Ch 8. The Vibrational and Rotational Spectroscopy of Diatomic Molecules MS310 Quantum Physical...

Ch 8. The Vibrational and Rotational Ch 8. The Vibrational and Rotational Spectroscopy of Diatomic MoleculesSpectroscopy of Diatomic Molecules

MS310 Quantum Physical Chemistry

- light interacts with molecules to induce transitions light interacts with molecules to induce transitions between states.between states.

- Absorption of electromagnetic radiation in the Absorption of electromagnetic radiation in the infrared and microwave regions of the spectrum.infrared and microwave regions of the spectrum.

- Transitions between eigenstates of vibrational and Transitions between eigenstates of vibrational and rotational energy induced by light rotational energy induced by light


8.1 An introduction to spectroscopy8.1 An introduction to spectroscopy

What is Spectroscopy? : see the ‘bond’ → a lot of information about the character of chemical bond and reactivity

Our focus in this chapter : rotation and vibration(Atomic spectroscopy : Ch 11, electron spectroscopy : Ch 15, NMR : Ch 18)

Bond length : rotational spectroscopyFrequency of characteristic oscillation with chemical bond : vibrational spectroscopy

Discrete energy level in Q.M → absorption and emission spectrum make individual peaks

Transition energy is given by || 21 EEh


Energy, wavelength of light(photon) and name of waveVisible light : very small range


Used wave in spectroscopy → from microwave to X-ray region : nine order of magnitude

In spectroscopy, we use the wave numberTherefore, energy is given by

Experimentally, only a few transition occur(not arbitrary chosen state) : selection rule

1~

~|| 12 hchEE


Electric and magnetic fields associated with a traveling light wave. → consider the dipolar diatomic molecule in time-dependent electric field


Interaction between classical harmonic oscillator in electric field.(arrow : direction of mass)

Oscillator absorbs energy in both the stretching and compression half cycles.

Real Q.M oscillator : similarElectric field effects 2 way : permanent and dynamic

Ex) polar HCl molecule has permanent dipole moment, and dynamic dipole moment can be generated.


8.2 Absorption, Spontaneous emission, and 8.2 Absorption, Spontaneous emission, and Stimulated emissionStimulated emission

Photon-assisted transition

Absorption : photon induces a transition to higher level

Spontaneous emission : excited state relaxes to lower level

Stimulated emission : photon induces a transition from

excited state to lower level


Spontaneous emission : random event, related to lifetime of excited stateAbsorption, Stimulated emission : related to radiation density ρ(ν)

In equilibrium, overall transition 1 to 2 and 2 to 1 must be same.

221221112 )()( NANBNB System is described by

Use blackbody spectral density function, we can obtain

3

32

21

212112

16,

cB

ABB


Ex) 8.1Derive the result by the (1) the overall rate of transition is zero at equilibrium, (2) ratio of N2 to N1 is governed by the Boltzmann distribution

Sol) overall transition rate is zero :

Boltzmann distribution is given by

Two expression must be same →

Lightbulb : incoherent photon source(random direction)Laser : coherent photon source(all photons are in phase and same direction)

221221112 )()( NANBNB

)( 21//

1

2

1

2 ggeeg

g

N

N kThkTh

1

18)( /3

3

21/

12

21

kThkTh ec

h

BeB

A

3

32

3

3

21

212112

168,

cc

h

B

ABB

8.3 An introduction to vibrational spectroscopy8.3 An introduction to vibrational spectroscopy


Vibrational frequency → depends on the 2 vibrating atoms at the end of bond other atoms affect much less degree → group frequency : characteristic frequency of bond

Can caculate the # of first excited state(N1) and # of ground state(N0) at 300K and 1000K by the Boltzmann distribution

Except the Br2, N1<<N0 acceptable even 1000K → absorption of light at characteristic frequency occur at n=0

Experimental result in Q.M harmonic oscillator : ∆n = nfinal – ninitial = +1


Use more higher sensitive instrument : ∆n = +2, +3, … can measure, but much weaker than ∆n = +1 → selection rule ∆n = +1 is not rigorously for anharmonic potential

Model of anharmonic potential : Morse potential

2)( ]1[)( eRRe eDRV

Bond energy D0 : respect to lowest allowed level

Energy level is given by : anharmonic correlation

oscillator harmonicfor valid also:

,)(constant force

2/potential, of bottom the to relative energy ondissociati :

2

2

k

dx

Vdk

DkD

exx

ee

22

)2

1(

4

)()

2

1( n

D

hnhE

en


Material-dependent parameter is given by this table. B : rotational constant, r : bond length

8.4 The origin of selection rule8.4 The origin of selection rule


Transition probability is not zero → transition dipole moment follow this condition

In real case, position depends on time.We can take Taylor expansion at the x=0(equilibrium position)

First term : permanent dipole momentSecond term : dynamic dipole moment(time-dependent)

We can think absorption occurs state of n=0 : reasonable

0)()()(* dxxx nxmmnx

...))(())(( 00 xx

xx dx

dtxtx


We can calculate the transition dipole moment

dxexxHxHdx

dAA

dxexHxHAA

xmxx

xm

xmxm

mx

2

0

2

)()(])[(

)()(

2/10

2/10

2/10

2/100

0

First integral : 0 by orthogonality, We only see the second integral

m=even, Hm(α1/2x) : even, m=odd, Hm(α1/2x) : oddTherefore, if m is even, overall integral becomes integral of odd function and it becomes zero.

→ value of integral may not zero if the transition of n=0 → m=2b+1, b=0,1,2,…


Figure of m=1,3,5Except the m=1 case, area of red and blue region is same.

00 mx

Therefore, only m=1 transition allowed. → ∆n=1 : selection rule for absorption ∆n=-1 : selection rule for emission

Vibrational excitation : only dμx/dx≠0H2, N2, O2 : μx0 =0, dμx/dx=0 → 99.93% of atmosphere doesn’t absorb the IR radiation

IR radiation only absorb by CO2, NOx, and hydrocarbon : they occur greenhouse effect


8.5 Infrared absorption spectroscopy8.5 Infrared absorption spectroscopyIR spectroscopy follows the Beer-Lambert Law

Incident light I0(λ) through the distance dl

Absorption related to dl, intensity, and concentration(also it

related to # of molecule), ε(λ) : molar absorption coeffieicnt

MleI

IMdl

I

dI

dlMIdI

)(

0 )(

)(,)(

)(

)(

)()()(


Ex) 8.4 ε(λ) of ethane : 40 (cm bar)-1 at 12μmCalculate the I(λ)/ I0(λ) when 10cm absorption cell length and 2.0ppm. Also, how long the cell length if I(λ)/ I0(λ)=0.90?

Sol)

0.19992.0

)bar)(10cm)10(2.0 bar) (cm40exp()(

)( 6--1)(

0

MleI

I

Therefore, it is difficult to measure.

Rearrange the equation, we can obtain

cm103.190.0ln)bar 100.2()bar cm 40(

1

)(

)(ln

)(

1 361

0

I

I

Ml

This order of length can make by the mirror and it uses to the gas detection.


How does ε(λ) depend on the wavelength?Consider the ketone molecule

Vibration frequency of carbonyl group : determined by the force constant of C=O bond.Force constant depends on the chemical bond between C and O and other alkyl groups affect much less!

Vibrational modes : depends on the degree of freedom total degree of freedom : 3n translational freedom : 3 rotational freedom : 3(nonlinear), 2(linear) vibrational freedom : 3n-6(nonlinear), 3n-5(linear)

Benzene : 30 vibrational modes, but only 20 distinct vibrational frequencies because of the degeneracy


See the IR spectroscopy of CH4 and CO

Case of CH4, 9 vibrational frequency expected. However,

there are only 2 peaks and a lot of additional frequencies.

Also, broadening in CO peak. Why?

→ rotational spectra


There are many rotational states with only 1 vibrational

transition, n=0 → n=1.

We can analyze the rotational spectra if we use the high-

resolution instrument.

In CH4 spectra, there are only 2 peaks instead of 9. Why?

Case of CH4, there are only 2 peaks

1306 cm-1 : 3 degenerate C-H bending modes

3020 cm-1 : 3 degenerate C-H stretching modes

Where are other 3 modes?

Use group theory, these modes are symmetric and doesn’t

satisfy the condition dμx/dx≠0. Therefore, these modes are

infrared inactive.

However, all modes of CH4 and CO are raman active.(section 8.8)


Consider the CO2 case.

2 C=O bonds are equivalent and expected there are only 1 peaks.

However, there are 2 peaks by the experiment. Why?

→ symmetric and antisymmetric stretching


Symmetric stretch : only depends on k1 ,

Asymmetric stretch : C moves → opposite direction of each oscillator, F=-(k1+2k2)x :

1

2

1 ksymmetric

21 2

2

1 kkricantisymmet

Symmetric and antisymmetric O-H stretching modes

8.6 Rotational spectroscopy8.6 Rotational spectroscopy


Rotational selection rule : ∆J=Jfinal – Jinitial=±1We see the Example 8.1

Ex) 8.1Use these eigenfunctions of rigid rotor, J=0 →J=1 transition is allowed and J=0 → J=2 is forbidden.

)1cos3()16

5(),(

cos)4

3(),(

)4

1(),(

22

10

2

2

10

1

2

10

0

Y

Y

Y


Sol) assume the electromagnetic field : z direction → μz=μcosθ

J=0 → J=1

J=0 → J=2

Therefore, J=0 → J=1 transition is allowed, J=0 → J=2 transition is forbidden. Also, μz

J0 is zero unless ∆MJ=0.

0

00

02

0

0 sin),())(cos,( dYYd JJz

03

3]

3

cos[

2

3sincos

4

30

3

0

22

0

10

ddz

0]4

1

4

1[

8

5]

2

cos

4

cos3[

8

5

sincos)1cos3(8

5

0

24

0

22

0

20

ddz


Consider the charged two particles in rotational motion.

Rotational absorption : nonzero ‘permanent’ dipole moment!(by contrast, in vibrational absorption, nonzero dynamic dipole moment)

We use the angular quantum number J instead l.(l used for orbital angular momentum)

Energy is given by the

)1()1(8

)1(2 2

02

2

20

2

JhcBJJJr

hJJ

rE

B : rotational constant


Simulated rotational spectroscopy


calculate the energy change of the transition ∆J=1 and ∆J=-1

hcBJJJr

JJr

E

JhcBJJr

JJr

E

JEJEE initialfinal

2)1(2

)1(2

)1(2)1(2

)2)(1(2

)()(

20

2

20

2

20

2

20

2

| ∆E+ | ≠ | ∆E- | : energy level not

spaced equally

Rotational energy not depends on

mJ : 2J+1 fold degenerate

Difference between ∆ν, but ∆(∆ν) is

same and its value is 2cB


In Real case : rotational and vibrational change simultaneously

∆Erotational << kT : many rotational peaks observe

Calculate the ratio of ∆Erot and ∆Evib

krk

r

E

E

vib

rot2

0

20

2

/

/

This ratio is 0.028(H2) and 0.00034(I2).

moment of inertia, force constant large

→ smaller ratio


Relative ratio of given J state

IkTJJkTJJ eJeg

g

n

nJ 2/)1(/)(

00

20 )12(

Degeneracy (2J+1) dominant for small J and large T

nJ/n0 decrease to 0 rapidly J increase

Moment of inertia increase : upper value of J increase HD case : 4 for 300K, 7 for 700KCO case : 13 for 100K, 23 for 300K, 33 for 700K


Higher frequency(∆J=+1) : R branchLower frequency(∆J=-1) : P branch

Center of spectrum : ∆J=0 : forbidden transition

For raman spectroscopy, selection rule becomes to ∆J=0, ±2


Higher resolution IR spectra of CO molecule

8.7 Fourier transform infrared spectroscopy8.7 Fourier transform infrared spectroscopy


FTIR : one pulse → same as many single-wavelength experiment(multiplex advantage) → short time!Instrument : Michelson interferometer


At first, we consider the one incident waveIncident light : amplitude A0ei(kx-ωt), intensity I0

Through the beam splitter S : 50% of light transmitted, and other 50% of light reflected and go to M2

Transmitted light : reflect by M1 and 50% is reflected by SReflected light : reflect by M2 and 50% is transmitted through S

These 2 waves make interference.)()(0)][()(0 )1(

2][

2)( tkyititdykitkyi DDD ee

Aee

AtA

δ(t) : phase difference from the path difference, Δd

)(2

)22(2

)( 21 tdSMSMt

intensity I : related to A*(t)A(t)

200

00 ),)(2

cos1(2

))(cos1(2

)( AItdI

tI

tI


Δd=nλ : constructive, measuredΔd=(2n+1)λ/2 : destructive, not measured

Measured signal : interferogram, single sine wave


After, consider the many incident waves(different frequencies)Amplitude is given by

)2

()(2

)1(2

)(tyi

tdi

j

jjD

jj eeA

tA

Measured intensity is given by

jjj tc

vItI ))

2cos(1)((

2

1)(

FTIR gives all frequency data simultaneously → use for determine of fraction of air

8.8 Raman spectroscopy8.8 Raman spectroscopy


Consider the oscillating electric field and characteristic frequency of molecule is νvib Electric field induce the induced magnetic moment, μinduced and it related to polarizability

tEE 2cos0

tEtinduced 2cos)( 0

Polarizability related to bond length xe+x(t) and we can expand by the Taylor-Mclaurin series

...)()()( exxe dx

dxxx

Consider the vibration of molecule

txtx vib2cos)( max


We can calculate induced dipole moment by these results

])22cos()22[cos()(2

12cos)(

]2cos)()([2cos)(

0max0

max0

ttExdx

dtEx

txdx

dxtEEt

vibvibxxe

vibxxeinduce

e

e

Therefore, allowed frequency is ν, ν-νvib, ν+νvib

ν : Rayleigh frequency

ν-νvib : Stokes frequency

ν+νvib : anti-Stokes frequency

Unless the dα/dx≠0, stokes and anti-stokes peak : zero.

It means raman active bond satisfies dα/dx≠0.

However, it is not related to dμx/dx≠0

→ can be raman active although IR inactive!


Stokes : n=0 to n=1Anti-stokes : n=1 to n=1Intensity of stokes and anti-stokes peak same?

kThkTh

kTh

ground

excited

stokes

stokesanti ee

e

n

n

I

I /2/

2/3

Range of 1000cm-1 to 3000cm-1 , ratio is 8x10-3 to 5x10-7 at 300K

Therefore, 2 peaks are quite different.

Raman and IR spectroscopy are complementary.

8.9 How does the transition rate between states8.9 How does the transition rate between states depend on frequency?depend on frequency?


Solution of time-independent Schrödinger equation : constant energy → it cannot describe transition state

Consider the 2-level system and E2>E1, wavefunction is written by

At t=0, system in ground state : a1=1, a2=0Write the initial hamiltonian as H0

Light turns on : electric field applied → permanent or dynamic dipole moment generateAssume the electric field along the x axis, time-dependent potential energy is given by

22112211

21

)()(),(

aaexaexatxtiEtiE

)(2

2cosˆ 2200int

titixx ee

EtEEH

: dipole approximation


We must solve time-dependent Schrödinger equation

2211int0 )()(),(,),(

),()ˆˆ(

tatatx

t

txitxHH

22201110ˆ,ˆ EHEH : trivial

Therefore, equation changes to

dt

tdai

dt

tdaiHtaHta

)()(ˆ)(ˆ)( 22

112int21int1

Multiply Ψ2* left side and integration

dxdt

tdaidx

dt

tdaidxHtadxHta 2

*2

21

*2

12int

*221int

*21

)()(ˆ)(ˆ)(

Ψ1 and Ψ2 are orthonormal, equation can be simply

dxHtadxHtadt

tdai 2int

*221int

*21

2 ˆ)(ˆ)()(


Assume a1(t) and a2(t) are small change(it means a1=1, a2=0 on the right side, but not left side!)Equation is change by

)ee(2

E

dx)x()x()ee(2

E

dx)x()x()ee(e2

E

dx)x(H)x(edxHdt

)t(dai

t)hEE(i

t)hEE(i

021x

1x*2

t)hEE(i

t)hEE(i

0

1x*2

ti2ti2t)EE(i

0

1int*2

t)EE(i

1int*2

2

1212

1212

12

12


We use dummy variable t’ and integrate it.

)hEE

e1

hEE

e1(

2

E

'dt)ee(2

Eia

12

t)hEE(i

12

t)hEE(i

021x

t

0

't)hEE(i

't)hEE(i

021x2

1212

1212


Unless the μx21 ≠ 0, a2(t) must be zero.

First term of a2(t) : stimulated emissionHowever, our focus is absorption : second term

period of oscillation becomes zero when E2 - E1 → hν(we assume E2 > E1, E1 - E2 → hν can neglect)Use L’Hôpital’s rule

ite

itlim

)hEE(d/)hEE(d

)hEE(d/)e1(dlim

hEE

e1lim)t(alim

)x('g

)x('flim

)x(g

)x(flim

t)hEE(i

0hEE

1212

12

t)hEE(i

0hEE12

t)hEE(i

0hEE2

0hEE

0x0x

12

12

12

12

12

1212

Resonance condition : E1 - E2 = hν In this case, a2(t) increase linearly.


E1 - E2 is slightly different to hν : do not resonance → transition probability almost zero : no transitionTherefore, transition occurs only the hν is equal or extremely close to E1 - E2.


Final goal : find the transition probability, a2*(t)a2(t)Neglect the first term, we can calculate easily

212

122

221202

*2 )(

]2/)[(sin][)()(

hEE

thEEEtata x

→ when resonance condition, a2*(t)a2(t) increases as t2

Plot a2*(t)a2(t) when 40ps, 120ps, and 400ps

time increase → height increases as t2 and width decreases as 1/t.

Therefore, there are no transition without resonance, E1 - E2 = hν.

Why height of peak decrease when time increase? → uncertainty principle


a2*(t)a2(t) : closely related to observed in an absorption spectraThen, what is measured by the ‘real’ instrument?

Intrinsic linewidth of vibrational spectra : less than 10-3 cm-1

Resolution of instrument : ~ 0.1cmIn this case, measurement peak broaden and no information about the intrinsic linewidth by the measurement → ‘inhomogeneous broadening’


-Light interacts with molecules to induce transitions between states and molecular spectroscopy were described.

- It was discussed the absorption of electromagnetic radiation in the infrared and microwave regions of the spectrum.

- Light of these wavelengths induces transitions between eigenstates of vibrational and rotational energy.

- The frequency at which energy is absorbed or emitted is related to the energy levels.

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Ch 8. The Vibrational and Rotational Spectroscopy of Diatomic Molecules MS310 Quantum Physical...

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Transcript of Ch 8. The Vibrational and Rotational Spectroscopy of Diatomic Molecules MS310 Quantum Physical...