lectures in spectroscopy electronic spectroscopyintroduction vibronic transitions rotational...

Introduction Vibronic transitions Rotational structure

Lectures in Spectroscopy

Electronic Spectroscopy

K. Sakkaravarthi

Department of PhysicsNational Institute of Technology

Tiruchirappalli – 620 015Tamil Nadu

India

Email: [email protected]

K. Sakkaravarthi Lectures in Spectroscopy 1/35


My sincere acknowledgments toFundamentals of Molecular Spectroscopy, 4th Ed.,C.N. Banwell, McGraw-Hill, New York (2004).

Molecular Structure and Spectroscopy,G. Aruldhas, Prentice Hall of India, New Delhi (2002).

Introduction to Atomic Spectra,E. H. White, McGraw-Hill, New York (2005).

Many other free & copyright internet resources.



1 IntroductionElectronic transitions

2 Vibronic transitionsVibrational coarse structureProgressions & SequencesDissociation & Predissociation

3 Rotational fine structureRotational fine structuresFortrat parabolae



SpectroscopyThree types of spectra in molecular transitions!



Spectroscopy...Three types of spectra in molecular transitions!



Rotational/MW Spectroscopy

Low energy EMR (MW/Far-IR) can change rotationallevels only!



Vibrational/IR Spectroscopy

Medium energy EMR (Near-IR) can change vibrationallevels and rotational sublevels!!



Electronic SpectroscopyHigh energy EMR (UV/Vis) can change electroniclevels along with vibrational and rotational sublevels!!!



Electronic Spectroscopy...High energy EMR (UV/Vis) can change electroniclevels along with vibrational and rotational sublevels!!!



Electronic Spectra in MoleculeEach electronic state has several vibrational levels whichthemselves contain a large number of rotational sub-levels.

Transitions between different electronic states falling in thevisible/UV region of EMR spectrum.Separation between electronic levels ≥ 106 cm−1.

Electronic transitions also change both vibrational (coarsestructure) & rotational (fine structures) levels.Each electronic state (ground or excited state) has its ownpotential function characterized by the equilibriuminternuclear distance re and dissociation energy Do or De.So, electronic spectra give information about rotationalconstants & vibrational frequencies.

All molecules show electronic spectra, includinghomonuclear molecules.



Vibrational coarse structure due to Electronic transitionsw.k.t. Total Molecular energy ε

total= ε

el+ ε

vib+ εrot .

For better understanding of vibrational coarse structure, weomit rotational effects now! So, ε

total= ε

el+ ε

vib!!

εtot = εel

+(v + 1

2

)ν̄e −

(v + 1

2

)2xe ν̄e +

(v + 1

2

)3ye ν̄e − · · ·

cm−1.Transition between two Electronic levels ε′

el& ε′′

el:

ν̄v′v′′ = (ε′el− ε′′

el) +

[(v′ + 1

2

)ν̄ ′e−(v′ + 1

2

)2x′

eν̄ ′e

]−[(

v′′ + 12

)ν̄ ′′e−(v′′ + 1

2

)2x′′

eν̄ ′′e

]cm−1.

Lower state ′′ & Upper state ′.No specific selection rule for vibrational changes.∵ Transition occurs across electronic states which havedifferent vibrational levels.More population in ground state. ∴ Transitions fromground state have more intense spectral lines.



Vibrational coarse structure due to Electronic transitionsMost of the molecules will be, initially, in the v = 0 state ofthe ground electronic state el = 0.Several spectral lines for single electronic transition whichhas multiple vibrational level changes: Progression(v′, v′′) can be (0,0), (1,0), (2,0), (3,0), etc.

Ex.: (0,0) Transitionν̄00 = (ε′

el− ε′′

el) + 1

2 ν̄′e− 1

4x′eν̄ ′e− 1

2 ν̄′′e

+ 14x′′eν̄ ′′ecm−1.

From electronic spectral band, we can findvibrational frequency ν̄e ,anharmonicity constant xe , andseparation between two electronic states ∆ε

el.

Probability of transition depends on the relative position ofpotential energy internuclear curves (or Morse curve).



Vibrational coarse structure due to Electronictransition from ground state



Vibrational analysis: Deslandres TableFor a clear understanding of electronic transition, weneglect energy for least vibrational state (consider NILzero-point energy)!Vibrational energyE(v) = (v + 1

2)v̄e − (v + 12)2v̄exe + (v + 1

2)3v̄eye + · · ·

Let v = 0 state E(0) = 12 v̄e −

14 v̄exe + 1

8 v̄eye + · · · = 0.⇒ E0(v) = vv̄e−v2v̄exe−vv̄exe+v3v̄eye+ 3

2vv̄eye+ 34v

2v̄eye+·⇒ v

(v̄e − v̄exe + 3

4 v̄eye)− v2

(v̄exe − 3

2 v̄eye)

+ v3v̄eye + · · ·⇒ E0(v) = vv̄0 − v2v̄0x0 + v3v̄0y0 + · · ·

Energy for any electronic transitions:ν̄v′v′′ = ν̄00 + E′0(v′)− E′′0 (v′′).

Deslandres Table: For all electronic transitions



Deslandres Table:Energy of electronic transitions among different vibrational levels

First difference transition: E0(v + 1)− E0(v)∆E(v + 1/2) = ν̄0 − ν̄0x0 − 2ν̄0x0v.

Second difference transition: E0(v + 1)− E0(v)∆2E(v + 1) = ∆E(v + 11

2)−∆E(v + 12).



Progressions & SequencesWavenumber (frequency/energy) difference betweenadjacent electronic levels (higher state v′ & lower state v′′).When vibrational level is increasing, the populationdistribution decreases. So, intensity of spectra varies duringtransition among the two levels!

v′ progression: For same vibrational ground/lower level v′′,different energy/intensity spectra based on various v′.

v′′ progression: For same vibrational ground/lower level v′,ifferent energy/intensity spectra based on various v′′.

Sequences: The diagonal levels (0,0), (1,1), (2,2), (3,3), etc.,



Franck-Condon PrincipleIntensity distribution along the progression determineswhich transitions are sufficiently intense (most probable).

Franck: Classical description for vibronic transitions!1. No zero-point energy.2. Only vertical transitions(transitions without change of nuclear geometry).3. Not much change in the intensity of band spectra.

Condon: Transitions are explained in Q.M. viewpoint!!1. Explicit zero-point energy.2. Vertical transitions with energy (horizontal*) shift.3. Significant differences in the intensity of band spectra.

* Upper potential energy (Morse) curve is appreciablydisplaced horizontally from the lower.



Franck: Classical view to Electronic transitionsNo zero-point energy: Vertical transitions only!

Vibronic transition when (a) r′e = r′′e and (b) r′e > r′′e



Probability distribution as wave function

Franck-Condon PrincipleThe most probable vibronic transition is a vertical transitionbetween positions on the vibrational levels of the upper andlower electronic state at which the vibrational wave functionshave maximum values.



Franck-Condon: Quantum view to Electronic transitionsWith zero-point energy: Vibrational wave function

Vibronic transition when (a) r′e = r′′e and (b) r′e > r′′e .



Vibronic transition for (a) r′e = r′′e , (b) r′e > r′′e & (c) r′e >> r′′e .K. Sakkaravarthi Lectures in Spectroscopy 24/35


Franck-Condon: Quantum view to Electronic transitions

Typical intensity distribution along a vibrational progressionwhen (a) r′e = r′′e and (b) r′e > r′′e

The most intense vibronic transition is from the groundvibrational state to the vibrational state lying vertically aboveit. Transitions to other vibrational levels also occur, but withlower intensity.



Dissociation (The breaking of bonds)Dissociation energy: Energy required to separate astable diatomic molecule AB in the v = 0 state into twounexcited atoms A and B!Dis. energy: De = D0 + ε0 cm−1

Energy of dissociated prod-ucts Eex = D′e −D′′e

ν̄cont. = D′′0 + Eex cm−1.ν̄cont. = D′0 + ν̄00 cm−1,

Combining the above two Eqs.D′′0 = D′0 + ν̄00 − Eex cm−1.



Dissociation...The vibrational structure of a band terminates at a certainenergy. Absorption occurs in a continuous band above thisenergy limit because the final state is an unquantizedtranslational motion.Dissociation limit: The frequency at which vibrationalstructure is replaced by continuous absorption.Locating the dissociation limit helps to find bonddissociation energy.Vib. energy: εv =

(v + 1

2

)ν̄e −

(v + 1

2

)2ν̄exe cm−1.

Energy diff. ∆ε = εv+1 − εv = ν̄e[1− 2xe(v + 1)] cm−1.∆ε decreases steadily from v = 0 to v = 1, 2, 3, · · · .∆ε = 0 at any vmax. ie. vmax = 1

2xe− 1.

So, Dis. energy De = ν̄e4xe− 1

4xeν̄e ≈ν̄e

4xecm−1.

If we know ν̄e & xe, we can get De.



PredissociationBreaking of molecules for energy < De!!Dissociation that occurs before the dissociation limit is reached.

Here the vibrational structure disappears but resumes athigher photon energies.When a molecule is excited to a vibrational level, itselectrons may undergo a reorganization (internalconversion: a radiationless conversion to another state).An internal conversion occurs at the point of intersection ofthe two molecular potential energy curves, because therethe nuclear geometries of the two states are the same.The state into which the molecule converts may bedissociative, so the states near the intersection have a finitelifetime, and hence their energies are imprecisely defined.As a result, the absorption spectrum is blurred in thevicinity of the intersection.



Predissociation...When the incoming photon brings enough energy to excitethe molecule to a vibrational level high above theintersection, the internal conversion does not occur (thenuclei are unlikely to have the same geometry).Consequently, the levels resume their well-defined,vibrational character with correspondingly well-definedenergies, and the line structure resumes on thehighfrequency side of the blurred region.



Schematic for predissociation



Rotational fine structures:Electronic+Vibrational+Rotational Spectra in Molecule.Including the vibrational energy, total molecular energyε′tot = ε′el + ε′vib +B′J ′(J ′ + 1) cm−1,ε′′tot = ε′′el + ε′′vib +B′′J ′′(J ′′ + 1) cm−1.Transition frequency/energyν̄ = ε′el − ε′′el + ε′v − ε′′v +B′J ′(J ′ + 1)−B′′J ′′(J ′′ + 1) cm−1.(or) ν̄ = ν̄v′v′′ +B′J ′(J ′ + 1)−B′′J ′′(J ′′ + 1) cm−1.Here v′v′′ corresponds to any vibronic transitions,Ex.: (0,0), (1,0), (0,1), (2,0), (0,2), etc.

Spectrum has different ‘bands’ based on J !! (∆J = J ′− J ′′)(i) R branch +1, (ii) P branch −1 & (iii) Q branch 0.ν̄P = ν̄v′v′′ − (B′ −B′′)(J ′ + 1) + (B′ −B′′)(J ′′ + 1)2 cm−1.ν̄Q = ν̄v′v′′ + (B′ −B′′)(J ′ + 1) + (B′ −B′′)(J ′′ + 1)2 cm−1.ν̄Q = ν̄v′v′′ + (B′ −B′′)J ′′2 − (B′ −B′′)(J ′′) cm−1.



Rotational vibronic transitions for diatomic molecule



Fortrat parabolae: Nature of band spectraFrequencies/energies of P,Q,R band spectraν̄P,R = ν̄v′v′′ + (B′ −B′′)p+ (B′ −B′′)p2 cm−1.ν̄Q = ν̄v′v′′ + (B′ −B′′)q2 − (B′ −B′′)q cm−1.Here p = ∓(J ′ + 1) and q = J ′′.dν̄P,R

dp = 0 = B′ +B′′ + 2(B′ −B′′)p.

Band head Phead = −(B′+B′′)2(B′+B′′) .

ν̄P,R − ν̄v′v′′ = −(B′+B′′)4(B′+B′′) .



SummaryFrom the present series of lectures, we have learned theprinciple, explicit description and detailed analyses ofelectronic transitions in molecules by including thevibrational and rotational effects!



Thank You


lectures in spectroscopy electronic spectroscopyintroduction vibronic transitions rotational...

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