Lectures in Spectroscopy - !வணக்கம்! WELCOME · 2018-11-17 · Introduction MW...
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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
www.ksakkaravarthi.weebly.com
K. Sakkaravarthi Lectures in Spectroscopy 1/119
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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
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Introduction MW spectroscopy IR spectroscopy
Vibrational (IR) spectroscopy
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Introduction MW spectroscopy 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.
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Introduction MW spectroscopy IR spectroscopy
IR Absorption Spectrometer: Schematic-1
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Introduction MW spectroscopy IR spectroscopy
IR Absorption Spectrometer: Schematic-2
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Introduction MW spectroscopy IR spectroscopy
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).
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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 + · · ·
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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.
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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.
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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.
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Introduction MW spectroscopy IR spectroscopy
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, ...
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
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.
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Introduction MW spectroscopy IR spectroscopy
Vibration of Diatomic Molecule as Harmonic Oscillator
General: Anharmonic vibration of molecules
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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, · · ·
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Introduction MW spectroscopy IR spectroscopy
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!!!
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Introduction MW spectroscopy IR spectroscopy
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
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Introduction MW spectroscopy IR spectroscopy
Vibrating (Anharmonic) Diatomic Molecule...
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.
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Introduction MW spectroscopy IR spectroscopy
Vibrating (Anharmonic) Diatomic Molecule...
Fundamental & Overtone transitions
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Introduction MW spectroscopy IR spectroscopy
Vibrating (Anharmonic) Diatomic Molecule...
Fundamental & Hotband transitions
K. Sakkaravarthi Lectures in Spectroscopy 86/119
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Introduction MW spectroscopy IR spectroscopy
Vibrating (Anharmonic) Diatomic Molecule...
Model spectrum withFundamental, Overtone & Hotband transitions
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Introduction MW spectroscopy IR spectroscopy
Vibrating (Anharmonic) Diatomic Molecule...
K. Sakkaravarthi Lectures in Spectroscopy 88/119
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Introduction MW spectroscopy IR spectroscopy
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, · · · .
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator...
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator...
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator...
K. Sakkaravarthi Lectures in Spectroscopy 91/119
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator...
K. Sakkaravarthi Lectures in Spectroscopy 92/119
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator...
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator: Transitions...Spectral Lines for Fundamental band v = 0 → v = 1
Maximum intensity for m =√
kT2hcB
+ 12.
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator: Transitions...
Vibrational spectra of CO
Spectra with LOW & HIGH resolution IR spectrometers.
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Introduction MW spectroscopy IR spectroscopy
Vibrating Diatomic Rotator: Transitions...Vibrational spectra of CO
CO spectra with HIGH resolution IR spectrometers.K. Sakkaravarthi Lectures in Spectroscopy 98/119
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Introduction MW spectroscopy IR spectroscopy
Fundamental band spectra of HCl molecule
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Introduction MW spectroscopy IR spectroscopy
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
)
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Introduction MW spectroscopy IR spectroscopy
Rotational energy levels of vibrational state v = 0 → 1
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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!
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
Vibrations of Polyatomic Molecules...
Dipole moment change in CO2 molecule
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
Vibrations of Polyatomic Molecules...
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
Vibrations of Linear Polyatomic Molecules...
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Introduction MW spectroscopy IR spectroscopy
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.
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Introduction MW spectroscopy IR spectroscopy
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Introduction MW spectroscopy IR spectroscopy
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!
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