′
′
′?
5.33 Lecture Notes,Vibrational-Rotational Spectroscopy
Vibrational-Rotational Spectrum of Diatomic
Absorption of mid-infrared light (~300-4000 cm
-1
),
Molecules can change vibrational and/or 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
E
field
=?E
vib
+?E
rot
=ω = E
f
E
i
= E
(
v′,J ′
)
E
(
v′′,J ′′
)
1
v =
ω
=
v
0
(
v′+
2
)
+ BJ ′( J ′+1)
v
0
(
v′′+
1
2
)
+ BJ ′( J ′+1)
2πc
′At room temperature,typically v=0 and?v = +1,
vv
0
=+B
J ′( J ′+1)? J ′′( J ′′+1)
Now,since higher lying rotational levels can be populated,we can have,
=+1 J ′= J ′′+1
vv
0
+ 2B ( J ′′+1) R? branch
J
=
=?1 J ′= J ′′?1 vv
0
2BJ ′′
P? branchJ
=
"
"
"
0
ν
v’’=0
v’=1
J’’=0
J’’=1
J’’=2
J’’=3
J’=0
J’=1
J’=3
J’=4
ν
4B 2B 2B 2B
+4B+2B +8B
2B 2B
-4B -2B-6B
0
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”),
(,) (,) ( )′ ′ ′ ′′ ′′?=? ≈NN v J N v J N J
( (
() () exp( / )
2 exp( 1 / )
′′
′′ ′′∝
′′ ′′ ′′=? +
J
NJ g J E kT
J hcBJ J kT
Q branch,
J = 0
v = v
0
′
) )
1
+
5.33 Lecture Notes,Vibrational-Rotational Spectroscopy Page 2
N
J''
g
J''
thermal population
Rotational Populations at Room Temperature for B = 5 cm
-1
0 5 10 15 20
Rotational Quantum Number J''
So,the vibrational-rotational spectrum should look like equally spaced lines about ν
0
with sidebands peaked at J’’>0,
ν
0
Overall amplitude from vibration transition dipole moment
Relative amplitude of rotational lines from rotational populations
In reality,what we observe in spectra is a bit different,
ν
0
Vibration and rotation aren’t really independent!
5.33 Lecture Notes,Vibrational-Rotational Spectroscopy Page 3
Two effects,
1) Vibration-Rotation Coupling,
For a diatomic,As the molecule vibrates more,bond stretches
→ I changes → B dependent on v,
=BB
e
α
(
e
v +
1
2
)
vibrational-rotational coupling constant!
2) Centrifugal distortion,As a molecule spins faster,the bond is pulled apart
→ I larger → B dependent on J
BB
e
D
e
J ( J +1)=
centrifugal distortion term
So the energy of a rotational-vibrational state is,
E
= v
0
( v +
1
2
)+ B
e
JJ +1)?α
e
( v +
1
2
)( JJ +1))? D
e
[
J (J +1
]
2
( (
=c
5.33 Lecture Notes,Vibrational-Rotational Spectroscopy Page 4
Vibrations of Polyatomic Molecules – Normal Modes
Remember that most of the nuclear degrees of freedom are the vibrations!
3n?6 nonlinear
3n?5 linear
C.O.M,
It was clear what this motion was for diatomic (only one!).
fixed
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 → especially for
spectroscopy,
The local modes aren’t generally independent of others! What we want and what
best represents what we observe are,normal modes” that are independent of one
another,The motion of one doesn’t influence the other,
EXAMPLE,CO
2
linear,3n?5 = 4 normal modes of vib,
local modes normal modes
stretch
symmetric stretch
bend
Doubly
O C O
degenerate
O C O
O C O O C O
Not
bend asymmetric stretch
independent
O C O
O C O O C O
O C O
(+) (?) (+)
Which normal modes are IR active?
symmetric stretch asymmetric stretch bend
O
C
O
O C O
O C O
O C O
Perpendicular to
μ?μ
symmetry axis
μ
= 0?Q
Q (?J =?1,0,+1)
Q
not IR active IR active IR active
5.33 Lecture Notes,Vibrational-Rotational Spectroscopy Page 5