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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