The rigid rotor is a fundamental model in quantum mechanics and spectroscopy describing a rotating system where the distance between constituent particles remains constant. For GATE 2026, understanding the rigid is essential for solving problems on rotational spectra, energy quantization (\(E_J\)), and selection rules in diatomic molecules.
What is a Rigid Rotor?
A rigid rotor is an idealized mechanical model used to describe systems that rotate without changing their shape. In this model, the distance between the particles (bond length in molecules) is fixed and does not change during rotation. This assumption simplifies the mathematics of rotational motion significantly.
The rigid rotor approximation is the first step in understanding molecular spectroscopy. Whether you are analyzing a classical dumbbell system or a quantum rigid, the core concept remains that the rotating body does not stretch or compress. For diatomic molecules, this means the bond length (\(r\)) is strictly constant. This model allows physicists and chemists to predict rotational energy levels and spectral lines with high accuracy for heavy molecules or low rotational states.
Classical vs. Quantum Rigid Rotor
In classical mechanics, a rigid rotor can have any energy value depending on its angular velocity. However, the quantum rigid behaves differently. Its energy is quantized, meaning the rotor can only exist in specific, discrete energy states. This distinction is vital for interpreting rotational spectra in physical chemistry.
Moment of Inertia in Rotational Dynamics
The moment of inertia (\(I\)) is the rotational equivalent of mass in linear motion and is a defining property of any rigit. It measures the system’s resistance to changes in its rotational velocity. For a diatomic molecule acting as a rigid rotor, the moment of inertia is calculated using the reduced mass (\(\mu\)) and the bond length (\(r\)).
The formula for the moment of inertia of a diatomic rigid rotors is:
Where the reduced mass (\(\mu\)) is given by:
Here, \(m_1\) and \(m_2\) are the masses of the two atoms. A higher moment of inertia implies that the spacing between energy levels will be smaller. Understanding \(I\) is crucial because it appears in the denominator of the rotational energy equation, inversely affecting the energy spacing.
Schrödinger Equation for the Quantum Rigid Rotor
The quantum rigid rotor is solved by applying the time-independent Schrödinger equation to a particle moving on the surface of a sphere. Since the radius \(r\) is fixed, the Hamiltonian only contains the kinetic energy term associated with angular momentum.
For a rigid rotor, the potential energy (\(V\)) is zero because the rotation is free. The Hamiltonian operator is proportional to the square of the angular momentum operator (\(\hat{L}^2\)).
The solutions to this equation are the Spherical Harmonics, \(Y_{J,M}(\theta, \phi)\). These wavefunctions describe the probability distribution of the rigid rotor. The quantization of angular momentum in the quantum rigid rotor leads directly to the quantization of energy.
Rotational Energy Levels
The rotational energy of a rigid rotor is quantized and depends on the rotational quantum number, \(J\). As the system rotates faster, it jumps to higher integer values of \(J\) (\(J = 0, 1, 2, …\)).
The energy expression for a quantum rigid rotor is:
In spectroscopy, we often express this rotational energy in wavenumbers (\(cm^{-1}\)) using the rotational constant (\(B\)).
Where the rotational constant \(\bar{B}\) is:
Key takeaways for GATE:
- The ground state (\(J=0\)) has zero energy.
- The spacing between adjacent energy levels increases as \(J\) increases.
- The gap between levels is \(2B, 4B, 6B, …\)
- This specific spacing pattern is the fingerprint of a rigid rotor in rotational spectra.
Rotational Spectra and Selection Rules
Rotational spectra arise from transitions between the quantized energy levels of a rigid rotor. However, not all molecules exhibit a rotational spectrum. For a molecule to interact with microwave radiation and show a pure rotational spectrum, it must possess a permanent dipole moment.
Selection Rules
For a rigid rotor, the transitions are governed by strict selection rules. A transition is allowed only if the change in the rotational quantum number (\(\Delta J\)) is:
This means the rigid rotor can only jump to the immediate next higher or lower energy level. It cannot skip levels (e.g., \(J=1\) to \(J=3\) is forbidden).
Spectral Line Position
When a transition occurs from \(J\) to \(J+1\), the wavenumber of the absorbed radiation is:
This results in a series of equidistant spectral lines spaced by \(2\bar{B}\). Analyzing the separation between these lines allows us to calculate bond lengths and the moment of inertia of the rigid rotor.
Isotopic Effect on Rotational Spectra
When an atom in a rigid rotor molecule is replaced by a heavier isotope, the rotational spectra shift. This happens because the substitution changes the mass, thereby altering the moment of inertia.
Since \(I = \mu r^2\), increasing the mass increases \(I\).
Since \(B \propto 1/I\), increasing \(I\) decreases the rotational constant \(B\).
Therefore, the energy levels of the heavier isotopic rigid rotor are lower, and the spectral lines become more closely spaced. This “isotopic shift” is a powerful tool used to determine atomic masses and confirm the rigid rotor geometry.
Classification of Molecules: Rotors
While diatomic molecules are the simplest rigid rotor systems, polyatomic molecules are classified based on their moment of inertia about three principal axes (\(I_A, I_B, I_C\)).
Spherical Top
A spherical top molecule has all three moments of inertia equal (\(I_A = I_B = I_C\)). Examples include \(CH_4\) and \(SF_6\). These highly symmetric molecules do not have a permanent dipole moment, so they do not show a pure microwave rotational spectra despite being a rigid rotor mechanically.
Symmetric Top (Prolate and Oblate)
In a symmetric top, two moments of inertia are equal.
- Prolate Rotor: (\(I_A < I_B = I_C\)). Shaped like a cigar (e.g., \(CH_3Cl\)). The rotation along the principal axis is easier.
- Oblate Rotor: (\(I_A = I_B < I_C\)). Shaped like a frisbee or discus (e.g., \(C_6H_6\)).
The energy equation for a symmetric rigid rotor is more complex, involving a second quantum number, \(K\).
Asymmetric Top
An asymmetric top has all three moments of inertia different (\(I_A \neq I_B \neq I_C\)). Most molecules, like \(H_2O\), fall into this category. The quantum rigid rotor solution for these systems is mathematically complex and does not follow a simple analytical formula like the linear rotor.
Critical Perspective: The Non-Rigid Rotor Limitation
While the rigid rotor model is excellent for introductory physics and low-energy states, it is fundamentally flawed at high rotation speeds. This is known as the “Non-Rigid Rotor” effect or Centrifugal Distortion.
Real chemical bonds are not rigid rods; they are like stiff springs. As a rigid rotor spins faster (higher \(J\)), the centrifugal force pulls the atoms apart, causing the bond length \(r\) to increase.
Since \(r\) increases, the moment of inertia (\(I\)) increases.
Since \(E \propto 1/I\), the actual energy levels are slightly lower than what the rigid rotor formula predicts.
To correct this, we introduce the Centrifugal Distortion Constant (\(D\)):
Critical Insight: If you rely solely on the rigid rotor equation for high \(J\) values in GATE problems, your answer will be incorrect. Always check if the question provides a value for \(D\). If \(D\) is given, the rigid rotor assumption must be modified.
Solved Numericals for GATE
Practicing numericals is the only way to master the rigid rotor for GATE. Below are solved examples covering standard patterns.
Problem 1: Determining the Rotational Constant
Question: The spacing between successive lines in the rotational spectra of a diatomic rigid is found to be \(20 cm^{-1}\). Calculate the rotational constant \(B\) and the position of the first line.
Solution:
For a rigid rotor, the spacing between adjacent lines is constant and equal to \(2B\).
$$B = 10 \, cm^{-1}$$
The first spectral line corresponds to the transition \(J=0 \to J=1\).
$$\bar{\nu}_{0\to1} = 20 \, cm^{-1}$$
Answer: The rotational constant \(B\) is \(10 cm^{-1}\) and the first line appears at \(20 cm^{-1}\).
Problem 2: Energy of a Specific Level
Question: Calculate the energy of the \(J=3\) level for a quantum rigid with a rotational constant \(B = 2 cm^{-1}\).
Solution:
Using the energy formula for a rigid rotor:
Substitute \(J=3\) and \(B=2 cm^{-1}\):
$$E_3 = 2 \times 3(4)$$
$$E_3 = 24 \, cm^{-1}$$
Answer: The energy of the third level is \(24 cm^{-1}\).
Problem 3: Moment of Inertia Calculation
Question: A diatomic molecule acts as a rigid rotor. If the bond length is \(1.2 \text{\AA}\) and the reduced mass is \(1.6 \times 10^{-27} kg\), find the moment of inertia.
Solution:
Convert bond length to meters: \(r = 1.2 \times 10^{-10} m\).
Formula for moment of inertia: \(I = \mu r^2\).
$$I = (1.6 \times 10^{-27}) \times (1.44 \times 10^{-20})$$
$$I \approx 2.304 \times 10^{-47} \, kg \cdot m^2$$
Answer: The moment of inertia is \(2.304 \times 10^{-47} \, kg \cdot m^2\).
Summary for GATE Aspirants
To succeed in GATE 2026, treat the rigid not just as a formula, but as a gateway to quantum mechanics. Remember that the rigid explains the quantization of angular momentum and the equal spacing of spectral lines. Watch out for questions asking about the prolate rotor or spherical top, as these test your theoretical depth beyond simple diatomics. Finally, always be aware of the non-rigid limitation when dealing with high energy levels. Mastering the rigid rotor ensures you secure marks in the physical chemistry and molecular physics sections.
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