Molecular Rotors in Rotational Spectroscopy

Classification of Molecular Rotors in Rotational Spectroscopy

Important for: CSIR-NET, GATE, SET, and other competitive exams.

The rotation of a three dimensional body may be quite complex and it is convenient to resolve it into rotational components about three mutually perpendicular direction through the centre of gravity - the principal axes of rotation. Thus a body has three principal moments of inertia, one about each axis, usually I_a, I_b, and I_c.

Molecules are classified as rigid rotors based on their three principal moments of inertia (I_a, I_b, I_c), conventionally ordered as I_a ≤ I_b ≤ I_c. Rotational constants are defined as A = h/(8π²c I_a), B = h/(8π²c I_b), C = h/(8π²c I_c) in cm⁻¹ (A ≥ B ≥ C).

Summary Table

Type	Moments of Inertia	Rotational Constants	Energy Levels (Rigid Rotor)	Examples	Microwave Spectrum
Linear Molecules	Ia ≈ 0, Ib = Ic	A >> B = C	EJ = B J(J+1) (hc cm⁻¹)	CO2, HC≡CH, OCS, HCl, CO	Yes (if μ ≠ 0)
Symmetric Tops	Two equal	Two equal	EJ,K = B J(J+1) + (A - B) K²	See sub-types	Yes (if μ along symmetry axis)
• Prolate	Ia < Ib = Ic	A > B = C	E = B J(J+1) + (A - B) K²	CH3Cl, CH3CN, CH3F
• Oblate	Ia = Ib < Ic	A = B > C	E = B J(J+1) + (C - B) K²	BF3, NH3, C6H6
Spherical Tops	Ia = Ib = Ic	A = B = C	EJ = B J(J+1)	CH4, SF6, CCl4	No (μ = 0; weak due to distortion)
Asymmetric Tops	Ia ≠ Ib ≠ Ic	A > B > C	No closed form; numerical (Wang basis)	H2O, H2CO, NO2, CH3OH	Yes (complex spectrum)

1. Linear Molecules

• Atoms arranged linearly; rotation about bond axis has negligible moment of inertia (I_a ≈ 0).
• Treated as rigid rotor with single rotational constant B.
• Energy levels:
  F(J) = BJ(J+1)   (in cm⁻¹)
• Degeneracy: (2J+1)
• Selection rule: ΔJ = ±1 → lines at 2B(J+1)
• Non-polar (e.g., CO₂, N₂) show no pure rotational spectrum.

2. Symmetric Tops

• Two moments equal; possess C₃ or higher symmetry axis.
• Quantum numbers: J (total angular momentum), K (projection along symmetry axis, |K| ≤ J).
• Energy levels:
  F(J,K) = BJ(J+1) + (A-B)K²
• Degeneracy: (2J+1) for K=0; 2(2J+1) for K≠0
• Selection rules: ΔJ = ±1, ΔK = 0
• Spectrum similar to linear molecules (spacing 2B).

Prolate Symmetric Tops

• Cigar-shaped; symmetry axis has smaller inertia.
• I_a < I_b = I_c → A > B = C
• Examples: CH₃F, CH₃Cl, CH₃C≡CH

Oblate Symmetric Tops

• Disc-shaped; symmetry axis has larger inertia.
• I_a = I_b < I_c → A = B > C
• Examples: NH₃, BF₃, benzene (C₆H₆)

3. Spherical Tops

• All moments equal; high symmetry (tetrahedral/octahedral).
• Energy:
  F(J) = BJ(J+1)
• Degeneracy: (2J+1)²
• No permanent dipole → no pure rotational microwave spectrum (weak lines due to centrifugal distortion).
• Examples: CH₄, SF₆, SiH₄, OsO₄, CCl₄

4. Asymmetric Tops

• All three moments different; most molecules belong here.
• No analytical energy expression; use basis of symmetric top functions and diagonalize Hamiltonian.
• Ray's asymmetry parameter κ = (2B - A - C)/(A - C) (-1 for prolate limit, +1 for oblate limit).
• Complex spectra; near-symmetric cases approximate symmetric top behavior.
• Examples: H₂O (near prolate), H₂CO, SO₂, CH₃OH, CH₃=CH-Cl

Practice Exam Questions (MCQs)

Suitable for CSIR-NET, GATE, SET, JAM, etc.

1. Which of the following molecules is classified as a spherical top rotor?

(a) CO₂
(b) NH₃
(c) CH₄
(d) H₂O

Answer: (c) CH₄
Explanation: Spherical tops have I_a = I_b = I_c (e.g., tetrahedral/octahedral symmetry). CH₄ is tetrahedral.

2. Linear molecules are considered as a special case of:

(a) Oblate symmetric top
(b) Prolate symmetric top
(c) Spherical top
(d) Asymmetric top

Answer: (b) Prolate symmetric top
Explanation: In linear molecules, I_a ≈ 0 → A → ∞, making it an extreme prolate case (A >> B = C).

3. Which molecule will NOT show a pure rotational microwave spectrum?

(a) HCl
(b) CH₃Cl
(c) CO₂
(d) H₂O

Answer: (c) CO₂
Explanation: CO₂ is linear but non-polar (μ = 0). Pure rotational transitions require a permanent dipole moment.

4. Benzene (C₆H₆) is an example of:

(a) Prolate symmetric top
(b) Oblate symmetric top
(c) Spherical top
(d) Asymmetric top

Answer: (b) Oblate symmetric top
Explanation: Planar hexagonal structure → I_⊥ (out-of-plane) smaller than in-plane → oblate (A = B > C).

5. The Ray's asymmetry parameter κ is +1 for:

(a) Prolate symmetric top
(b) Oblate symmetric top
(c) Linear molecule
(d) Spherical top

Answer: (b) Oblate symmetric top
Explanation: κ = (2B - A - C)/(A - C); κ → -1 (prolate limit), κ → +1 (oblate limit).

6. For a prolate symmetric top, the rotational energy is given by:

(a) BJ(J+1) + (C - B)K²
(b) BJ(J+1) + (A - B)K²
(c) AJ(J+1) + (B - A)K²
(d) CJ(J+1)

Answer: (b) BJ(J+1) + (A - B)K²
Explanation: Standard expression for prolate (A > B = C).

7. Spherical top molecules show very weak or no microwave absorption primarily because:

(a) They have very large moments of inertia
(b) They lack a permanent dipole moment
(c) K quantum number is undefined
(d) Energy levels are highly degenerate

Answer: (b) They lack a permanent dipole moment
Explanation: High symmetry (e.g., T_d, O_h) results in μ = 0.

8. Which of the following is an asymmetric top?

(a) CH₃F
(b) BF₃
(c) SO₂
(d) SF₆

Answer: (c) SO₂
Explanation: Bent structure → all three moments of inertia different.

9. The rotational spectrum of symmetric tops appears similar to that of linear molecules because:

(a) Selection rule ΔK = ±1
(b) Selection rule ΔK = 0
(c) A ≈ B ≈ C
(d) K-dependent term vanishes in transitions

Answer: (b) and (d)
Explanation: ΔK = 0 → transitions depend only on J → spacing 2B, independent of K.

10. Methyl chloride (CH₃Cl) is a:

(a) Linear molecule
(b) Oblate symmetric top
(c) Prolate symmetric top
(d) Spherical top

Answer: (c) Prolate symmetric top
Explanation: C_3v symmetry, elongated along C–Cl axis → I_a < I_b = I_c.