Electronic transitions: Vibrational and rotational structure

Transcription

1 Electronic transitions: Vibrational and rotational structure

2 An electronic transition is made up of vibrational bands, each of which is in turn made up of rotational lines

3 Vibrational structure Vibrational band positions of an electronic transition are obtained from the usual vibrational energylevel expression ν v -v = T e +ω e (v +1/)-ω e x e (v +1/) ω e (v +1/)+ω e x e (v +1/) +... Energy separation between the potential minima of the two electronic states Separation between vibrational energy levels in the ground and excited states

4 The intensities of the vibrational bands Population of the vibrational levels: N v =N tot exp(-e v /kt) the number of molecules vibrating with v frequency which can be found in the ground electronic state at temperature T, E v =h ω energy of the vibration, k Boltzman constant [ Intrinsic strength of the electronic transition Squared overlap integral of the two vibrational wavefunctions --Franck-Condon factor M = = ψ ev ev ψ * e' R = e μ ψ e e = ψ ψ v ψ e'' * e' v' dτ μψ el BO e'' v'' dτ ψ * v' q ψ = v'' v' v'' dτ ψ N approximation * e' v' + ( μ + μ ) ψ e ψ * e' ψ N e'' dτ e'' v'' el dτ ψ * v' μ N ψ v'' dτ wf electronic are orthogonal =0 wf vibrational are orthogonal within electronic state but not witrhin different el states

5 Franck Condon principle Electronic transitions are occur very quickly s Vibrational, rotational and translational motions are frozen in such a short time. The electronic transition occur vertically at the initial r value. Vertical electronic transition on energy-level digram Kinetic energy is conserved

6 Predictions of the intensities from the Franck-Condon principle Vibrating molecule spend more time at the classical inner and outer turning points of the vibrational motion than in the middle. Thus the transitions are approximated as occuring at the turning points. CN -0 strong in absorption 0-18 strong in emission

7 Vibrational structure is organized into sequences and progressions Sequence: 0-0, 1-1, - etc are strong if there is the optimal overlap of the vib. wf (r e ~ r e ) Progression: 3-1, -1, 1-1, 0-1 1)upper state progressions connect into the same lower vibrational level )lower state progression connect to the same upper vibrational level (r e very different to r e ) The vibrational bands of an electronic band system can be organized into Deslandres Table: Dim array of vibrational band energies

8 Rotational Structure of electronic transitions of diatomics Three types of transitions are possible: 1. ΔΛ=0, Λ =Λ = Σ +/- - 1 Σ +/- transitions have P and R branches (ΔJ=+/-1) these are parallel transitions, with the transition dipole moment lying along the z-axis. ΔΛ=+/ Σ +/- - 1 Π, 1 Π 1 Δ, etc. transitions have a strong Q-branch and similiar P,R branches these are perpendicular transitionsand have a transition dipole moment perpendicular to the molecular axis 3. ΔΛ=0, Λ =Λ different from Π 1 Π, 1 Δ 1 Δ transitions are characterized by a weak Q branches and strong P and R branches Nonsinglet transition are more complex and include effects of the spin and orbital angular momenta on the rotational structure Hund s cases a,b,c, d

9 The total power emitted by an excited rovibronic state One can obtain following equation from expression for the Einstein A factor (multiply by hv, n J number density of excited states and substitution of q v v IR e I S J /(J +1) for Iμ 10 I P J ' J '' = 16π 3ε c nj ' ν (J' + 1) 4 J ' J '' q v' v'' R e S J '' Excited state population in molecules [1/m 3 ] Is the transition frequency in [Hz] F-C factor Electronic transition dipole moment [Cm] Rotational line strength Hönl-London factor

10 Hönl-London factors are derived from the properties of symmetric top wavefunctions

11 Expressions for PQR branches v P = v 0 ( B' + B") J" + ( B' B") J" v R = v 0 + B' + (3B' B") J" + ( B' B") J" v Q = v 0 + ( B' B") J"( J" + 1) B <B spacing between the lines in the P branch increases as J increases and for the R branch decreases as J increases; at some point, it will pile up and then turn around forming edge structure- band head (characterisitic edge structure due to the overlap of many rotational lines); band is red degraded (degraded to longer wavelengths) B >B spacing between the lines in the P branch decreases as J increases forming band head (blue degraded band), R branch increases as J increases;

12 Fortrat parabola A Fortrat parabola is helpful in visualizing the rotational structure of a vibrational band: v P,R =v 0 +(B +B )m+(b -B )m v Q =v 0 +(B -B )m(m+1) The head will occur in the Fortrat parabola when dv/dm=0 m H =-(B +B )/(B -B ) with the head-origin separation v H -v 0 =-(B +B ) / (4(B -B )

13

14 B ~ Σ X ~ 4 4 Σ u g R P R

15

16 Polyatomic molecules vib-rot structure Within BO approximation, the separation of vibrational and electronic motion leads to the concept of associating electronic states with potential energy surfaces (PES) For polyatomic molecule the potential energy function is a function of 3N-6(or 5) Internal coordinates, expressed in terms of normal modes. The simple one dimensional curve is replaced by a multidimensional PES for each polyatomic electronic state. The solution of the Schrödinger equation for nuclear motion on each PES of polyatomic molecule provides the corresponding vibrational frequencies and anharmonicities for each electronic state: G d r r s ( v) = ωr ( vr + ) + xr, s( vr + )( vs + ) + gtt' ltlt' r r, s> r t, t' > t d d

17 Convention ~ ~ A X 1 v' v' 1 0 v' ',v'', etc 10 or 130 for the desription of electronic transitions for description of transitions between different electronic states Selection rules Within the BO and normal mode approximations the trnsition moment integral is comprised of an electronic transition dipole moment Me e and a product of 3N-6 (or 5) overlap integrals: M e' v' e'' v'' = M e' e'' ψ v ' 1 * ψ '' dq1 ψ ' * ψ ' dq... v 1 v v For totally symmetric vibrations Δν i = 0, ± 1, ± For nonsymmetric vibrational modes Δ ν i = ±, ± 4, ± 6

18 Non-BO effects Vibronic coupling: the Herzberg-Teller Effect (transitions of Benzene) nonsymmetric vibrational transitions can occur in electronic transition with the selection rule ν = ±1, ± 3... Δ i The total vibronic symmetry must be examined: Γ el Γ vib = Γ vibronic Jahn-Teller effect : any nonlinear molecule in an orbitally degenerate electronic state will always distort in such a way as to lower the symmetry and remove the degenerac J-T effect violets the selection rule ν = ±, ± 4 Δ i Renner-Teller Effect: is the interaction of vibrational and electronic angular momenta in linear molecule. The levels associated with bending modes are shifted in energy by an interaction that couples Vibrational motion to electronic motion for states in which Λ 0 ( Π, Δ,...)

19 R-T effect The R-T effect occurs because the double orbital degeneracy is lifted as a linear molecule bends during vibrational motion. As the linear molecule bends, the two potential curves V+, V- ( p-orbital in the plane of the molecule and the p-orbital out of the plane) become distinct. The combined vibrational and electronic motion on these two potential surfaces mixes the zeroth-order vibrational and electronic wavefunctions associated with electronic configurations. The coupling of electric and vibrational motion will also be characterized by new quantum number: K=Λ+l And vibrational symmetries are obtained from the direct product of the vibrational symmetry With the electronic orbital symmetry Γ el Γ vib = Γ vibronic

20

21

22