4.3 Harmonic Oscillator and Rigid Rotor

Quantum Harmonic Oscillator

Schrödinger Equation and Energy Levels

The quantum harmonic oscillator models a particle in a quadratic potential well. For chemistry, the most important application is describing small-amplitude vibrations of a diatomic molecule around its equilibrium bond length.

The time-independent Schrödinger equation for the one-dimensional case is:

$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

where $m$ is the particle mass, $k$ is the force constant (a measure of bond stiffness), and $x$ is the displacement from equilibrium.

Solving this equation yields quantized energy levels:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega, \quad n = 0, 1, 2, \ldots$

where $\omega = \sqrt{k/m}$ is the angular frequency of the oscillator. Two features to note:

• The energy levels are evenly spaced, each separated by $\hbar\omega$.
• The ground state ($n = 0$) has energy $\frac{1}{2}\hbar\omega$, not zero. This zero-point energy is a direct consequence of the Heisenberg uncertainty principle: confining a particle to a potential well guarantees some minimum kinetic energy.

For a diatomic molecule, you replace $m$ with the reduced mass $\mu = \frac{m_1 m_2}{m_1 + m_2}$, so $\omega = \sqrt{k/\mu}$.

Wave Functions and Probability Density

The wave functions take the form:

$\psi_n(x) = N_n H_n(\alpha^{1/2}x) \exp(-\alpha x^2/2)$

where $N_n$ is a normalization constant, $H_n$ is the $n$th Hermite polynomial, and $\alpha = m\omega/\hbar$.

Key properties of these wave functions:

• The probability density $|\psi_n(x)|^2$ gives the likelihood of finding the particle at position $x$. For $n = 0$, this is a Gaussian centered at the equilibrium position, meaning the particle is most likely found near the center of the well.
• Each wave function $\psi_n$ has exactly $n$ nodes (zero crossings). More nodes means higher energy.
• The wave functions are orthonormal: any two different states integrate to zero (orthogonal), and each state integrates with itself to one (normalized).
• The probability density extends beyond the classical turning points (where $\frac{1}{2}kx^2 = E_n$). This quantum mechanical tunneling into the classically forbidden region is something the classical oscillator cannot do.

Harmonic Oscillator in Spectroscopy

Vibrational Spectroscopy and the Harmonic Oscillator Model

Vibrational spectroscopy probes molecular vibrations by measuring the absorption or emission of infrared (or Raman-scattered) radiation. The harmonic oscillator model approximates these vibrational energy levels well when vibrations are small in amplitude and the potential energy surface is close to parabolic near equilibrium.

The selection rule for vibrational transitions in the harmonic approximation is:

$\Delta n = \pm 1$

Only transitions between adjacent levels are allowed. This means the harmonic oscillator predicts a single absorption frequency for each vibrational mode:

$\nu = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$

This frequency depends on two things: the force constant $k$ (stronger bonds vibrate faster) and the reduced mass $\mu$ (heavier atoms vibrate slower). For example, the C-H stretch has a higher frequency than the C-C stretch because hydrogen is much lighter, giving a smaller $\mu$.

A molecule must have a changing dipole moment during vibration to absorb infrared radiation. Homonuclear diatomics like $\text{N}_2$ are IR-inactive for this reason.

Anharmonicity in Real Molecules

Real molecular potentials aren't perfectly parabolic, especially at large displacements. This anharmonicity causes several deviations from the harmonic model:

• Overtones appear: transitions with $\Delta n = \pm 2, \pm 3, \ldots$ become weakly allowed. These show up at roughly 2x, 3x the fundamental frequency, but with much lower intensity.
• Combination bands arise when a molecule simultaneously undergoes transitions in two or more vibrational modes, producing absorption at sum or difference frequencies.
• Energy level spacing decreases at higher $n$, rather than staying constant.

The Morse potential provides a more realistic model:

$V(x) = D_e\left(1 - e^{-\beta x}\right)^2$

where $D_e$ is the well depth (dissociation energy) and $\beta$ controls the width. Unlike the harmonic potential, the Morse potential allows for bond dissociation at high vibrational energies and correctly predicts the convergence of energy levels near the dissociation limit.

Rigid Rotor Energy Levels

Schrödinger Equation and Energy Levels

The rigid rotor models the rotation of a diatomic molecule by treating it as two masses separated by a fixed bond length. The molecule rotates freely, but the bond doesn't stretch or compress.

The Schrödinger equation in spherical coordinates is:

$-\frac{\hbar^2}{2I} \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} \right]\psi = E\psi$

where $I$ is the moment of inertia, and $\theta$ and $\phi$ are the polar and azimuthal angles. The operator in brackets is the angular part of the Laplacian, which you may recognize as being proportional to $\hat{L}^2$, the squared angular momentum operator.

The solutions are the spherical harmonics $Y_J^{m_J}(\theta, \phi)$, and the quantized energy levels are:

$E_J = \frac{J(J+1)\hbar^2}{2I}, \quad J = 0, 1, 2, \ldots$

The moment of inertia is:

$I = \mu R_e^2$

where $\mu$ is the reduced mass and $R_e$ is the equilibrium bond length. Notice that $E_J$ is not linearly spaced: the gap between adjacent levels grows with $J$.

Degeneracy and Rotational Constants

Each energy level $E_J$ is $(2J+1)$-fold degenerate. This degeneracy comes from the magnetic quantum number $m_J$, which can take values $-J, -J+1, \ldots, J-1, J$. Physically, these correspond to different orientations of the angular momentum vector in space, all with the same energy (in the absence of an external field).

The rotational constant $B$ is defined as:

$B = \frac{\hbar}{4\pi I}$

so the energy levels can be written compactly as $E_J = hBJ(J+1)$, and $B$ is typically reported in units of $\text{cm}^{-1}$ or Hz. Molecules with larger moments of inertia (heavier atoms or longer bonds) have smaller $B$ values and more closely spaced rotational levels.

In real molecules, centrifugal distortion causes the bond to stretch slightly at high rotational speeds. This is corrected by adding a term:

$E_J = hBJ(J+1) - hDJ^2(J+1)^2$

where $D$ is the centrifugal distortion constant ($D \ll B$). The effect is a slight decrease in level spacing at high $J$.

Rotational Transition Selection Rules

Selection Rules and Transition Frequencies

For a molecule to undergo a pure rotational transition (absorb or emit microwave radiation), it must possess a permanent dipole moment. Homonuclear diatomics like $\text{O}_2$ and $\text{N}_2$ have no permanent dipole and are therefore microwave-inactive.

The selection rule for the rigid rotor is:

$\Delta J = \pm 1$

This arises because the transition dipole moment integral vanishes unless $J$ changes by exactly one unit. The resulting transition frequencies are:

$\nu = 2B(J+1)$

where $J$ is the quantum number of the lower state. This predicts a rotational spectrum consisting of equally spaced lines separated by $2B$. By measuring that spacing, you can determine $B$, and from $B$ you can extract the bond length $R_e$.

Transition Intensities and the Boltzmann Distribution

Not all rotational lines have the same intensity. The intensity depends on the population of the initial state, which at thermal equilibrium follows the Boltzmann distribution:

$N_J \propto (2J+1) \exp\left(-\frac{E_J}{k_BT}\right)$

Two competing factors shape the population:

• The degeneracy factor $(2J+1)$ increases with $J$, favoring higher states.
• The exponential factor $\exp(-E_J/k_BT)$ decreases with $J$, depopulating higher states.

The result is that population peaks at an intermediate $J$ value, and the rotational spectrum shows a characteristic intensity envelope that rises, reaches a maximum, and then falls off. The $J$ value at maximum population is approximately:

$J_{\text{max}} \approx \sqrt{\frac{k_BT}{2hB}} - \frac{1}{2}$

At higher temperatures, the distribution shifts toward higher $J$ values, and more lines appear in the spectrum. Analyzing relative line intensities provides a way to determine the rotational temperature of a sample.