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Molecular SpectroscopyThe Diatomic Rigid Rotator

General Aspects of Molecular Spectroscopy

The rigid diatomic Molecule

m1 m2

r1 r2

m1 r1 m2 r2=r0

THE MOMENT OF INERTIA IS I = μ r02

Reduced mass

Rotational Energy Level

• EJ = h2

• In terms of wavenumber εJ = EJ/hc cm-1

εJ = BJ(J+1), where B = h2 cm-1

B is called as the rotational constant.

8 π2 I J (J+1) joules, where J = 0, 1, 2,…..

8 π2 I c

Intensities of Spectral Lines• Intensities depends on number of molecules

in a particular energy level Boltzmann Distribution

NJ/NO = EXP (-EJ/kT) = EXP(-Bhc J(J+1)/ kT)

NJ/NO ≈ 0.98

There are as many molecules in the J=1 state, at equilibrium as in the J=0, at T=300K (room temperature). Further, there is more rapid decrease of NJ/NO with increasing J and with larger B.

EXP(

-Bhc

J(J+

1)/

kT)

100

B = 10 cm-1

B = 5 cm-1

Degeneracy of the energy states

• The energy and angular momentum of a rigid rotator are E = ½ I ω2

P = I ωP = (2EI)1/2 = [J(J+1)]1/2 units

• eg., for J=1, P=(2)1/2

11Spacing between adjacent rotational levels j and j-1,

Rotational Spectroscopy

(1) Bohr postulate

/h hc (2) Selection Rule

11 ( )1 ( )

jj absorptionj emission

Isotopic Shift12CO

13CO

321

TYPICALLY ROTATIONAL SPECTRUM SHOWS LINES CORRESPONDING TO 1-100 cm-1

The Vibrating Diatomic Molecule (eg., HCl)

req

H’

H’’

H’’ BOND LENGTH

APPROX. POTENTIAL

REAL/ EFFECTIVE POTENTIAL

Morse Potential

THE PARAMETERS

• E = ½ k (r – req)

• ωosc = 1/2π (k/μ)1/2 Hz ωosc = ωosc/c cm-1

• Allowed vibrational EnergiesEν = (ν + 1/2) ωosc

• Selection rule : Δν = ±1

Vibrations in Rigid rotator !!

req

ν = 0123

ENER

N1/NO ≈ 0.008

The molecular population Dies of very fast

Ignore all possible Vibrational states

Vibrational SpectroscopyVibrational selection rule

11 ( )1 ( )absorptionemission

ΔJ = -1 P

ΔJ = +1 R

ΔJ = +2 S

ΔJ = -2 O

ΔJ = 0 Q

Δν

Vibration-Rotation Spectra

Infrared spectrumΔJ = ±1

Raman spectrumΔJ = 0 , ±2

Vibration-Rotation Infrared Spectrum of HCl

• νvib is different for H35Cl and H37Cl molecules due to the slight difference in their reduced masses.

972.03635

ClH35

974.03835

ClH37

Vibration-Rotation Infrared Spectrum of HCl

• The lines due to H35Cl transitions are more intense because the isotopic abundance ration of H35Cl to H37Cl molecules is 3:1.

Vibration-Rotation Infrared Spectrum of HClB2B2B2B2 B2 B2 B2 B2B4

Band centerH35Cl

Band centerH37Cl

Vibration-Rotation Infrared Spectrum of HCl

• The rotational constant B slightly decreases as going to higher vibrational levels. This results in decrease of the gaps between transition lines as one goes to higher frequencies.

B2B2B2B2 B2 B2 B2 B2B4

Vibration-Rotation Infrared Spectrum of HCl

• The rotational constant B slightly decreases as going to higher vibrational levels. This results in decrease of the gaps between transition lines as one goes to higher frequencies.

Vibration-Rotation Infrared Spectrum of HClB2B2B2B2 B2 B2 B2 B2B4

Approximation of B values

RAMAN SPECTROSCOPYNobel in 1930

Discovery of Raman Effect (Raman Scattering)

(7 November 1888 – 21 November 1970)

WHAT IS RAMAN SPECTROSCOPY ?

“Raman spectroscopy is the measurement of the wavelength and Intensity inelastically scattered light from molecules.”

May be used to understand chemical composition and molecular structure.

Used in CMP and chemistry to understand the vibrational, rotational and other low-frequency modes of a system.

ScatteringIn addition to being absorbed and emitted by atoms and molecules, photons may also be scattered (approx. 1 in 107 in a transparent medium). This is a molecular effect, which provides another way to study energy levels.

ELASTIC INELASTICSYSTEM REMAINS IN THE SAME QUANTUM STATE

Resulting in CHANGE IN THE QUANTUM STATE

Scattering

• νin = νout • Rayleigh scattering– Ein = Eout = h ν

• νin ≠ νout Raman scattering–Ein ≠ Eout

VIRTUAL STATES

EXCITED STATES

GROUND STATE

RAMAN SCATTERING SELECTION RULES

RAMAN SCATTERING SELECTION RULES

Rotational Raman

Some fine points• Inelastic scattering can be in analogy with a ball

bearing hitting a drum so that it starts to oscillate at its natural frequency. Similarly if the drum is already oscillating and the ball bearing hits it at the right phase, it may get reflected at a higher energy.

• In order to be Raman active, a molecular rotation or vibration must cause some change in the component of molecular polarizibility.

• Strokes lines (those scattered with a lower frequency than the incident radiation) are generally more intense than the anti-stroke lines, because the former is accompanied by an increase in molecular energy.