Rotational-Vibrational Spectrum of a Diatomic Molecule

″

J

0

1

2

3

4

5

6

7

′

J

0

1

2

3

4

5

6

7

Within the harmonic and rigid rotor approximations, the rotational-vibrational energy levels (in wavenumbers) of a diatomic molecule are given by

E

n,J

=(n+1/2)

ν

+J(J+1)

B

, where

n

,

J

are the vibrational and rotational quantum numbers, respectively,

ν

is the harmonic vibrational constant, and

B

is the rotational constant. The right panel shows the ground and first excited vibrational states, labeled

″

n

and

′

n

, respectively, with their associated manifolds of rotational states labeled

″

J

and

′

J

. Allowed transitions must obey the selection rules

Δn=±1

,

ΔJ=±1

. The left graphic shows transitions where

Δn=+1

,

ΔJ=+1

(referred to as the

R

-branch and shown in blue), and

ΔJ=-1

(referred to as the

P

-branch and shown in red). The arrows shown on the right indicate a transition between a pair of energy levels. The corresponding transition frequency is highlighted in the spectrum on the left.