文档介绍：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 =∆Evib + ∆Erot
=ω= E f − Ei = E (v′, J ′) − E ( v′′, J ′′)
ω 1 1
v = = v0 (v′+ ) + BJ ′( J ′+1)− v0 (v′′+ ) + BJ ′′( J ′′+1)
2πc 2 2
At room temperature, typically v=′′ 0 and ∆v = +1:
′′′′′′
v=+v0 B J ( J +1) − J ( J +1)
Now, since higher lying rotational levels can be populated, we can have:
∆=J +1 J ′= J ′′+1 v= v0 + 2B ( J ′′+1) R − branch
P − branch
∆=J −1 J ′= J ′′−1 v= v0 − 2BJ ′′
J’=4
J’=3
J’=1
v’=1 J’=0
J’’=3
J’’=2
J’’=1
v’’=0 J’’=0
∆J = 0
2B 2B 4B 2B 2B 2B
" " Q branch:
v = v0
-6B -4B -2B +2B +4B +8B "
ν 0 ν
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 ′′)
NJ()′′∝ g ()J ′′ ex p( − E J′′/ kT )
= ( 2 J ′′+ 1) exp( − hcBJ ′′( J ′′+ 1) / kT )