Vibrational SpectroscopyTutorial

Ewa S. KirkorUniversity of New Haven

Sources of Information

General Principles and Instrumentation:

Principles of Instrumental Analysis, by Douglas A. Skoog, F. James Holler, Timothy A. Nieman

Inorganic:

Infrared and Raman Spectra of Inorganic and Coordination Compounds : Theory and Applications in Inorganic Chemistry (Volume A) by Kazuo Nakamoto

Infrared and Raman Spectra of Inorganic and Coordination Compounds : Applications in Coordination, Organometallic, and Bioinorganic Chemistry (Volume B) by Kazuo Nakamoto

Organic:

The Handbook of Infrared and Raman Characteristic Frequencies of Organic Moleculesby Daimay Lin-Vien, et al

Excitation of Raman Spectra

A Raman spectrum can be obtained by irradiating a sample of carbon tetrachloride with an intense beam of an argon ion laser having a wavelength of 488.0 nm (20492 cm-1). The emitted radiation is of three types:

1. Stokes scattering 2. Anti-stokes scattering 3. Rayleigh scattering

ElectronicGround State

1st ElectronicExcited State

Exci

tatio

n En

ergy

, s (c

m–1

)

Vib.states

4,000

25,000

0 IR

2nd ElectronicExcited State

s

s semit

fluor

esce

nce

Impu

rity

semit

fluor

esce

nce

UV/VisFluorescence

semits

ElasticScattering(Raleigh)

Main Optical Transitions: Absorption, Scattering, and Fluorescence

Vibrational Spectroscopy: Classical Treatment

• Number of peaks related to degrees of freedom

DoF = 3N - 6 (bent) or 3N - 5 (linear) for N atoms• Energy related to model of harmonic oscillator

• Selection rules related to symmetry Rule of thumb: symmetric=Raman active, asymmetric=IR active

Example of gasses:

Raman: 1335 cm–1

IR: 2349 cm–1

IR: 667 cm–1

CO2

s or s c

2k(m1m2)

m1m2

Raman + IR: 3657 cm–1

Raman + IR: 3756 cm–1

Raman + IR: 1594 cm–1

H2O

Raman Spectroscopy: Overview• A vibrational spectroscopy

- IR and Raman are the most common vibrational spectroscopies for assessing molecular motion and fingerprinting species

- Based on inelastic scattering of a monochromatic excitation source

- Routine energy range: 200 - 4000 cm–1

• Complementary selection rules to IR spectroscopy

- Selection rules dictate which molecular vibrations are probed

- Some vibrational modes are both IR and Raman active

• Great for many real-world samples

- Minimal sample preparation (gas, liquid, solid)

- Compatible with wet samples and normal ambient

- Achilles Heal is sample fluorescence

EM wave Induced polarization is the key in Raman!

• Group assignments identify characteristic vibrational energy

Raman Spectroscopy:reference assignments

ElectronicGround State

1st ElectronicExcited State

Exci

tatio

n En

ergy

, s (c

m–1

)

Vib.states

4,000

25,000

0 IRs

s semit

2nd ElectronicExcited State

Raman∆s=semit–s

s ±∆s ∆s

Resonance Raman∆s=semit–s

Raman Spectroscopy: Absorption, Scattering, and Fluorescence

Stokes Anti-Stokes

Excitation Energy, s (cm–1)

Raman Spectroscopy: For graphene or CNTs, you pick the Laser ExcitationIn

tens

ity

11,000 13,000 15,000 17,000 19,000 21,000

Near IR785 nm

Visible514 nm

–∆s +∆s –∆s +∆s

Stokes Anti-StokesStokes Anti-Stokes

Visible630 nm

Molecular VibrationsThe infrared region of the electromagnetic spectrum is in the energy range of molecular vibrations, approximately from 250 cm-1 to 4000 cm-1.

The potential for a vibrating system is:

Let l – l0 = x, the atomic displacement, and the non-zero differential coefficients are equal to k and γ, respectively; then:

Infrared Spectrum of a DiatomicUsing the energy levels from the quantum harmonic oscillatorAnd applying the Bohr Frequency Condition:

Which has a selection rule Δv = ± 1, so that for absorption ofinfrared radiation:

So that the absorbed frequency is:

Which is more commonly expressed in wavenumbers:

This produces the fundamental vibrational frequency.

The frequency range is from 150 cm-1 to 4500 cm-1 formost molecules. However, these are not the only frequenciesobserved in a vibrational spectrum. This will be examinedwhen spectroscopy is discussed.A dipole is required for infrared radiation to be absorbed.

The Vibrating MoleculeRecall that the potentialfor the harmonic oscillatoris given by:

Which describes a parabolic potential. Thisharmonic potential is unrealistic since it does notpermit “decomposition.”

The potential energy can be expanded around the equilibriumgeometry of the molecule:

And remembering that the zero of the potential energy isarbitrary and the slope is zero at the equilibrium bond length:

which may be more simply written as:

The first term is clearly the harmonic one that predicts onlyone absorption frequency for the Δv = ± 1 vibrational transitions.

This is in clear disagreement with experiment where all diatomicmolecules have more than one vibrational absorption. There isone very intense absorption, called the fundamental, but there are many less intense absorptions that are called overtones.These arise because the molecule is not vibrating harmonically.

This anharmonicity is expressed by the higher order terms in the expansion which when used in the Schroedinger Equationgive the vibrational energy as:

Where xe is the anharmonicity constant, with values normallyon the order of 10-2. Usually, however, the xeve value is theone that is tabulated.

For the anharmonic oscillator there is no effective selection rule and any change of the vibrational quantum number is possible. For transitions from the zero-point energy level:

Vibrations and Normal Coordinates

Number of vibrational degrees of freedom:

Non-linear molecule 3N - 6 Linear molecule 3N - 5

The potential can be expanded about the vibrational displacementcoordinates: q1, q2, … qN

A new set of coordinates, {Qj } – the normal coordinates ornormal modes, may be defined that eliminates all of the cross-terms in the potential expression so that:

Which produces the Hamiltonian operator:

Symmetries of Normal Modes

SymmetricStretch

AsymmetricStretch

And is then of B2 symmetry under C2v.

Vibrational Energy Diagram of a Triatomic Molecule

Gas Phase Spectrum

• Type of observed spectrum depends upon the resolution at

which the data is obtained.RESOLUTION

(a) Low

(b) Medium

(c) Medium

(d) High

Raman Scattering Spectroscopy

• Sample is transparent ( 100% transmissive) to the incident radiation which is of one wavelength

• Intensity is not directly dependent on the transition moment

• Scattered light is commonly observed from a direction normal to the incident light beam

The Bohr Frequency condition is the basis of the observed frequencies.

The intensity of the scattered light arises from the change in the polarizability of the molecue with vibration, rotation or both.

The intensity of the scattered light is also proportional to the fourth power of the intensity of the incident (exciting) light.

The Raman Effect

When a molecule or atom is subjected to an external electric field a dipole is induced. If the external field is oscillatory,

2sin)( tEtE o

Then the induced dipole also oscillates:

tEt o sin)(

Here α is the polarizability of the molecule which is a scalar for an isotropic molecule. If the molecule is anisotropic, then:

zzzyzyxzxz

zyzyyyxyxy

zxzyxyxxxx

EEEEEEEEE

Where: E2 = Ex2 + Ey

2 + Ez2

The polarizability tensor is symmetric so αij = αji .

Further, axes can always be chosen so that there are only diagonal elements to the tensor:

''''

''''

''''

zzzz

yyyy

xxxx

EEE

Or as a tensor equation: μ = α E

The equations define the polarizability ellipsoid where the length from the center of the volume to any point on its surface is a length proportional to (αdir)-½ where αdir is the polarizability in that direction. The equation of the ellipsoid for the diagonalized axial system is:

12'''

2'''

2''' zyx zzyyxx

The polarizability can be developed in a Taylor seriesexpansion around the polarizability of the molecule in itsequiblibrium geometry, αe :

...21 2

2

2

QQ

QQ

ooe

If the molecule is vibrating at some frequency vo then Q mustalso be dependent on time:

tQQ o sin

Neglecting all but the first two terms in the expression for thepolarizability gives:

tQQ o

oe sin

So the induced dipole moment is:

ttQQ

EtE ooo

ooe sinsinsin

The first term represents the Rayleigh (elastic) scattering sinceit is at the frequency of the light. The second term represents the Raman ( inelastic ) scattering and its nature can be seen afterexpansion of the term using a trigonometric identity:

])(cos)(cos[21 tt

QQE oo

ooo

The molecule clearly does not have to have a permanent dipole to display Raman scattering.

Depolarization Ratio

kThvo

S

A eII /

4

0

Totally symmetric vibrations can be distinguished by the depolarization ratio, ρl :

||II

l

For totally symmetric modes: 0 < ρl < ¾Where for non-totally symmetricmodes ρl = ¾ which are said to bedepolarized.

Vibrational Potential Functions

• Force constants are obtained from spectra• Vibrational frequencies must be used• To obtain the force constants simplifying

assumptions must be made regarding the potential energy

• Each set of assumptions leads to a different force field model

General Quadratic Potential

Neglect of cubic and higher terms and consideration of only internal (vibrational) displacement coordinates lead to:

63

1',''2

N

tttttt SSFV

The F are the various force constants and the S are the vibrational displacement (internal) coordinates. Not all of the force constants are independent since they may be related by symmetry.

In SO2 a convenient set of internal coordinates could be r1 and r2 for stretches (displacements) along the two S-O bonds and α for the distortion of the bond angle on vibration. This permits the potential energy to be written as:

22311321122

332

2222

111 2222 rFrFrrFFrFrFV

Where symmetry dictates that F11 = F22 and F13 = F23. This means there are only four independent force constants for this molecule instead of six.

For large molecules the number of force constants may still be larger than the number of fundamental frequencies making unique solution impossible. Isotopic substitution is then used but this may also be insufficient.

Model Force Fields• Central Forces

– Assume forces keeping atoms in their equilibrium positions act only along lines joining pairs of atoms

– Assume every pair of atoms is connected by such central forces

– Most applicable to ionic systems– Not used very much

1(2 rFV r

)()(2 231

223

212

23

22

21 sssFrrrFV sr

For ammonia:

• Valence Forces– Forces are taken in directions that resist the

extension, compression or torsion of bonds– Forces between non-bonded atoms are not

directly considered– Number of force constants is usually less than

the number of fundamental frequencies• Allows force constants to be calculated from some

of the frequencies and use the remaining for checks• Deviations may be as high as 10%

)()(2 231

223

212

23

22

21 FrrrFV r

For ammonia:

Raman spectrum of Graphene and Graphite

• Raman spectrum shows characteristic dependence on thickness of graphite film

• It allows identification and comparison of single, bi… layers

• Evolution of Raman lines is directly connected to electronic structure and energy of excitation

Raman effect

Kkk si

si

Energy and

Momentum conservation

usually K~0, because BZ>>k

i e

E

kK(max)

Phonon band

Photon dispersion

Setup

Spectrometer

Laser633nm or 514nm

0.04-4 mWSample

Raman effect in Graphene

G-Band

K

•Most prominent line•Relative Intensity enhances with the number of layers•shift~1/n;chemical doping?

11580 cm

D-Band-Double Resonance• Phonon momentum at edge of Brillouinzone• 1 and 2 phonon processes• General character: wavelength dependence and difference for

changing number of layers

Single phonon processInduced by defects

Two phonon process

D-Band

4th order transition1. e excitation2. e-phonon scattering3. defect scattering4. E-hole recombination

To mention: influence of number of layers

~1eV>>E(phonon)

2D-Band

Wavelength dependent

Dependent on number of layers

4th order transition1. e excitation2. e-phonon scattering3. Phonon with opposite momentum4. E-hole recombination

2 phonon process

•Line shape and position sensitive to the number of layers

Graphene bi layer

•2 inequivalent sublattices•Splitting into 4 bands

Graphene bi-layerDifference to single layer and bulk graphite

Level splitting due to splitting in electron bands

• Objective will be to identify the vibrational frequencies of CCl4 and to determine which modes are totally symmetric by comparison with the IR spectrum.

• Stokes frequency shifts will be measured

Exercise

For a symmetrical tetrahedral molecule, the valence force potential function is:

2V = k(R12 + R2

2 + R32 + R4

2) + kδ(δ122 + δ13

2 +δ142 +δ23

2 + δ242 + δ34

2 )

And the valence force frequencies are:

Experimental Procedure

Raman Spectrum of CCl4

Assigned Raman spectrum of CCl4

Data Analysis

• Assign fundamentals using fact stretching vibrations ( v1 and v3 ) are at higher energies than bending ones.

• Compare with the IR spectrum to assign v1which is of totally symmetric symmetry.

• The v3 band appears as a doublet due to Fermi Resonance ( combination or overtone falls near a fundamental )

• Using values of other fundamentals, deduce which combination/overtone is causing the Fermi resonance.

• Tabulate the Stokes/Anti-Stokes frequencies and intensity ratios.– Calculate the sample temperature– Compare to room temperature. Explain any

difference.• Calculate valence force field constants from

first two relations and from second two. Compare and rationalize if a correction term is required or not.

• Remember the most accurate intensities are obtained by integration.

• Using the relative isotopic abundances of Cl, calculate the relative intensities expected for the components of the band found near 460 cm-1.– Examine the frequency spacings in these

components– What modifications to the valence force field

equations would allow you to account for these?

ElectronicGround State

1st ElectronicExcited State

Exci

tatio

n En

ergy

, s (c

m–1

)

Vib.states

4,000

25,000

0

fluor

esce

nce

IRs

s semit

2nd ElectronicExcited State

Raman∆s=semit-s

s ∆sflu

ores

cenc

eIm

purit

y

Fluorescence = high and variable background

= Trouble

Raman Spectroscopy: Absorption, Scattering, and Fluorescence

Stokes Anti-Stokes

1000 2000 3000

Raman Shift (cm-1)

Ram

an In

tens

ity

Without Bleaching

After 2 hours Bleaching

Poly diallyl phthalate (dirty!) lex = 514.5 nm

Raman Spectroscopy: Coping with Fluorescence

1. 785 nm laser line excites many fewer fluorophores

2. Possibly photobleach with long exposure to laser irradiation.

Raman Spectroscopy: Summary

1. Raman is a vibrational spectroscopy complementary to IR- Good for fingerprinting, probing molecular symmetry

2. It is Scattering-based, …gas, liquid, or solid- Virtually always use Stokes lines due to stronger signal

3. You need to pick excitation energy (laser line)- 785 nm: Fluorescence less probable; Lower Raman signal- 514 nm: Fluorescence more probable; Resonance more likely; Higher signal

4. Other things not talked about- Quantum origins of selection rules and scattering cross-section- The intensity and position of Raman transitions in CNT is excitation wavelength dependent- SERS: Surface Enhanced Raman Spectroscopy