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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 7

Molecular Spectroscopy: Rotational, vibrational and

Roto-vibrational Spectroscopy of diatomic Molecules

Giuseppe Pileio

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Learning Outcomes

By the end of this lecture you will be able to:

!! Understand the fundamentals of pure rotational spectroscopy

!! Understand the fundamentals of pure vibrational spectroscopy of diatomic molecules

!! Understand the fundamentals of roto-vibrational spectroscopy of diatomic molecules

!! Extract molecular information from these molecular spectra

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Moment of inertia and rotor classification

The moment of inertia is the inertia of a rigid rotating body with respect to its rotation

Different axes of rotation will have, in general, different moment of inertia. Thus, the moment of inertia is better represented by a tensor:

Ijk ! !i!1

N

mi""ri2"jk $ rij"rik# Ixx Ixy IxzIyx Iyy IyzIzx Izy Izz

! principal frame !Ixx 0 0

0 Iyy 0

0 0 Izz

Rotor (molecules) classification

1.! Linear

2.! Symmetric

3.! Spherical

4.! Asymmetric

Izz ! 0 and Ixx ! Iyy

Izz ! Ixx ! Iyy

Izz ! Ixx ! Iyy

O2, CO, CO2

NH3, CHCl3

CH4, SF6

H2O, NO2

Izz ! Ixx " Iyy or Izz # Ixx " Iyy

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Pure Rotational Spectroscopy

Rotational Spectroscopy is the study of the interaction between a Micro-Wave (MW, 1-1000 GHz or 0.03-33 cm-1) radiation and the matter

The interaction is between the radiofrequency and the molecular electric dipole moment

! " e.r! + - r" !I ~

then if the molecule does not have a permanent dipole moment it cannot have rotational spectrum

! !i,el"e"!j,el"%el"! !i,vib"!j,vib"%vib

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Group Theory: o! If the molecule has a dipole moment it should lie on its principal axis o! Homonuclear diatomic molecules cannot have a permanent dipole since the two atoms equally attract electrons o! Molecules with a centre of inversion cannot have a permanent dipole o! Spherical molecules cannot have a permanent dipole

"+ "- Heteronuclear diatomic

"+ "+

"- Non-centrosymmetric

Homonuclear diatomic

"+

"+ "+

"+

"-

Spherical

Rigid Rotor

Classical theory "

Rotational Energy: linear molecules

r E !1

2"I 2

I ! r2

!m1"m2

m1 % m2

# QuantumTheory

EJ ! B J"!J # 1" J ! 0, 1, 2, ...B !

!2

2"I

E

J=1

J=0

J=2

J=3

6B

4B

2B

re

r

r

Restoring Force

F=-k(r-re) Non-rigid rotor

Classical theory:

E !1

2"Ie

2 $1

2" Ie

k4

QuantumTheory

EJ ! B J"!J # 1" $ D J2"!J # 1"2B !

!2

2"I

D ! !4

2#I3#k

J=1 J=0

J=2

J=3

rigid non-rigid

E

J=1

J=0

J=2

J=3

6B

4B

2B

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Pure Rotational Spectra of linear molecule

To understand how a pure rotational spectrum of a linear molecule will look like we need:

1) Selection rules, i.e. when ! !i" !j"$ & 0 !J " #12) frequencies, i.e. !E " #J$1 % EJ " h

!E " #J$1 % EJ " h

!E " 2'B !J $ 1" Rigid rotor This means that if we take a pure rotational spectrum of a diatomic heteronuclear molecule it is made by a series of peaks spaced 2B in the rigid hypothesis of whose space decreases as J increases for the more realistic case of a non-rigid rotor

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3) intensities, i.e. population of energy levels

nj

ni!gj

gi"e#$E!kT

gJ ! 2"J # 1

Rotational levels are degenerate since for each J there are 2J+1 levels with mJ=0, 1, 2, , J

nJ

n0! !2"J # 1""e$BJ"!J#1"#kT

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Pure Vibrational Spectroscopy

Vibrational Spectroscopy is the study of the interaction between an infra-red (IR, 400-4000 cm-1) radiation and the matter

Such a radiofrequency stimulates molecular vibrations, i.e. nuclei move from their original position with the center of mass unmoved

Although the interaction is, as in rotational spectroscopy, between the radiation and the dipole moment, the molecule does not need to have a permanent dipole moment since some vibrations can generate instantaneous dipoles

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In other words, the requirement this time is that the dipole moment is a function of the vibrational coordinates i.e. the vibration has to create a dipole

"+ "- "+

Group theory: a D!h molecule cannot have a permanent dipole since has an inversion center

"+ "- "+

a symmetric stretch of the two bonds do not generate dipoles

"+ "- "+

the asymmetric stretch of the two bonds do generate a net dipole

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Vibrational Spectroscopy of diatomic molecules

We first consider the simplest case of a diatomic molecule where the only vibrational motion can be the stretch of the molecular bond

re

r

q Harmonic oscillator

Classical Theory "

q ! !r " re"F ! "k q

E !1

2#k q2

# QuantumTheory

Ev ! v "1

2#h v ! 0, 1, 2, ...

!1

2##

k

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Actually, the harmonic symmetric potential is a poor representation for diatomic molecules. In fact, when q decreases the atoms start to repel and if it increases the molecule start to dissociate. This suggest an asymmetric potential need to be used

Anharmonic oscillator

Classical Theory " Morse Potential E ! De"!1 # e#q"2QuantumTheory:

Ev ! v "1

2#h e % v "

1

2

2

#h e#xe

e !

#

De

2#xe !

h e

4#De

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Pure vibrational spectra of diatomic molecules

To understand how a pure vibrational spectrum of a diatomic molecule will look like we need:

1) Selection rules, i.e. when ! !i" !j"$ & 0 !v " #12) frequencies, i.e. !E " #v $ E0 " h

v " e$v % e$xe$v$!v & 1"3) intensities, i.e. population of energy levels

nv

n0!e"!v#1

2"$h#kT

e"12$h#kT ! e"vh#kT

Transition from v>0 are unlikely since those levels are scarcely populated

Transition from v=0 to v=1 is the fundamental, v=0 to v=2 is the first overtone, v=0 to v=3 is the second overtone and so on.

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Roto-vibrational spectroscopy of diatomic molecules

However, since the IR frequencies are bigger that MW then rotational transitions are also stimulated during a vibrational transition

This make possible to observe the rotational fine structure of a vibrational transition

Of course this can be done if there is enough resolution to see the small splitting of rotational lines (~2B i.e. few cm-1) underneath the bigger split of vibrational lines (~h# i.e. 102-103 cm-1) " low pressure gas phase

~h#!

$v=1!

$v=0!$J=0!

$J=1!

$J=2!

E

~4B!

~2B!

$J=0!

$J=1!

$J=2!~4B!

~2B!

Red pure rotational; Black pure vibrational; Blue vibro-rotational

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Ev,J ! Be J"!J # 1" $ D J2"!J # 1"2# v #

1

2"h e $ v #

1

2

2

"h e"xe

$e" v #1

2"J"!J # 1"

Ev,J ! Bv J"!J # 1" $ D J2"!J # 1"2 # v # 12

"h e $ v #1

2

2

"h e"xe

Bv ! Be " e$ v %1

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Rotational constants are different for different vibrational levels

Roto-vibrational energies of diatomic molecules

non-rigid rotor

anharmonic oscillator

vibro-rotational coupling

e "3# !3#e

2# re2#De

#1

re(

1

2#re2

with

and also:

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Roto-vibrational spectra of diatomic molecules

To understand how roto-vibrational spectra look like we also need:

1) Selection rules: in this case both rotational and vibrational rules need to be satisfied, i.e. !v " #1, !J " #1

! Transition are indicated as (v,J) -> (v,J) ! in the fundamental band v = 0->1 all transitions are (0,J)->(1,J1)

P"!J" # h e % J !B0 & B1" & J2 !%B0 & B1"R"!J" # h e % 2 B1 % J2 !&B0 % B1" % J !&B0 % 3 B1"

Q"!J" # h e % J !J & 1" !B0 % B1"o! (0,0)->(1,0) is called Q branch observed only in linear polyatomic and symmetric molecules (two moment of inertia are equal)

o! (0,J)->(1,J-1) is called P branch (frequency lower than !e)

o! (0,J)->(1,J+1) is called R branch (frequency higher than !e)

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P R Q

~B0,B1

no Q band

Two close lines due to the two isotopes of Cl

(m=35,m=37)

Be !!2

2" re2

"m1#m2

m1 $ m2

HCl

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2) Intensities: the form of a vibrational band is due to the two rotational bands (P and R) which, in turn, have the characteristic shape of a rotational spectrum

Temperature effect: the R branch at low temperatures is higher then P branch since P transitions originates from levels with higher J that are less populated at low temperatures

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Vibro-rotational Spectra: Information Content

The difference between a P and a R transition that end at the same value of J is proportional to B0 only

P " E#!1, J $ 1" $ E#!0, J" " h e $ J !1 & J" B0 & !$1 & J" J B1R " E#!1, J $ 1" $ E#!0, J $ 2" " h e $ !2 & !$3 & J" J" B0 & !$1 & J" J B1P $ R " 2 !1 $ 2 J" B0

CO

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The difference between a P and a R transition stating from the same value of J is proportional to B1 only

The center of the spectrum is the pure vibrational frequency which gives the potential parameters

e "

%

De

2%

Bvs contain info on inter-nuclear distance in the equilibrium or in the vibrational excited state

Bv ! Be " e$ v %1

2

Be !!2

2" re2

P " E#!1, J $ 1" $ E#!0, J" " h e $ J !1 & J" B0 & !$1 & J" J B1R " E#!1, J & 1" $ E#!0, J" " h e $ J !1 & J" B0 & !1 & J" !2 & J" B1P $ R " $2 !1 & 2 J" B1

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What did we learn in this lecture?

! The basics of rotational, vibrational and roto-vibrational spectroscopy

! The rigid and non-rigid rotor description of molecular rotations

! The harmonic and anharmonic oscillator description of molecular vibration in diatomic molecules

! The expected spectral patterns

! The info encoded in these spectra