Rotation-Vibrational Spectra
The Vibrating Rotator

The inertia moment Ie of non-vibrating diatomic molecule is Ie = µre2.

There is averaging of squared distance r when vibrating happens, Iv = µ< r2>; due to this fact Iv will be bigger (µ: reduced mass).

Obvious calculation, r = re ± s :

Iv = µ<r2> = µ<(re ± s)2> = µre2± 2<µsre> + µ<s2> = µre2 + µ<s2>

(± 2 <µsre> mean noise) that is apparently bigger than µre2 .

Therefore the rotational constant B will depend on vibrations:

with vibrations: Bv = h/4πcµrv2 without vibrations Be = h/4πcµre2.

Bv < Be

One can written down the following relation for the Bv desciption:

Bv = Be - αe(v + ½) and Dv = De - βe(v + ½)

where αe, βe are small correction quantities.

Example HCl:
Be | Bv = 0 | Bv = 1 | Bv = 2 | Bv = 3 / cm-1
Be = 10,5934 | 10,44 | 10,13 | 9,83 | 9,52
De = 5,32 · 10-4 |  |  |  |
αe = 0,3072 |  |  |  |
βe = 7,51 · 10-6 |  |  |  |

We will neglect the electron rotation around molecular axis. The rovibronic state energy is then [all in cm-1]:

E = Evib + Erot = we(v + ½) - wexe(v + ½) + BvJ(J + 1) - DvJ2(J + 1)2

The energy difference between v' ← v, v' = v + 1 (without J-change) is as follows neglecting centrifugal tension D:

ΔE = | we - 2wexe(v + 1) | + {Bv'J'(J' + 1) - BvJ(J + 1)}
| ½¾¾¾¾¾¾½ ¯ |
| n0 |

For ΔJ = + 1, the so-called R-Branch, J + 1 ← J obtain:

n = n0 + 2Bv' + | (3Bv' - Bv) | J - | (Bv - Bv') J2 | where J = 0, 1, 2, ...
| (>0) |  | (>0) |

For ΔJ = -1, the so-called P-Branch, J - 1 ← J obtain:

n = n0 - (Bv + Bv') J - (Bv - Bv') J2 with J = 1, 2, 3, ...

The separation between two successive lines J, J+1 is

Dn (R-Branch) = 2(2Bv' - Bv) - 2(Bv - Bv') J (decrease)
Dn (P-Branch) = 2Bv + 2(Bv - Bv') J (increase)

The separation between two lines is 2B if Bv » Bv' . However, the distance between 1st transition in R-Branch and P-Branch is:

Dn (R-P-Branch) ≈ 4Bv

Almost all diatomic molecules show such spectra with R- and P-Branch. As example one can see HCl spectrum here. The line branching follows from application of natural isotope mixture since the reduced mass µHCl changes on about 0,15%. It essentially gives the vibrational frequency change, Dw» 2cm-1.

Vibrational level Ev = (v + ½) h w0 | Rotational level Er = B · J (J + 1) |
Vibrational and rotational energy levels of diatomic molecule | P-branch R-branch ΔJ = −1 ΔJ = +1
Rovibronic transition in diatomic molecule

There are also molecules for which there are additional lines in a free space between P- and R-Branch. It is the so-called Q-Branch, for ΔJ = 0 transitions. Certainly, the rotation should be kept, i.e. such transitions are possible only when rotational axis is changed (K¹ 0) and when electronic motion works in that way to compensate changes, correspondingly. That's why Q-Branch is only possible in rovibronic spectra of diatomic molecules when electronic motion gives its contribution into the total kinetic moment.

Projection of electronic rotation on the molecular axis: Λ = 0, 1, 2, ...

We assign this kinetic moment projection on the internuclear binding axis as quantum number Λ. For Λ = 0 everything is the same as it'd been before. The total molecular kinetic moment (nuclear rotation + electron) is |J| ≥ |Λ|.
The energy levels are identical to that of the symmetric top.

EJ = | BJ(J + 1) + | (A - B) Λ2
| nuclear rotation |
|  | electron motion >> B

The selection rules for Λ ¹ 0 change:

ΔJ = ± 1 it is also ΔJ = 0 possible.

Example is NO; obviously it is also spin phenomenon here and detailed description is a bit complicated. If we neglect spin, how will Q-Branch look like?

E(v',J) - E(v,J) = n0 + Bv'J(J+1) + (A-Bv')Λ2 - BvJ(J+1) - (A-Bv)Λ2

n0 (Q-Branch) = n0 - (Bv-Bv')J(J+1) + (Bv-Bv')Λ2

Since Bv ≈ Bv' all Q-lines should lay very close to each other.

The selection rules for symmetric top are as follows for the so-called
Parallel band (ΔK = 0)
for K = 0, ΔJ = ± 1 (as diatomic molecule)
for K ¹ 0, ΔJ = 0, ± 1 (as diatomic with electr.)

Vertical band (ΔK = ± 1)
ΔJ = 0, ± 1

The intensity of rovibronic lines depends on transition probability and the occupation number of states. If only v = 0 is populated then →

I ˜ NJ · |µJ'J|2

I ˜ NJ · SJ

SJ: Hoenl-London factors

For a simple rotator transition probabilities for high rotations do not depend on transition, i.e. the occupation number in vibrational ground state (J-distribution) essentially defines intensity. After Boltzmann one will obtain:

NJ ˜ (2J+1) e-Erot/kT = (2J+1)e-BJ(J+1)/kT

We can estimate the occupation number maximum if the separation between rotational levels << kT that is the always the case for T ≈ 300 K. Then we set J as variable quantity of x and try to find maximum:

dNx/dx = 0 d[(2x+1)e-Bx(x+1)/kT] /dx = 2e-Bx(x+1)/kT - B/kT(2x + 1)2e-Bx(x+1)/kT = 0

→ xmax = -½ + (kT/2B)½ = Jmax

Jmax ≈ (kT/2B)½ Example: Jmax(HCl) ≈ 2 - 3; Jmax(I2) ≈ 50

When we have thermal distributions we can determine temperature from line intensities for instance in stratosphere. In chemical reactions products are made up very often when there is absolutely another than Boltzmann distribution. We can obtain the occupation number of the initial state from the line intensity (if we know the transition probability). In many reactions where HF is produced the HF is obtained from the excited vibrational states. .