Molecular Spectroscopy

Transitions in different regions of the electromagnetic spectrum

Region Microwave Far IR IR UV-vis

Frequency(Hz) 109-1010 1011-1013 1013-1014 1014-1016

Wavelength/m 3×10-1-3 ×10-3 3×10-3-3 ×10-5 3×10-5- 6.9×10-7 6.9×10-7-2×10-7

Wavenumber/cm-1 0.033-3.3 3.3-330 330-14500 14500-50000

Energy/J.molecule-1 6.6×10-25 6.6×10-23 6.6 ×10-21 2.9 ×10-19

- 6.6×10-23 -6.6 ×10-21 -2.9 ×10-19 -1.0 ×10-18

Molecular Process Rotation Rotation Vibration Electronicof polyatomic of small transitionmolecule molecule

most molecules vibration 400-4000 cm-1

Consider a mass m, connected to a wall by a spring, suppose thatno further gravitational force is acting on m; the only force is due tothe springAssume: F=-k(l-lo)=-kx Hooke’s Law

k= force constant of the spring

= Fma

Classical Harmonic Oscillator

2/1

21

2

2

2

2

2

2

mk

A isnt displaceme initial if cos)(

cossin)(

0

)(

��

���

�=

=

+=

=+

−=

−−=

ω

ω

ωω

tAtx

or

tctctx

kxdt

xdm

kxdt

xdm

llkdt

ldm o

2

2

2V(x)

0at x 0 V(x) Choose2

)(

)()(

xk

Cxk

xV

CdxxFxVdxdV

F

kxF

=

==

+=

+−=�−=

−=

�

2

32

33

32

2

2

)(21

~)(

..)(61

)(21

)(

...)(!3

1)(

!21

)()()(

xkxV

xxkxV

lldl

Vdll

dlVd

lldldV

lVlV o

ll

o

ll

oll

o

ooo

++=

+−���

����

�+−��

�

����

�+−�

�

���

�+====

γ

Taylor Expansion:

We choose V(lo)=0since a min exists,1st derivative vanishes

called equilibrium bond length, no force is acting

Anharmonic terms are neglected Small x

V(x)

xlo

length, no force is actingthere

Force constant=curvature of the curve at the minimum

Small x(small displacement)

Physically important at RT

Rotational/Vibrational Spectroscopy:Infra Red (IR) region: 330-14500 cm-1; 3 ×10-5- 6.9 ×10-7 nm

υ=0

υ=1

υ=2

υ=3

2/12/1

;21

;2

21

21

���

����

�=���

����

�==

��

���

� +=��

���

� +=

µω

µπνπνω

ωυνυυ

kk

hE �

a. Consider a diatomic harmonic oscillator:

1±=∆υTransitions between vibrational levels involve absorptionof light subject to the selection rule:

1±=∆υ•Gross Selection Rule: the dipole moment of a molecule must change during a vibration

e.g., antisymmetric stretch of CO2

C OO

υ=0

υ=1

υ=2

υ=3

2/1

21

���

����

�=

=∆

µπν

ν

k

hE

obs

obs

1±=∆υ

IR vibrational spectrum: 1 line

In wavenumbers, define the vibrationalTerm, G(υ)=E /hc

You can obtaink, a measure of bondstrength

Term, G(υ)=Eυ/hc

2/1

21~~

~21

)(

���

����

�=�=

��

���

� +=

µπννν

νυυ

kc

c

G

b. Consider a diatomic rigid rotor:

cIIB

jjBjF

RI

eneracyjjjI

E

e

j

22

e2

2

8h

B~

;8

hB constant Rotational

~

);1(~

)(Term Rotational

length Bond:R inertia; ofmoment

deg12g ; 0,1,2...j );1(2

ππ

µ

===

+==

==

=+==+= �

Transitions between various rotational levels of a rigid rotor resulting from

In wavenumber (cm-1)

Resulting in the microwave rotational spectrum consisting of equally spaced lines with separation = 2B

Transitions between various rotational levels of a rigid rotor resulting fromabsorption of radiation are governed by selection rules: ∆J=±1• and Gross Selection Rule: the molecule must have a permanent dipolemoment. (N2 no microwave rotational spectrum)

J=0J=1

c. Consider a diatomic molecule: capable of vibrating and rotating. Within the rigid rotor/harmonic oscillator approximation:

...3,2,1,0

...3,2,1,0

)1(~~

21

)()(~

,

==

++��

���

� +=+=

j

jjBjFGE j

υ

νυυυ

υ=0

υ=1

υ=2

υ=3

A molecule is in a particular vibrational and rotational level.

J=0J=1

11

±=∆+=∆

j

υ

A molecule is in a particular vibrational and rotational level.To each vibrational level, there corresponds a series of rotationalenergy levels where the molecule can be.

13

1

10

1B~

:are valuestypical

−

−

≈≈

cm

cm

υSpacing between vibrational levelsis 100-1000 times the spacing between rotational levels.

Absorption- Selection Rules:Harmonic oscillator/rigid rotor approximation.

1,2....j ~

2~~~)1(~Branch P 1

0,1,2....j )1(~

2~

)1(~~

21

)2)(1(~~

23~~

)1(~

Branch R 11-j and 1j toj fromn transitionumber, quantum rotational initial theis j

,1,1

,1,1

=−=−=−=∆

−=∆=++=

+−��

���

� +−+++��

���

� +=−=+=∆

+=∆+

−+

++

jBEEj

j

jB

jjBjjBEEj

j

jjobs

jjobs

νυ

ν

νυνυυ

υυ

υυ

P branch R branchRough sketch

More closelyspaced

Further apart

Q-branchNot observed∆j=0 (forbidden)

Low frequencyP branch, ∆J=-1

High frequencyR branch, ∆J=1

HBr: 10 cm-1 separation between lines

cm-1

Rough sketch

vibrational/rotational interactions:which explains the difference in spacing observed between the P-branchand R-branch.

Examination of spectra shows that R-branch lines are more closelyspaced with increased energy (as ν increases, R branches becomemore closely spaced, and lines of P-branch are further apart withdecreasing ν)

8~

)1(~~

21

)()(~

22

,

e

j

Rc

hB

jjBhjFGE

µπ

νυυυ

=

++��

���

� +=+=

tabulatedare ~

,~21~~~

)1(~~

21

)()(~

~~ increasingwith increasingith slightly w decreases

~ increasingith slightly w increases

8

,

e

ee

ee

j

e

e

B

BB

jjBhjFGE

BB

RB

R

Rc

α

υα

νυυ

νν

µπ

υ

υυ

υ

��

���

� +−=

++��

���

� +=+=

=�

�

increases lines Pbetween spacing increases, j as

decreases lines Rbetween spacing increases, j as ~~

1,2...j ;)~~

()~~

(~)1(~0,1,2...j ;)

~~()

~~3(

~2~)1(~

10

1

211,1,1

2111,1,1

�<

=−++−=−=−=∆

=−+−++=−=+=∆

=→=

−

+

o

oojojP

oojojR

BB

jBBjBBEEj

jBBjBBBEEj

νν

ννυυ

Overtones are observed in vibrational spectraAnharmonic oscillator

υ=0

υ=1

υ=2

υ=3Re

44332 .....2462

=���

����

�=

+++=

Rj

j

i dRVd

xxxk

V

γ

γγ intensity 1 sintensitie ...3,2.....3,2,1

±=∆<<±±=∆±±±=∆

υυυ

x is the displacement of the nuclei from the equilibrium position =R-Re. k is Hooke’s law force constant

υ=0

1constant ity anharmonic~

,....2,1,0 ; .....21~~

21~)(

2

<<=

=+��

���

� +−��

���

� +=

e

eee

x

xG υυνυνυ

If the anharmonic terms are included in the hamiltonian- perturbation theorycan be used:

IR spectrum:A major line: called fundamental, ∆υ=1Also lines of weaker intensities: ~at integral multiples of the fundamental; called overtones,or harmonics. 1st overtone or 2nd harmonic ∆υ=2

2nd overtone or 3rd harmonic ∆υ=3

( ) ,....3,2,1 1~~~)0()(~ =+−=−= υυυνυνυν eeeobs xGG

Selection Rule for anharmonic oscillator:

intensity 1 sintensitie ...3,2.....3,2,1

±=∆<<±±=∆±±±=∆

υυυ

0→ν:

Vibrations of Polyatomic molecules can be understood using the harmonicoscillator approximation- introduction of normal coordinates

Molecule of N nuclei: 3N coordinates, 3 for each nucleus�3N degrees of freedom. (3 for center of mass coordinates)

Linear NonlinearTranslational 3 3Rotational 2 3Vibrational 3N-5 3N-6

:ldimensiona-multi behavior, oscillator harmonic

=Nvib

,....2,1,0each )21

(

)()...()(),.....,(

ˆ21

2H

21

:modes) normalor scoordinate (normal Qi with expression following the toleads This

....21

.....21

)0,...,0,0(),....,,(

:ldimensiona-multi behavior, oscillator harmonic

1

2121

1,

1

2

12

22

vib

1

2

1 11 1

2

21

=+=

=

=+−=

=∆

+=+��

�

�

��

�

�

∂∂∂=−=∆

=

===

=

= == =

j

N

jjjvib

NvibvibvibNvib

N

jjvib

N

jjj

N

j jj

N

jjj

ji

N

i

N

jij

N

i

N

j jiN

vib

vibvib

vibvibvib

vib

vib vibvib vib

vib

hE

QQQQQQ

HQFdQd

QFV

qqfqq

VVqqqVV

υυν

ψψψψµ�

The consequence: under harmonic oscillator approximation, vibrational motion of a polyatomic molecule appears as Nvib independent harmonic oscillators. Each (in the absence of degeneracies) will have an independent ν:

Gross selection rule: the dipole moment must vary during the normal mode motion for the mode to be infrared active,otherwise it is infrared inactive

H2O 3N-6=3 normal modes

Symmetric stretchν13650 cm-1

H2O

Asymmetric stretchν33760 cm-1

Bendν21600 cm-1

dipole moment varies during each of these normal modes of vibrations, they are all infrared active

CO2 3N-5= 4 normal modes

Symmetric stretch, IR inactiveν1

Asymmetric stretch,ν32349 cm-1

IR active

µ oscillatesParallel to axis

⊗ ��

IR active

Bend (doubly degenerate)ν2 667 cm-1

IR active

Parallel to axis

µ oscillatesperpendicular to axis

µ oscillates parallel to internuclear axis or perpendicular

Different vibration-rotation spectra:

Parallel oscillation: like diatomic molecule leading to P and R branches∆ν = +1 (absorption)∆J = �1 (parallel band)

Called parallel band

P R

Perpendicular oscillation:∆ν = +1 (absorption)∆J = 0,�1 (perpendicular band)

Called perpendicular band

∆J = 0 called Q bandSuch as bending vibration of HCN

P

Q

R

Nonlinear molecule, with N atoms, has 3N-6 vibrational degrees of freedom

ν1Symmetric stretch

ν2Bend:Lowest

ν3Unsymmetric stretchHighest frequencySymmetric stretch Lowest

frequency Highest frequency

ν1, ν2, ν3: are called the fundamentalsWeak bands may be observed: Overtones: 2νi, 3νiCombination overtones: νi ± νj ; 2νi ± νjBecause of coupling of anharmonicities to normal vibrations

Binary overtones and binary combinations are more likelyto be observed than the others

Valence force Model

l from lengths bondin changes are r and r2 anglein change is

])([21

21

222

211

αδ

δδkrrkU ++=

212222

1223

2

sin2

12

cos2

1)(4

sin2

14

kmkm

mk

mm

OO

OS

O

δααννπ

ανπ

���

���

++���

���

+=+

���

����

�+=

2212

221

4

22122

221

2

21216

sin2

12

cos2

1)(4

lk

mk

mm

lk

mm

mmk

mm

OS

O

S

O

OOS

O

δ

δ

ννπ

ααννπ

���

����

�+=

���

����

�++��

�

����

�+=+

2α=119.5o, l =0.1432 nmmo= 15.5995, mS= 31.972obtain the values of kl and kδ/l2 in Nm-1

Vibrational partition function. Calculating heat capacity at constant volume.

Tq

TT

RvibC

RrotC

RtransC

vibv

v

v

ln)(

~23

)(~

23

)(~

2 ��

���

�

∂∂

∂∂=

=

=

kThvue

euRvibC

ee

kT

hq

q

qq

ii

iu

ui

v

kTh

kThiHOi

HOi

N

i

HOivib

i

i

i

i

/)1(

)(~

1

)21

(exp

mode normalith for thefunction partition oscillator harmonic

2

2

/

2/

0

63

1

=−

=

−=

�

�

�

� +−=

=

=

∏

−

−

−

−∞

=

−

=

ν

ν

υ

νυ