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 )
