibilities: Vibrational transitions between vibrationally excited states Σ Σ, Π Σ or Π Π, Δ Δ Transition between fundamentals, overtones or hot bands Transitions between combination bands Vibrational spectra of symmetric tops and asymmetric tops
Fermi and Coriolis perturbations Generally, the regular energy pattern predicted by the term expression G(vi) rearly exists and deviations from the regular pattern are called perturbations ----a Fermi resonance----: Interaction between vibrational levels of the same symmetry two states of the same symmetry are in close proximity and are coupled by a non-zero anharmonic interaction term
Coriolis perturbation Coriolis effects (first order)can split energy levels of the same symmetry, and Second order Coriolis effects are possible between states of different symmetry Coriolis interactions involve rotational motion See examples in P. Bernath, Spectra of atoms and molecules Similarly, one can there also find an introduction to the spectroscopy describing a fluxional behavior in molecules