Spectroscopy/Rotational spectroscopy
Subject classification: this is a chemistry resource. 
Type classification: this is a lesson resource. 
 A molecule can store energy in its tumbling motion.
 This is only relevant in the gas phase where molecules are in continual motion and are free to rotate unhindered.
 Rotational motion at the molecular level is quantized in accordance with quantum mechanical theory.
 Transitions between discrete rotational energy levels give rise to the rotational spectrum of the molecule (microwave spectroscopy).
We will study:
 classical rotational motion, angular momentum, rotational inertia
 quantum mechanical energy levels
 selection rules and microwave (rotational) spectroscopy
 the extension to polyatomic molecules
Classical rotational motion[edit]
Rotational energy is kinetic energy associated with a tumbling motion.
Consider a particle with mass m, rotating with angular velocity ω, at a distance R from a given axis.
 This particle has angular momentum J
 J = I ω where J is the magnitude of the angular momentum vector, and I is the moment of inertia (the rotational equivalent of mass).
Where do we use rotational spectroscopy?
The 2.45 GHz frequency used by microwave ovens is the most ideal one for causing water molecules to rotate at their fastest possible rate. There is also the added bonus that this frequency is not used for communication, so microwave ovens do not interfere with cell phones, wireless internet, televisions and so on.
Diatomic molecules: the rigid rotor[edit]
Remember that a molecule does not distort under rotation.
The axis passes through the centre of mass (C):
Moment of inertia:
Using equations 1, 2 and 3:
μ is the reduced mass.
Example: For ^{1}H^{35}Cl  (Remember that u = atomic mass unit = 1.661x10^{27} kg)
Rotational energy levels[edit]
Classical:
Quantum mechanical: solve the Schrödinger equation for the rigid diatomic rotor.
 E_{J} is measured in Joules
 J is the rotational quantum number (= 0, 1, 2, ...)
Converted to wavenumbers:  Rotational constant: 
 The units of ε_{J} and B = cm^{1}
Below are the rotational energy levels for diazenylium, N_{2}H^{+}.
N_{2}H^{+}  
J = 0 → ε_{J} = 0  0 
J = 1 → ε_{J} = 2B cm^{1}  ≈3 cm^{1} 
J = 2 → ε_{J} = 6B cm^{1}  ≈10 cm^{1} 
Important points to notice:
 The energy of the ground state (J = 0) is zero  because the molecule is not rotating.
 As J increases, the molecules rotate more and more quickly  as a result, the energy levels are more widely spaced apart.
 The energy level separations are compatible with the microwave region of the EM spectrum.
But what about spectroscopic transitions between energy levels? And what about the selection rules for microwave spectroscopy?
Gross selection rule[edit]
Gross Selection Rule: The requirement for a permanent dipole.
For example, take a rotating HCl molecule. The fluctuation in its dipole component has an identical form to the fluctuation in the electric field of EM radiation (see the electromagnetic wave in Lesson 1).
As a result, the electric field of the EM wave exerts a torque on the dipole of the HCl molecule.
This means that energy can be absorbed or emitted, giving rise to a rotational spectrum.
Gross Selection Rule: molecules with permanent dipoles are microwave active (the molecule must be polar), e.g. heteronuclear diatomics  HCl, CO, NO, etc. Homonuclear diatomics are microwave inactive (e.g. O_{2}, N_{2}, etc.) In other words, a dipole must be present in the molecule for you to get a rotational spectrum. 
Transitions between energy levels[edit]
Specific Selection Rule: During a transition, the rotational quantum number must change by 1 unit only, i.e. ΔJ = ±1 (angular momentum is conserved)
In other words, only transitions between neighboring energy levels are possible. 
Below is an example rotational spectrum.
The top part shows the rotational energy levels, ε_{J}. The bottom part shows the microwave spectrum as observed from an experiment.
Each absorption (red arrow) complies with ΔJ = +1.
As a result of the equations in the box above, you get a series of spectral lines, each separated by 2B.
For CO, Δε = 3.86 cm^{1}. For HCl, Δε = 21.18 cm^{1}.
You'd get the same values for emission spectroscopy, except ΔJ = 1.
After obtaining a microwave spectrum from experiment, you measure n_{exp} in cm^{1}. (The distance between spectral lines is Δn_{exp}). From this you can work the rotational constant B, because Δv_{exp} (this is the intensity of each spectral line) = 2B. Using the formulae in the green boxes further up this page you can work out B and I. Now you can calculate molecular properties (such as R, μ) to a high degree of accuracy.
Rotational energy level population[edit]
Note that the most intense band is not the first line in the spectrum. Why?
Remember the 3 factors from Lesson 1:
 Amount of sample
 Population of energy states
 Selection rules
We need to consider number 2  the Boltzmann distribution AND degeneracy.
N_{f} and N_{i} are the population of molecules in energy levels e_{f} and e_{i} with degeneracy g_{f}and g_{I}.
From quantum mechanics, the degeneracy of each e_{J} level = 2J + 1
J = 0  1 (nondegenerate) 
J = 1  3 (3fold degenerate) 
J = 2  5 (5fold degenerate) 
J = 3  7 (7fold degenerate) 
Population of excited state relative to ground state:
Initial state:  J = 0  e_{i} = e_{0} = 0  g_{i} = g_{0} = 1 
Final state:  J  e_{f} = e_{J} = BJ(J + 1)  g_{f} = (2J+1) 
Δε = ε_{J}  ε_{0} = BJ(J + 1) →
Example for J = 1 (at T = 300 K):
B = 2 cm^{1}  N_{f}/N_{i} = 2.94 
B = 10 cm^{1}  N_{f}/N_{i} = 2.70 
The population of the J = 1 level decreases as the energy level separation (i.e. 2B) increases.
Degeneracy (the preexponential term) moves the maximum population away from J = 0. You can use calculus to determine the most populated level  the maximum which occurs when .
For example, at 300K, B = 5 cm^{1} → k_{B}T = 208.5 cm^{1} → J_{max} = 4 (which corresponds to the 4→5 transition).
Nonrigid rotor[edit]
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We have assumed so far that the bond length remains fixed during rotation of the molecule  this is the rigid rotor model. However, as the molecule rotates the atoms are subject to centrifugal forces which stretch the bonds  this is the nonrigid rotor model.
Hooke's law states for an elastic bond:
 F = restoring force (N)
 r = bond length; r_{eq} = equilibrium bond length (m)
 k = force constant (Nm^{1})
 c = vibrational frequency (cm^{1})
 m = reduced mass (kg)
Nonrigid rotor model for diatomic molecules:
Nonrigid rotor model for diatomic molecules  Centrifugal distortion coefficient 
Centrifugal distortion leads to lowering of the given energy level (at high J). Consequently, spectral lines cluster together at high J and are no longer equally spaced.
Polyatomic molecules[edit]
The majority of molecules are not diatomic  they can possess rotation around more than one axis.
The classical expression for energy of a body rotating about axis a is
With similar expressions for the other axes, it follows that:
Polyatomic molecules are categorised according to their moments of inertia about 3 perpendicular axes:
 Linear  one moment of inertia equals zero: I_{a} = I_{b}, I_{c} = 0. Molecules include CO_{2}, OCS, C_{2}H_{2}.
 Spherical rotor  three equal moments of inertia: I_{a} = I_{b} = I_{c}. Molecules include CH_{4}, SiH_{4}, SF_{6}.
 Symmetric rotor  two equal moments of inertia: I_{a} = I_{b} ≠ I_{c}. Molecules include NH_{3}, CH_{3}Cl, CH_{3}CN.
 Asymmetric rotor  three different moments of inertia: I_{a} ≠ I_{b} ≠ I_{c}. Molecules include H_{2}O, CH_{3}OH, H_{2}CO.
Calculation of B: Example
Calculate the value of the rotational constant B for a molecule of carbon monoxide. The bond length (R) of CO is 0.113 nm.

Lesson 3. Vibrational Spectroscopy 