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Dalian ROA Lectures June-July 2010

Lecture 1

Introduction to Vibrational Spectroscopy

Basic Principles, Force Fields, Normal Modes and IR Spectral Measurement

Outline

• Definitions of IR and Raman Spectroscopy• Vibrational Frequencies • Vibrational Force Fields • Vibrational Normal Modes• Measurement of IR Spectra

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Definitions of IR and Raman Spectroscopy

OverviewA. Vibrational spectroscopy –nuclear vibrations in

molecules are excited within the ground electronic state of the molecule

B. Transition frequency in IR region between 10 and 12,800 cm-1

1. IR absorption a direct resonance between transition frequency and photon frequency2. Raman scattering, scattered radiation shifted from the incident laser frequency by vibrational transition frequency

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Vibrational Energy LevelsInteraction of a Molecule with Radiation During Vibrational ExcitationInfrared Absorption Raman Scattering

Absorption of Infrared Radiation Scattering of Incident Laser Radiation

IRRaman

Raman Scattering Energy Level Diagram

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Vibrational Frequencies

Molecular Vibrations

A. For a diatomic molecule or fragment, the vibrational stretching frequency is

B. For a polyatomic molecule there are 3N-6 vibrational normal modes

C. In the harmonic approximation

D. If rotational transitions are included

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fνπ μ

= 1 2

1 2

m mm m

μ =+

0( 1/ 2)vibE h nν= +

0( 1/ 2) ( 1)vibE h n BhJ Jν= + + + 28hB

Iπ=

5

E. Anharmonicity

F. Calculation of normal modes must be done with computer programs.

1. In typical molecules, the vibrational modes are highly coupled

2. Determinations are now done using ab initio theoretical calculations for all but the most complex molecules

G. Fundamentals, overtones and combination bands1. Fundamentals are single quanta transitions2. Overtones are two or more vibrational quanta3. Combination bands are mixtures of different

quanta

20 0( 1/ 2) ( 1/ 2)vibE h n x nν ⎡ ⎤= + − +⎣ ⎦

H. Wavenumbers

1. IR spectra are given either in microns or wavenumbers in cm-1

I. Vibrational spectra are so complex that any molecule is unambiguously characterized by its vibrational spectrum

1 4( / ) 10 /( / )ν λ μ− =cm m

110,000 1μ− =cm m 15,000 2μ− =cm m

/ 1/ν ν λ= =c

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J. Regions of the Infrared Spectrum, Table 1.

1. Near-IR 12,800 to 4000 cm-1

2. Mid-IR 4000 to 200 cm-1

3. Far-IR 200 to 10 cm-1

Vibrational Force Fields

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( ) = ( )m m m, ,Ψ Ψr R r RH EMolecular Schrödinger Equation

= E N EE EN NN E NT T V V V H T+ + + + = +H

Adiabatic Approximation ( ) ( ) = ( )( )υ υψ φΨ ≅ Ψ r r RR Rr R, A

emA

e e, ,

Electronic Schrödinger Equation

Born-Oppenheimer Approximation( 0) =ψ r ReNAT ,

( ) υΨ ≅ + ΨeA

m NEH TH

( ) +( ) ( ) ( )( ) ( )=υ υ υ υψ ψφ φ φψr r R rR R R R RNA A A

E e e e e eA

e eH , , ,ET

( ) = ( ) ( )ψ ψR R Rr rA A AE e e eH , E ,

( ) ( )( ( )) = ( )υ υ υψ ψφ φ= + R Rr r R RA AE e e e e e

ANH , ,T E

Most Important Equation in Quantum Chemistry

Born-Oppenheimer Schrödinger Equation

Electronic Wave Equation Most Important Equation in Quantum Chemistry

( ) = ( ) ( ) = ( ) ( )ψ ψ ψR R R R Rr r rA A A A AE e e e e eH , E , , E

=( ) ( ) ( ) (( ) + ( ) ( ))υ υ υ υψ φ φψ ψ φR R R Rr r r rR R RA A A Ae e e e N e

Ae e e, E ET, , ,

( ) +( ) ( ) ( )( ) ( )=υ υ υ υψ ψφ φ φψr r R rR R R R RNA A A

E e e e e eA

e eH , , ,ET

Nuclear Wave Equation (from B-O Schrödinger Equation)

( ) + ( ) ( )) =( υ υ υ υφ φ φR R R RAe e e

Ae eN EE T

) ( )=( )( υ υ υφ φ⎤⎦+⎡⎣ R R RAe e

AN e eE ET

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Vibrational Potential Energy of Ground Electronic State

0 0

23 3

01 , 1

1( ) = ( ) ...2 ′

′= = ′

⎛ ⎞ ⎛ ⎞∂ ∂+ Δ Δ Δ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠∑ ∑

= =

R R R + R RR R R

A AN Ng gA A

g g J J JJ J JJ J JR R R R

E EE E

Energy at any Nuclear Position

Energy at Equilibrium Nuclear Positions

Slopes of Energy Surface

( )RAgE

0( )RAgE

0

3

1=

⎛ ⎞∂⎜ ⎟⎜ ⎟∂⎝ ⎠

∑=

R

ANg

J J R R

E

Curvatures of Energy Surface, Force Constants

0

23

, 1

12 ′= ′

⎛ ⎞∂⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

∑=

R R

ANg

J J J J R R

E

Vibrational Normal Modes

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Vibrational Wavefunction and Normal Modes

Normal Coordinate Transformation

Vibrational Potential in Normal Coordinates

21 ( ) ( ) ( )

2 υ υ υφ φ⎡ ⎤+ =⎢ ⎥⎣ ⎦∑ R R R R3N

A AJ J g g g g

JM E E

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, 1 0

1 1 ( ) ( )2 2

φ φ′′= ′ =

⎡ ⎤⎛ ⎞∂+ Δ ⋅ ⋅Δ =⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑R R R R RR R

A3N Ng A

J J J J gv gv gvJ J J J J R

EM E

23 3

, 1 , 0 00

1( ) =2

′

′= ′= ==

⎛ ⎞∂⎛ ⎞ ⎛ ⎞∂ ∂⋅ ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

∑ ∑ R RQR R

a b

AN N-6gA J J

g a bJ J Q Q a J J bQ QR

EE Q Q

Q Q

3 3

0

==

⎛ ⎞∂Δ =⎜ ⎟∂⎝ ⎠

∑ ∑RR SN -6 N -6

JJ a Ja a

a aa Q

Q QQ

S-vector SJa

Normal Mode Potential and Wave Equation

Vibrational Wavefunction in Normal Coordinates - Diagonal

Vibrational Potential in Normal Coordinates continued

232 2

,20

1 1( ) =2 2

=

⎛ ⎞∂=⎜ ⎟⎜ ⎟∂⎝ ⎠

∑Qa

AN -6gA

g a g a aQ a Q

EE Q k Q

Q

3 32 2

,1 1 ( ) ( )2 2 υ υ υφ φ⎡ ⎤

+ =⎢ ⎥⎣ ⎦∑ ∑ Q Q

a a

N -6 N -6A

a g a a g g gQ Q

Q k Q E

Each Normal Mode De-Coupled from All Other Normal Modes

2 2

, ,1 1 ( ) ( )2 2 υ υ υφ φ⎡ ⎤+ =⎢ ⎥⎣ ⎦

Aa g a a g a g a g aQ k Q Q E Q

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Solutions of the Normal Mode Vibrational Wavefunction

Solution of the Wavefunction

Classical to Quantum Mechanical Wavefunction

/= = − ∂ ∂a a aQ P i Q

2 22

, ,2

1 ( ) ( )2 υ υ υφ φ⎡ ⎤∂− + =⎢ ⎥∂⎣ ⎦

a A ag a a g a g a g a

a

k Q Q E QQ

, ( 1/ 2)υ υ ω= +A

g a aE 1/2

,( )ω =a g ak

Vibrational energy levels equally spaced in the harmonic approximation

Transitions Between Vibrational Normal Mode Levels

Transitions from g0 to g1 and also g1 to g0

Transition from gv to gv’. Matrix element from right to left

1/ 2 1/21

( ) ( )0 2υ υφ φ υ

ω′

⎡ ⎤ ⎛ ⎞⎛ ⎞= + ⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦a ag a a g a

a

Q Q Q

1/ 2

1 0 0 1( ) ( ) ( ) ( )2

φ φ φ φω

⎛ ⎞= = ⎜ ⎟

⎝ ⎠a a a ag a a g a g a a g a

a

Q Q Q Q Q Q

Upper option for increase in vibrational levelLower option for decrease in vibrational levels

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Infrared and Raman Vibrational Intensities

00

( ) ( ) ...μμ μ=

⎛ ⎞∂= + +⎜ ⎟∂⎝ ⎠

aa Q

Q Q QQ

IR intensities proportional to the change in dipole moment of the molecule with respect to nuclear coordinate, aQ

Raman intensities are proportional to the change in the polarizability of the molecule with respect to aQ

00

( ) ( ) ...αα α=

⎛ ⎞∂= + +⎜ ⎟∂⎝ ⎠

aa Q

Q Q QQ

1/ 2

1 0 1 00 0

( )2

α αφ α φ φ φω

= =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

g g g a ga a aQ Q

Q QQ Q

Measurement of IR Spectra

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Fourier Transform InstrumentationA. The basic instrument consists of the following

elements, Fig. 1

1. Broadband IR thermal source, SiC glower2. Collimation optics3. Beamsplitter, KBr, ZnSe, Ge4. Interferometer assembly 5. Sample delivery and focusing optics6. Optical filters if desired7. Sample position8. Detector focusing optics9. Detector, DGTS, MCT, InSb, InGaAs, Ge10. Processing electronics, digital interferogram11. Computer, Fourier transformed transmission

Interferometer Operation

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Interferometer Signals

2626Position Space and Inverse Position Space

Wavenumber/

/ cm

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Interferogram for Air Background

200 300 400 500Data point

-20000

-10000

0

10000

Inte

nsity

Interferogram for Air Background

0 1000 2000 3000 4000 5000 6000 7000 8000Data point

-20000

-10000

0

10000

Inte

nsity

Interferogram for Air Background

Comparison of Interferograms for Sample and Background

200 300 400 500 600Data point

-15000

-10000

-5000

0

5000

Inte

nsity

Interferogram for Polystyrene Film

200 300 400 500Data point

-20000

-10000

0

10000

Inte

nsity

Interferogram for Air Background

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Instrument Background with Sample Absorption

3900 3400 2900 2400 1900 1400 900 400Wavenumbers (cm-1)

0

10000

20000

30000

40000In

tens

ity

Instrument Background (I0)Transmission Spectrum of Sample (I) with

I(ν)

Polystyrene Film IR Spectra

4000 3500 3000 2500 2000 1500 1000 500Wavenumbers (cm-1)

0.0

0.5

1.0

1.5

2.0

Abs

orba

nce

Polystyrene Film Sample Absorbance

4000 3500 3000 2500 2000 1500 1000 500Wavenumbers (cm-1)

0

25

50

75

100

% T

rans

mitt

ance

Polystyrene Film Sample Transmittance

0% ( ) 100 ( ) / ( )ν ν ν= ×T I I [ ]10 0( ) log ( ) / ( ) ( )ν ν ν ε ν= − =A I I Cl

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Figure 1Figure 1

A) Schematic diagram of a Michelson interferometer

B) Signal registered by the detector D, the interferogram

C) Spectrum obtained by Fourier transform (FT) of the interferogram

S=Radiation source; Sa=Sample cell; D=Detector; A=Amplifier; M1=Fixed Mirror; M2=Movable mirror; BS=Beam Splitter; x=Mirror displacement

B. Resolution

1. Depends on path difference maximum2. Apodization function3. Rayleigh criterion: to resolve two lines

separated by d, the path difference max must be at least 1/d

C. Fourier transformation carried out by the FastFourier Transform algorithm of Cooley and Tukey

1. Apodization needed to remove “feet” Fig. 22. Apodization functions include: nothing, box

car, sinc function, triangular, Happ-Genzel3. No apodization gives the best resolution4. Happ-Genzel has low intensity side lobes

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D. Advantages of Fourier Transform IR Spectrometers

1. Jacquinot’s throughput advantage, no slit2. Fellgett’s, the multiplex advantage3. Connes’ accuracy advantage, the mirror

position is determined by HeNe reference to .005 microns or less than 0.01 cm-1

4. As a results, FT-IR can be measured in seconds rather than minutes and spectral subtractions can be carried out without frequency errors

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‘Corner-Cube’ Michelson FT-IR

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Manufacturers of FT-IR Spectrometers

ABB Bomem AnalyticsThermo NicoletBrukerPerkin ElmerVarian (Digilab)

Nicolet 6700 FT-IR

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ABB Bomem MB100 FT-IR spectrometer

IV. Interpretation of IR spectra – Group Frequencies

A. Basic types of vibrational motion, Table 2

1. Stretching, sym and antisym2. Angle bending, in-plane, out-of-plane3. Wagging4. Twisting5. Rocking

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Table 2.Table 2.

Commonly used symbols and descriptions or different vibrational forms

IV. Interpretation of IR spectra – Group Frequencies

B. Intensity designations, Table 3

1. Vs,s,m,w,vw, sh,b,sr,v

C. Methyl and methylene groups

1. Methyl, str-as 2960 cm-1, str-s 2870 cm-1, def-as 1465 cm-1, def-s, 1375 cm-1

2. Methylene, str-as 2920 cm-1, str-s 2850 cm-1, def 1470 cm-1, wagging 1350-118 cm-1, twisting 1300 cm-1, rocking 720 cm-1

D. Alkene groups, =CH above 3000 cm-1

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E. Aromatic RingsF. Triple Bonds and Cumulated Double BondsG. EthersH. Alcohols and PhenolsI. AminesJ. Azo CompoundsK. Nitro CompoundsL. Carbonyl CompoundsM. AmidesN. LactamsO. ThiolsP. Sulfides and DisulfidesQ. Sufones

V. Applications of IR spectroscopy

A. Transmission1. Long path for gases2. Liquids and solution, 5 to 500 micron

pathlength3. Windows, KBr, CaF2, BaF2, KRS-5 (ThBrI)4. Fixed path, variable path, heated, flow-

through.

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B. External Reflection, RA or IRRAS

1. Different geometeries, Fig. 3

Figure 3.Figure 3. Near-normal and grazing angle incidence( )1θ ( )2θ

Figure 4.Figure 4. Phase shift of the reflected beam occurring at grazing angle incidence with perpendicular (180ο shift) and parallel (90o

shift) polarized light

External Reflection, RA or IRRASGrazing angle, selection rules, only vibrations with transition

moments normal to the surface are seen, Fig. 4

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Figure 5Figure 5

Ray diagram of the wafer/disk checker attachment for recording spectra of thin films on large samples, e.g.,

M1-M6=mirrors(reproduced by permission of Harrick Scientific Corporation. Ossing, NY 10562)

IRRAS penetrates sample film, reflects and repenetratesbefore being detected, Fig. 5

Otherwise the IR is true specular reflectance

D. Internal Reflection, ATR

1. Reflection angle must be below the critical angle2. Depth of penetration of evanescent wave depends

on the internal reflection angle

the higher index of refraction is

12 2 1/2

122 (sin )pdn

λπ θ

=−

11nλλ = 2

121

nnn

=

1n

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Figure 6Figure 6Schematic representation of total internal reflection with: A) Single reflection; B) Multiple reflection IRE (internal reflection element)

n1=Refractive index of the internal reflection element;

n2-=Refractive index of the sample with

Of incidence; dp=Depth penetration

2 1; Anglen n θ⟨ =

3. The geometry is given in Fig. 6

Figure 7Figure 7Schematic drawing of an

FT-IR measurement system utilizing the Deep Immersion Probe Model DPR-124 mounted in a batch reaction vessel [68]

a) FT-IR spectrometer;

b) Optical transfer elements;

c) Detector assembly;

d) Reaction vessel;

e) Mixing blade;

f) ATR sensing head

(reproduced by permission of Axiom Analytical Inc., Irvine CA 92614)

3. ATR can also be used for difficult process applications using optical transfer elements, Fig. 7

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D. Diffuse Reflectance, DRIFT, Fig. 8

1. The main advantage is little or no sample preparation.

2. Powders and samples with rough surfaces3. Drift spectra are affected by particle size, grinding

helps if this is a problem4. Spectral intensities measured and interpreted in

terms of the Kubelka-Munk equation with s, the scattering coefficient and R infinity, diffuse reflec.

2(1 ) 2.303( )2

R acf RR s

∞∞

∞

−= =

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Figure 8.Figure 8.

Ray diagram of the Praying Mantis diffuse reflectance attachment

EM=Ellipsodalmirror;

PM=Planar mirror;

S=Sample

(reproduced by permission of Harrick Scientific Corporation, Ossining NY 10562)

Diffuse Reflectance, DRIFT, Fig. 8

D. Photoacoustic, PA, microphonic detection, Fig.9

1. Useful for difficult samples.

2. Virtually no sample preparation

3. Christiansen Effect does not matter

4. Rapid scan FT frequencies modulate the intensity at excellent frequencies for PA effect

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D. Photoacoustic, PA, microphonic detection, Fig.9

Figure 9.Figure 9. Schematic of a photoacoustic cell