Lectures in Spectroscopy - !வணக்கம்! WELCOME · 2018-11-17 · Introduction MW spectroscopy IR spectroscopy Vibration of Diatomic Molecule as Harmonic Oscillator A mass

Introduction MW spectroscopy IR spectroscopy

Lectures in Spectroscopy

Rotational (Microwave) &Vibrational (Infra-Red)

Spectroscopy

K. Sakkaravarthi

Department of Physics

National Institute of Technology

Tiruchirappalli – 620 015

Tamil Nadu

India

[email protected]

www.ksakkaravarthi.weebly.com

K. Sakkaravarthi Lectures in Spectroscopy 1/119


My sincere acknowledgments to

Fundamentals 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).

and many other free & copyright internet resources.



Outline of Talk

1 IntroductionSpectroscopyIR SpectroscopyMW Spectroscopy

2 MW spectroscopyRigid RotatorNon-Rigid RotatorLinear Polyatomic MoleculesSymmetric Top MoleculesMW Spectrometer

3 IR spectroscopyIR spectrometerVibration of Diatomic MoleculeVibrating Diatomic RotatorVibrations of Polyatomic Molecules



Vibrational (IR) spectroscopy



Vibrational (IR) spectroscopy

Review: Energy of a molecule

Total energy = Electronic+Vibrational+Rotational.

Vibrational energy E = hf(v + 1/2),f frequency of vibration, v vibrational quantum no.For a transition b/w v and v + 1 state: ∆E = hf .

Rotational energy E = ~2

2Il(l + 1).

For a transition b/w l and l − 1 state: ∆E = −~2l

I.

Change in energy : ∆Etot = hf − ~2 lI

. ∆l = −1.

(or) ∆Etot = hf + ~2 lI

. ∆l = +1 (from l to l + 1)

So, three spectral lines with different energiescorresponding to ∆l = −1, hf and ∆l = +1.



IR Absorption Spectrometer: Schematic-1



IR Absorption Spectrometer: Schematic-2



Vibration of Diatomic Molecule as Harmonic Oscillator

A mass m connected to a spring (having force constant k)undergoes simple harmonic motion, with fundamental

frequency f = 12π

√

km

.

If we consider two atoms are connected by an elastic bondand an external force stretches the bond according toHook’s law (stretching is proportional to the force applied).

Bond length changes from its equilibrium re to r (r > re).




Restoring force (F = −k(r − re)) tries to bring theatoms to their original equilibrium state.

So, the potential energy U(r) =∫

Fd(r− re) ⇒12k(r− re)

2.

Assumption: Bond is compressed (or stretched) & releasedand executes a harmonic motion.

The fundamental frequency of such harmonic oscillation

ν0= 1

2π

√

kµ.

Here µ is reduced mass µ = m1+m2

m1m2.

The comparison of molecular vibrations toharmonic motions is not accurate, because the bonddisplacement is not uniform/same.

Bond length: r >> re ⇒ dissociates & r << re ⇒ repels.

Molecular potential energy is not same to that of HO.So, the validity of harmonic oscillator model?

Molecular vibrations should be treated as an Anharmonicoscillator. K. Sakkaravarthi Lectures in Spectroscopy 72/119



Potential energy of vibrating atoms will have the influenceof higher order terms (polynomial as Taylor series) ofdisplacement.U(r) =

U(re) +(

dUdr

)

re∆r + 1

2!

(

d2Udr2

)

re∆r2 + 1

3!

(

d3Udr3

)

re∆r3 + · · ·




The first term is a constant and can be set to zero.

The second term is zero as the potential is a minimum.

Third term shows that systems are harmonic in first order.

Fourth & higher order terms are much smaller.

So, the harmonic approximation is good for smalldisplacements away from equilibrium.

Thus the potential energy becomes

U(r) = 12!

(

d2Udr2

)

re∆r2 + 1

3!

(

d3Udr3

)

re∆r3 + · · · .

⇒ U(r) = 12k∆r2 + 1

6γ∆r3 + · · ·

∆r: displacement.

For small displacements, higher order terms like ∆r3, ∆r4,∆r5, · · · become much smaller and negligible.

So, the final potential energy U(r) = 12k∆r2.

Energy is proportional to square of displacement.




PE curve: The SHO approximation is accurate for smalldisplacements (around r = re) only.

Curvature of PE curve shows force constant.More curve ⇒ higher force constant ⇒ higher frequency.

So, we should find the vibrational frequency/energy.

QM: Schrödinger equation model for molecular vibrations

Here ξ = r − re is the displacement.Solving the equation, we can obtain E

vib.




Quantized Vibrational energy levels of atoms in a molecule:Ev =

(

v + 12)hν

0, v = 0, 1, 2, · · · .

v: vibrational quantum no.

Energy (in wavenumber units): ǫv= Ev

hc=

(

v + 12

) ν0

ccm−1.

Lowest vibrational energy (for v = 0): ǫ0= 1

2ν̄0.

Zero-point energy: Molecules never have zerovibrational energy or never at rest. They vibratealways at least with frequency ǫ

0.



Vibration of Diatomic Molecule becomes Anharmonic

On a different perspective: Potential energy of anharmonicoscillator U(r) = De (1− exp[−a(r − re)])

2,Here De: dissociation energy.

Vibrational Energy :ǫv=

(

v + 12

)

ν̄e−

(

v + 12

)2x

eν̄e+(

v + 12

)3yeν̄e− · · · cm−1.

νe= ν̄

ec: equilibrium frequency of vibration.

xe

& ye: small change (anharmonic) in the bond from ∆r.

Neglecting higher order (retaining first order anharmonic).

ǫv=

(

v + 12

)

ν̄e−

(

v + 12

)2x

eν̄e

cm−1, v = 0, 1, 2, ...

Energy gap between two vibrational levels:ǫv+1

− ǫv= ν̄

e− 2x

eν̄e(ν + 1) cm−1, v = 0, 1, 2, ...




General: Anharmonic vibration of molecules

Exact zero-point energy ǫ0= 1

2ν̄e− 1

4x

eν̄e.

Vibrational energy ǫv=

[

1−(

v + 12

)

xe

] (

v + 12

)

ν̄e.




General: Anharmonic vibration of molecules



Selection Rules for Vibrational spectra

There should be a change in dipole moment for IRactive. So, a transition between two vibrational levels areallowed when

(

dµdξ

)

e6= 0: Dipole moment should change in the

equilibrium position.

∆v = ±1: The change in vibrational quantum number(harmonic approximation).

Note:Homonuclear molecules (Ex. N2, O2) have no dipole moment µ,so no possibility for ∆µ ⇒ IR inactive.

Vibrational spectra is possible for heteronuclear molecules only.Symmetry of the molecule is also restricts IR activity.



Selection Rules for Vibrational spectra...

There are some exemptions!!!

Some symmetric molecules may be IR active. Ex. CO2

Their asymmetric stretching/bending ⇒ ∆µ 6= 0.

For anharmonic oscillator type ∆v = ±1,±2,±3, · · ·



Vibrating Diatomic Molecule...

Previous discussions: Energy due to a transition from onevibration level to another.

Harmonic case (∆v = ±1): Ex. v → v + 1∆ǫ

v=

(

v + 1 + 12

)

ν̄0−(

v + 12

)

ν̄0= ν̄

0cm−1.

Energy gap is same (ν̄0) b/w any two allowed levels.

Anharmonic case: r − re is not fixed/uniform.Energy ǫ

v=

[

1−(

v + 12

)

xe

] (

v + 12

)

ν̄e

cm−1.

Energy gap (∆ǫv) is not same b/w two successive levels.

Here selection rule ∆v = ±1,±2,±3, · · · ⇒ level jumppossible!

Several spectral lines can be observed for transition b/wdifferent vibrational levels!!

As ∆v increases intensity decrease!!!



Vibrating (Anharmonic) Diatomic Molecule...

Boltzman distribution: at room temperature

Higher levels v > 0 have less than 0.01% of the totalpopulation.

So, transition from v > 0 to other higher levels arenegligible.Less possible: v = 1 → v = 2, v = 1 → v = 3, · · · ,v = 2 → v = 3, v = 2 → v = 4, · · ·

But, transition from ground state v = 0 to other levelsoccur the most.v = 0 → v = 1, v = 0 → v = 2, v = 0 → v = 3, · · ·

Boltzman distribution: at higher temperature

Higher levels v > 0 also can have sufficient population.

So, transitions from higher levels v > 0 are also occur.

These transitions are called Hot Bands!K. Sakkaravarthi Lectures in Spectroscopy 83/119



At room temperature

Fundamental absorption: v = 0 → v = 1∆ǫ

0→1=

(

1 + 12

)

ν̄e−(

1 + 12

)2ν̄ex

e−(

12ν̄e− 1

4ν̄ex

e

)

⇒ ν̄e(1− 2x

e)cm−1.

First overtone (jump): v = 0 → v = 2

∆ǫ0→2

=(

2 + 12

)

ν̄e−(

2 + 12

)2ν̄ex

e−(

12ν̄e− 1

4ν̄ex

e

)

⇒ 2ν̄e(1− 3x

e)cm−1.

Second overtone (jump): v = 0 → v = 3

∆ǫ0→3

=(

3 + 12

)

ν̄e−(

3 + 12

)2ν̄ex

e−(

12ν̄e− 1

4ν̄ex

e

)

⇒ 3ν̄e(1− 4x

e)cm−1.

From these one can get ν̄e

& xe.

At higher temperature

First Hot Band: (v = 1 → v = 2) ⇒∆ǫ

1→2= ν̄

e(1− 4x

e)cm−1.




Fundamental & Overtone transitions




Fundamental & Hotband transitions




Model spectrum withFundamental, Overtone & Hotband transitions






Vibrating Diatomic Rotator (vibration+rotation)

Molecular vibrations induce rotations!Molecular rotations induce vibrations!!

So, each vibrational level contain fine structures of energydue to rotations.

Total energy ǫtotal

= ǫrot

+ ǫvib

⇒ ǫJv

= ǫJ+ ǫ

v.

From the earlier discussionsǫJv

= BJ(J + 1)−DJ2(J + 1)2 + ν̄e− 2x

eν̄e(ν + 1) cm−1.

Here J, v = 0, 1, 2, 3, · · ·

Selection rules ∆J = ±1 & ∆v = ±1,±2, · · · .



Vibrating Diatomic Rotator...















Vibrating Diatomic Rotator: Transitions

Consider a transition b/w states J′

↔ J′′

and v → v + 1.

∆ǫJv

= B[J′

(J′

+ 1)− J′′

(J′′

+ 1)]−D[J′2(J

′

+ 1)2 −J

′′2(J′′

+ 1)2] +(

32ν̄e− 9

4x

eν̄e

)

−(

12ν̄e− 1

4x

eν̄e

)

cm−1.

⇒ ∆ǫJv

= ν̄e(1− 2x

e) +B(J

′

− J′′

)(J′

+ J′′

+ 1)]−D[J

′2(J′

+ 1)2 − J′′2(J

′′

+ 1)2] cm−1.

Based on the selection rule: ∆J = ±1: (′

vib. state)(i) ∆J = +1 ⇒ R branch∆ǫ

Jv= ν̄

0+ 2B(J

′′

+ 1)− 4D(J′′

+ 1)3 cm−1.(ii) ∆J = −1 ⇒ P branch∆ǫ

Jv= ν̄

0− 2B(J

′

+ 1) + 4D(J′

+ 1)3 cm−1.

Combining the above two P & R branch transitions:(∆ǫ

Jv)P,R = ν̄

0+ 2Bm− 4Dm3 cm−1 ⇒ ν̄

0+ 2Bm cm−1.

R branch: (J′

+1) = m > 0 & P branch: (J′′

+1) = m < 0.Centrifugal distortion D is very small.



Vibrating Diatomic Rotator: Transitions...R & P branch in Fundamental transition v = 0 → v = 1

IR spectra: R branch(m > 0) ⇒high freq. & Pbranch(m < 0) ⇒low freq.



Vibrating Diatomic Rotator: Transitions...Spectral Lines for Fundamental band v = 0 → v = 1

Maximum intensity for m =√

kT2hcB

+ 12.



Vibrating Diatomic Rotator: Transitions...

Vibrational spectra of CO

Spectra with LOW & HIGH resolution IR spectrometers.



Vibrating Diatomic Rotator: Transitions...Vibrational spectra of CO

CO spectra with HIGH resolution IR spectrometers.K. Sakkaravarthi Lectures in Spectroscopy 98/119


Fundamental band spectra of HCl molecule



Vibrating Diatomic Rotator: Transitions...

From the IR spectra, we can find

ν̄0

band centre/origin ⇒ anharmonicity.

B spectral spacing⇒ Rotational constant.

Centrifugal Distortion constant D. (But, here D << B).

Intensity of spectral line⇒ Population in a given vib. level.

Maximum intensity is possible for m =√

kT2Bhc

+ 12.

Transition Energy corresponding to maximum intensity:

(∆ǫJv)max.int = ν̄

0± 2B

(

√

kT2Bhc

+ 12

)

cm−1.

Energy gap b/w two maximum int. 4B

(

√

kT2Bhc

+ 12

)



Rotational energy levels of vibrational state v = 0 → 1



Influence of Asymmetry in Vibration-Rotation Spectra

Assumption so far: Vibration & rotation proceedindependently without much interaction. This givesconstant spacing (energy/frequency gap) b/w bands.

Reality: Vibration & Rotation are highly dependent andinfluence each other.

Asymmetric stretching of bonds changes the MoI, whichdecreases rotational constant continuously⇒ B can not be same for all vibrational states.⇒ Bv = h

8π2Ivc(or) Bv = Be −

(

v + 12

)

12B2e

ν̄.

Let us have different B for states v = 0 & v = 1: B0 & B1.ǫJv

= ν̄0+B1J

′

(J′

+ 1)−B0J′′

(J′′

+ 1)−DJ′2(J

′

+ 1)2 +DJ

′′2(J′′

+ 1)2 cm−1.ǫJv

= ν̄0+B1J

′

(J′

+ 1)−B0J′′

(J′′

+ 1) cm−1. (no D)

∴ (∆ǫJv)P,R = ν̄

0+ (B1 +B0)m+ (B1 −B0)m

2 cm−1.

R branches crowded (spacing decreases) than P branch.i.e. As frequency increases spacing decrease regularly.



Vibrations of Polyatomic Molecules

General: Molecule with n atoms has 3n degrees of freedom.

Actual: Translational & Rotational motions utilizes 3 each.

So, only 3n− 6 degrees of freedom: nonlinear molecule3n− 5 for linear molecule (if no rotation about bond axis)

These 3n− 6 or 3n− 5 degrees of freedom are responsiblefor stretching & deformation/bending vibrations .⇒ Internal/Normal/Fundamental vibrations.

Fundamental vibrations: Molecular motions are in phase &vibrate with same frequency.The centre of gravity is also same.

Ex.: Normal/Fundamental mode of vibrations inLinear molecule CO2 & Asymmetric molecule H2O!







Vibrations of Polyatomic Molecules...

Dipole moment change in CO2 molecule



Important Factors affecting Vibrations of Molecules

Anharmonicity: Potential energy functionDeviation from harmonic due to mechanical anharmonicity

Leads to unequal energy spacing & activation of overtones.Combination modes: simultaneous change in more modes.

Overtones & combinations: weak in Raman! stronger IR!!

Electrical anharmonicity: nonlinear changes in dipolemoment.

Fermi Resonances: Two or more levels of differentvibrations with nearly same/degenerate energy.

Perturbations in fundamental & first overtone of CO2.

Hydrogen Bonding: Molecules with H covalent bonds.Have different stretching/deformations resulting to a shiftin frequency, width & intensity of spectral lines.



Vibrations of Polyatomic Molecules...



Rotation+Vibration of Linear Polyatomic Molecules

Diatomic molecules: equally spaced for P & R branchvibrational lines.

Polyatomic molecules: Direction (both parallel orperpendicular) of vibrations affects change in dipolemoment.

Linear Molecules:(i) Parallel vibrations ∆J = ±1 & ∆v = ±1:P & R branch spectral lines.

(ii) Perpendicular vibrations ∆J = ±1, 0 & ∆v = ±1:P, R & Additional Q branch spectral lines for ∆J = 0.Q branch have frequency/energy in between P & R.













Vibrations of Linear Polyatomic Molecules...



Rotation+Vibration of Linear Polyatomic Molecules

Symmetric top Molecules:Total energy ⇒ ǫ

v(v + ǫ

J(J,K)) ⇒

ǫJv

=(

v + 12

)

ν̄e−(

v + 12

)2x

eν̄e+BJ(J + 1) + (A−B)K2

cm−1.Here J, v = 0, 1, 2, 3, · · · & K from J to −J .

Parallel vibrations ∆v = ∆J = ±1 & ∆K = 0:P, Q & R branch spectral lines (similar to linear).

Perpendicular vibrations ∆v = ∆J = ∆K = ±1:R branch: (∆ǫ

Jv)R = ν̄

0+ 2B(J + 1) + (A−B)(1± 2K).

P branch: (∆ǫJv)P = ν̄

0− 2B(J + 1) + (A−B)(1± 2K).

Q branch: (∆ǫJv)Q = ν̄

0+ (A−B)(1± 2K) cm−1.





Summary

From the present series of lectures, we have learned theprinciple, explicit description and detailed analyses onthe transitions due to different types of vibrational androtational characteristics of molecules by means ofspectroscopy!


Documents

Lectures in Spectroscopy - !வணக்கம்! WELCOME · 2018-11-17 · Introduction MW spectroscopy IR spectroscopy Vibration of Diatomic Molecule as Harmonic Oscillator A mass