Electronic transitions: Vibrational and rotational

structure

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

Vibrational structure

Vibrational band positions of an electronic transition are obtained from the usual vibrational energy- level expression

ν

v‘-v‘‘ = Te +ωe ‘(v‘+1/2)-ωe ‘xe ‘(v‘+1/2)2+ ....- ωe ‘‘(v‘‘+1/2)+ωe ‘‘xe ‘‘(v‘‘+1/2)2+...

Energy separation between the potential minima of the two electronic states

Separation between vibrational energy levels in the ground and excited states

The intensities of the vibrational bands

Population of the vibrational levels:Nv‘‘ =Ntot exp(-Ev‘‘ /kT)

the number of molecules vibrating with v‘‘ frequency which can be found in the ground electronic state at temperature T, Ev‘‘ =h ω‘‘ energy of the vibration, k Boltzman constant

Intrinsic strength of the electronic transitionSquared overlap integral of the two vibrational wavefunctions --Franck-Condon factor

ionapproximatBO

qR

dddd

ddM

veev

vve

vNveleeNvveleee

veNeveveveev

ψψψ

τψμψτψψτψψτψμψ

τψμμψτμψψ

=

=

+=

+==

−

∫∫∫∫

∫∫

'''2

2''

*'

2''

*'

2''

*'

2''

*'

2''''

*''

2''''

*''

2 )(

[

wf electronic are orthogonal =0

wf vibrational are orthogonal withinelectronic state but not witrhin different el states

Franck Condon principle

Electronic transitions are occur very quickly ----10-15 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

Predictions of the intensities from the Franck-Condon principle

Vibrating molecule spend more time at the classical inner and outer turningpoints of the vibrational motion than in the middle. Thus the transitions areapproximated as occuring at the turning points.

CN

2-0 strong in absorption 0-18 strong in emission

Vibrational structure is organized into sequences and progressions

Sequence: 0-0, 1-1, 2-2 etc are strong if there is the optimal overlap of the vib. wf (re ‘~ re ‘‘)

Progression: 3-1, 2-1, 1-1, 0-1 1)upper state progressions connect intothe same lower vibrational level2)lower state progression connect to the same upper vibrational level(re ‘ very different to re ‘‘)

The vibrational bands of an electronic band system can be organized into DeslandresTable: 2 Dim array of vibrational band energies

Rotational Structure of electronic transitions of diatomics

Three types of transitions are possible:

1.

ΔΛ=0, Λ‘‘=Λ‘=0

-----1Σ+/- -1Σ+/-

transitions have P and R branches (ΔJ=+/-1)these are parallel transitions, with the transition dipole moment lying along the z-axis

2. ΔΛ=+/-1----------------1Σ

+/- - 1Π, 1Π −

1Δ, 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 0 --------1Π−1Π, 1Δ−1Δ 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

The total power emitted by an excited rovibronic state

''2

''''''4'

30

3

''' )1'2(316

JevvJJJ

JJ SRqJn

cP ν

επ

+=

Excited state population in molecules [1/m3]

Is the transition frequency in [Hz]

F-C factor

Electronic transitiondipole moment [Cm]

Rotational line strength Hönl-London factor

One can obtain following equation from expression for the Einstein A factor(multiply by hv, n J‘ number density of excited states and substitution ofq v‘v‘‘ IRe I2S J‘‘ /(2J‘+1) for Iμ10 I2

Hönl-London factors are derived from the

properties of symmetric top wavefunctions

Expressions for PQR branches

)1"(")"'(

")"'(")"'3('2

")"'(")"'(

0

20

20

+−+=

−+−++=

−++−=

JJBBvv

JBBJBBBvv

JBBJBBvv

Q

R

P

B‘<B“------------spacing between the lines in the P branch increases as J“ increasesand for the R branch decreases as J“ increases; at some point, it will pile up and then turn around forming edge structure- band head (characterisiticedge 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;

Fortrat parabola

A Fortrat parabola is helpful in visualizing the rotational structure of a vibrational band:

vP,R =v0 +(B‘+B“)m+(B‘-B“)m2

vQ =v0 +(B‘-B“)m(m+1)

The head will occur in the Fortrat parabola when dv/dm=0

mH =-(B‘+B‘‘)/2(B‘-B‘‘)

with the head-origin separation

vH -v0 =-(B‘+B‘‘)2 / (4(B‘-B‘‘)

R

R

P

−− Σ−Σ gu XB 44 ~~

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:

'',

',

, )2

)(2

()2

()( ttttt

tts

sr

rrsr

srr

rr

r llgdvdvxdvvG ∑∑∑>>

+++++= ω

Convention

20

01

10

'''

''' 321,2,1

~~

−−−−

−

oretcXAvv

vv

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 integralis comprised of an electronic transition dipole moment Me‘e‘‘ and a product of 3N-6 (or 5)overlap integrals:

...** 21''''''''' ''2

'2

''1

'1

dQdQMM vvvveeveve ψψψψ ∫∫=

2,1,0 ±±=Δ iνFor totally symmetric vibrations

For nonsymmetric vibrational modes 6,4,2 ±±±=Δ iν

Non-BO effects

Vibronic coupling: the Herzberg-Teller Effect (transitions of Benzene)

nonsymmetric vibrational transitions can occur in electronic transition with the selection rule ...3,1 ±±=Δ iν

vibronicvibel Γ=Γ⊗Γ

The total vibronic symmetry must be examined:

Jahn-Teller effect : any nonlinear molecule in an orbitally degenerate electronic statewill always distort in such a way as to lower the symmetry and remove the degenerac

J-T effect violets the selection rule 4,2 ±±=Δ iν

Renner-Teller Effect: is the interaction of vibrational and electronic angularmomenta in linear molecule.

The levels associated with bending modes are shifted in energy by an interaction that couplesVibrational motion to electronic motion for states in which ,...),(0 ΔΠ≠Λ

R-T effectThe 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 theplane of the molecule and the p-orbital out of the plane) become distinct. The combined vibrational and electronic motion on these two potential surfacesmixes the zeroth-order vibrational and electronic wavefunctions associated with

electronic configurations.

The coupling of electric and vibrational motion will also be characterized by new quantumnumber:

K=Λ+lAnd vibrational symmetries are obtained from the direct product of the vibrational symmetryWith the electronic orbital symmetry

vibronicvibel Γ=Γ⊗Γ