Spec 5
5
Vibrational-Rotational Spectroscopy Vibrational-Rotational Spectrum of Heteronuclear Diatomic Absorption of mid-infrared light (~300-4000 cm -1 ): • Molecules can change vibrational and rotational states • Typically at room temperature, only ground vibrational state populated but several rotational levels may be populated. • Treating as harmonic oscillator and rigid rotor: subject to selection rules ∆v = ±1 and ∆J = ±1 =∆ +∆ field vib rot E E E ( ) ( ) , , f i E E Ev J Ev J ω ′ ′ ′′ ′′ = − = − ( ) ( ) ( ) ( ) 1 1 0 0 2 2 1 1 2 ′ ′ ′ ′′ ′′ ′′ = = + + + − + + + v v v BJ J v v BJ J c ω π At room temperature, typically v =0 ′ ′ and ∆v = +1: ( ) ( ) 0 1 1 v v B J J J J ′ ′ ′′ ′′ = + + − + Now, since higher lying rotational levels can be populated, we can have: 1 1 J J ∆ =+ ∆ =− 1 1 ′ ′′ = + ′ ′′ = − J J J J ( ) 0 0 2 1 2 v v BJ v v BJ ′ ′ = + + ′′ = − R branch P branch − −
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Transcript of Spec 5
Vibrational-Rotational Spectroscopy
Vibrational-Rotational Spectrum of Heteronuclear Diatomic Absorption of mid-infrared light (~300-4000 cm-1):
• Molecules can change vibrational and rotational states
• Typically at room temperature, only ground vibrational state populated but several rotational levels may be populated.
• Treating as harmonic oscillator and rigid rotor: subject to selection rules
∆v = ±1 and ∆J = ±1
= ∆ + ∆field vib rotE E E
( ) ( ), ,f iE E E v J E v Jω ′ ′ ′′ ′′= − = −
( ) ( ) ( ) ( )1 10 02 21 1
2′ ′ ′ ′′ ′′ ′′ = = + + + − + + + v v v BJ J v v BJ J
cωπ
At room temperature, typically v =0′′ and ∆v = +1:
( ) ( )0 1 1v v B J J J J′ ′ ′′ ′′= + + − + Now, since higher lying rotational levels can be populated, we can have:
11
JJ
∆ = +∆ = −
11
′ ′′= +′ ′′= −J JJ J
( )0
0
2 1
2
v v B J
v v BJ
′′= + +
′′= −
R branchP branch
−−
5.33 Lecture Notes: Vibrational-Rotational Spectroscopy Page 2
v’’=0
v’=1
J’’=0J’’=1J’’=2
J’’=3
J’=0J’=1
J’=3
J’=4
ν+4B+2B +8B
4B 2B 2B 2B2B 2B
-4B -2B-6B
0ν
P branch Q branch
E =02B6B
12B
J”
By measuring absorption splittings, we can get B . From that, the bond length! In polyatomics, we can also have a Q branch, where J 0∆ = and all transitions lie at
0ν = ν . This transition is allowed for perpendicular bands: ∂µ ∂q ⊥ to molecular symmetry axis.
Intensity of Vibrational-Rotational Transitions There is generally no thermal population in upper (final) state (v’,J’) so intensity should scale as population of lower J state (J”). ( , ) ( , ) ( )′ ′ ′′ ′′ ′′∆ = − ≈N N v J N v J N J
( ) ( )( ) ( )exp( / )
2 1 exp( 1 / )′′′′ ′′∝ −
′′ ′′ ′′= + − +JN J g J E kT
J hcBJ J kT
5.33 Lecture Notes: Vibrational-Rotational Spectroscopy Page 3
J''
NJ''
gJ''thermal population
0 5 10 15 20
Rotational Quantum Number
Rotational Populations at Room Temperature for B = 5 cm-1
So, the vibrational-rotational spectrum should look like equally spaced lines about 0ν
with sidebands peaked at J’’>0.
ν0
• Overall amplitude from vibrational transition dipole moment • Relative amplitude of rotational lines from rotational populations
In reality, what we observe in spectra is a bit different.
ν0 ν Vibration and rotation aren’t really independent!
5.33 Lecture Notes: Vibrational-Rotational Spectroscopy Page 4
J’
J’’ J’-1=J’’=J’+1
RP
Two effects: 1) Vibration-Rotation Coupling: For a diatomic: As the molecule vibrates more, bond stretches → I changes → B dependent on v.
( )12ve eB B α= − +
Vibrational-rotational coupling constant! 2) Centrifugal distortion: As a molecule spins faster, the bond is pulled apart → I larger → B dependent on J
( )1= − +e eB B D J J Centrifugal distortion term So the energy of a rotational-vibrational state is:
( ) ( ) ( ) ( ) ( ) 21 10 2 2v 1 v 1 1e e e
E v B J J J J D J Jhc
α= + + + − + + − +
Analysis in lab: Combination differences – Measure ∆∆E for two transitions with common state
Common J” ( ) ( )
( )
R P
e
E E E v 1, J J 1 E v 1, J J 1
3B 4J 22
′ ′′ ′ ′′∆ − ∆ = = = + − = = −
′′= − α +
B’
Common J’ ( )R P e1E E B 4J 22
′∆ − ∆ = − α +
B”
B’−B” = α
J’’
J’=J’’-1
J’=J’’+1
RP
5.33 Lecture Notes: Vibrational-Rotational Spectroscopy Page 5
Vibrations of Polyatomic Molecules – Normal Modes • Remember that most of the nuclear degrees of freedom are the vibrations! • It was clear what this motion was for diatomic (only one!). • For a polyatomic, we often like to think in terms of the stretching or bending of a
bond. — This “local mode” picture isn’t always the best for spectroscopy. • The local modes aren’t generally independent of others! The motion of one usually
influences others. EXAMPLE: CO2 linear: 3n−5 = 4 normal modes of vib.
O OCO OC
O OC
O OC
O OC
O OC
O OCO OC
local modes
stretch
bend
symmetric stretch
asymmetric stretch
bend
(+) (−) (+)
normal modes
g+Σ
u+Σ
uΠ
Molecules with linear symmetry: Σ : motion axially symmetric Π : motion breaks axial symmetry g/u : maintain/break center of symmetry Which normal modes are IR active?
symmetric stretch
O OCO OC
0Q
=∂
µ∂
asymmetric stretch
O OC
Q∂µ∂
not IR active IR active
O OC
bend
Q∂µ∂
IR active
3n−6 nonlinear 3n−5 linear
C.O.M. fixed
Not independent! C.O.M. translates.
Doubly degenerate
Perpendicular to symmetry axis (∆J = −1,0,+1)
