Page 131 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:37 9in x 6in b985-ch11 Elementary Physical Chemistry

116 Elementary Physical Chemistry

which has the dimensions of cm −1 (c is the velocity of light). The energy

of radiation of a transition from J to J is hν J←J = E J − E J , yielding a

transition frequency in s −1
ν J←J =(E J − E J )/h
2

=[J (J +1) − J{J +1)]h/(8π I)
=[J (J +1) − J{J +1)]B (11.2c)

∗
The transition frequency is often expressed in wave-numbers, ν = ν/c,
yielding values in cm −1 ,

∗
ν J←J =[J (J +1) − J(J +1)]B ∗ (11.3a)
where,
2
∗
B = B/c = h/(8π cI) (11.3b)
For pure rotation, the selection rule requires that ∆J be either +1 or −1.

11.3. Vibrational Selection Rules
The vibrational energy of a diatomic molecule (simulated by a harmonic
oscillator) is

v =1, 2,... (11.4)
E v =(v +1/2)hν o
where v is the vibrational quantum number and ν o is the vibration
frequency, often expressed in terms of wave-numbers, ν o ∗ = ν o /c,which
has the dimension of cm −1 . The energy can be expressed as

E v =(v +1/2)hcν ∗ o (11.5)

and the transition frequency, in wave-numbers cm −1 ,as

ν v←v (v)=(v − v)ν ∗ 0 (11.6)
The selection rule for harmonic oscillators requires that ∆v be +1 or −1.

11.4. Further Requirements

The foregoing selection rules are necessary but not sufficient conditions: