Rotational–Vibrational Spectroscopy
1. Introduction
Rotational–vibrational spectroscopy studies the combined rotational and vibrational
motions of molecules. It is mainly observed in the infrared (IR) region of the
electromagnetic spectrum and provides important information about molecular
structure, bond strength, and energy levels.
Diatomic molecules are commonly used to explain these concepts because of their
simplicity.
2. Molecular Vibrations
Molecules are not rigid; atoms vibrate about their equilibrium positions. For a diatomic
molecule, this vibration can be approximated using the harmonic oscillator model.
2.1 Harmonic Oscillator Model
The vibrational energy levels are given by:
Ev=(v+12)hνE_v = \left(v + \frac{1}{2}\right)h\nuEv=(v+21)hν
Where:
•
v=0,1,2,3,…v = 0, 1, 2, 3, \dotsv=0,1,2,3,… (vibrational quantum number)
•
hhh = Planck’s constant
•
ν\nuν = vibrational frequency
Key Points:
•
Energy levels are equally spaced
•
The lowest energy is not zero (zero-point energy)
•
Only transitions with Δv=±1\Delta v = \pm 1Δv=±1 are allowed (selection rule)
3. Molecular Rotation
Molecules can also rotate, and this is modeled using the rigid rotor approximation.
3.1 Rigid Rotor Model
The rotational energy levels are:
EJ=h28π2IJ(J+1)E_J = \frac{h^2}{8\pi^2 I} J(J+1)EJ=8π2Ih2J(J+1)
Where:
•
J=0,1,2,3,…J = 0, 1, 2, 3, \dotsJ=0,1,2,3,… (rotational quantum number)
•
III = moment of inertia
Key Points:
•
Energy levels increase with JJJ
•
Spacing between levels increases as JJJ increases
•
Selection rule: ΔJ=±1\Delta J = \pm 1ΔJ=±1
4. Rotational–Vibrational Transitions
In reality, vibration and rotation occur simultaneously. When a molecule absorbs IR
radiation:
•
It changes vibrational state
•
It also changes rotational state
Combined Energy
The total energy is:
E=Ev+EJE = E_v + E_JE=Ev+EJ
This leads to multiple closely spaced spectral lines rather than a single peak.
5. IR Spectrum Features
The rotational–vibrational spectrum consists of two main branches:
5.1 P-Branch
•
ΔJ=−1\Delta J = -1ΔJ=−1
•
Appears at lower frequency (longer wavelength)
5.2 R-Branch
•
ΔJ=+1\Delta J = +1ΔJ=+1
•
Appears at higher frequency (shorter wavelength)
Important Observation:
•
There is usually a gap between the P and R branches
•
This gap corresponds to the pure vibrational transition
6. Anharmonicity
Real molecules do not behave as perfect harmonic oscillators.
Effects:
•
Energy levels are not equally spaced
•
Higher transitions (Δv=±2,±3\Delta v = \pm 2, \pm 3Δv=±2,±3) become weakly
allowed
•
Bond breaking becomes possible at high energy
A more accurate expression includes anharmonic corrections.
7. Applications
Rotational–vibrational spectroscopy is widely used in:
•
Identifying molecules
•
Determining bond lengths
•
Studying molecular structure
•
Atmospheric chemistry (e.g., CO₂, H₂O detection)
•
Analytical chemistry (gas analysis)
8. Conclusion
Rotational–vibrational spectroscopy provides a detailed understanding of molecular
motion by combining vibrational and rotational energy transitions. The observed spectra
give insight into molecular structure, energy quantization, and real-world deviations
from ideal models.