4.3 Harmonic Oscillator and Rigid Rotor

Quantum mechanics gets wild when we look at molecules vibrating and spinning. The harmonic oscillator model helps us understand how molecules vibrate, while the rigid rotor model explains their rotation. These models are key to grasping molecular behavior.

By studying these models, we can predict and interpret molecular spectra. This knowledge is crucial for understanding chemical bonding, molecular structure, and how molecules interact with light. It's like peeking into the microscopic world of dancing molecules!

Quantum Harmonic Oscillator

Schrödinger Equation and Energy Levels

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• The quantum harmonic oscillator describes the motion of a particle in a quadratic potential energy well, such as a diatomic molecule undergoing small-amplitude vibrations
• The Schrödinger equation for the one-dimensional quantum harmonic oscillator is: $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$, where $m$ is the mass of the particle, $k$ is the force constant, and $E$ is the energy
• The energy levels of the quantum harmonic oscillator are quantized and given by the formula: $E_n = (n + \frac{1}{2})\hbar\omega$, where $n = 0, 1, 2, ...$ is the vibrational quantum number and $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency of the oscillator
• The ground state energy of the quantum harmonic oscillator is non-zero and equal to $\frac{1}{2}\hbar\omega$, a consequence of the Heisenberg uncertainty principle

Wave Functions and Probability Density

• The wave functions of the quantum harmonic oscillator are given by the Hermite polynomials multiplied by a Gaussian function: $\psi_n(x) = N_n H_n(\alpha^{1/2}x) \exp(-\alpha x^2/2)$, where $N_n$ is a normalization constant, $H_n$ is the $n$th Hermite polynomial, and $\alpha = \sqrt{\frac{m\omega}{\hbar}}$
• The probability density of finding the particle at a given position is proportional to the square of the absolute value of the wave function: $|\psi_n(x)|^2$
• The wave functions of the quantum harmonic oscillator are orthonormal, meaning that they are mutually orthogonal and normalized
• The nodes of the wave functions correspond to the classical turning points of the oscillator, where the potential energy equals the total energy

Harmonic Oscillator in Spectroscopy

Vibrational Spectroscopy and the Harmonic Oscillator Model

• Vibrational spectroscopy probes the vibrational energy levels of molecules by measuring the absorption or emission of infrared or Raman radiation
• The harmonic oscillator model can approximate the vibrational energy levels and transitions of diatomic and polyatomic molecules, provided that the vibrations are small in amplitude and the potential energy is well-approximated by a quadratic function
• The selection rule for vibrational transitions in the harmonic oscillator model is $\Delta n = \pm 1$, meaning that only transitions between adjacent vibrational energy levels are allowed
• The frequency of the absorbed or emitted radiation in a vibrational transition is given by $\nu = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, which depends on the force constant and reduced mass of the oscillator

Anharmonicity in Real Molecules

• Anharmonicity in real molecules leads to deviations from the harmonic oscillator model, such as the appearance of overtones and combination bands in vibrational spectra
• Overtones occur when a molecule absorbs or emits radiation corresponding to a transition between non-adjacent vibrational energy levels ($\Delta n = \pm 2, \pm 3, ...$)
• Combination bands arise when a molecule simultaneously undergoes transitions in two or more vibrational modes, resulting in the absorption or emission of radiation at frequencies that are the sum or difference of the individual mode frequencies
• The Morse potential is a more accurate model for the potential energy of a diatomic molecule, accounting for anharmonicity and the dissociation of the molecule at high vibrational energies

Rigid Rotor Energy Levels

Schrödinger Equation and Energy Levels

• The rigid rotor is a model system that describes the rotational motion of a diatomic molecule, assuming that the bond length remains constant during rotation
• The Schrödinger equation for the rigid rotor 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, respectively
• The energy levels of the rigid rotor are quantized and given by the formula: $E_J = \frac{J(J+1)\hbar^2}{2I}$, where $J = 0, 1, 2, ...$ is the rotational quantum number
• The moment of inertia depends on the masses and equilibrium bond length of the diatomic molecule: $I = \mu R_e^2$, where $\mu$ is the reduced mass and $R_e$ is the equilibrium bond length

Degeneracy and Rotational Constants

• The degeneracy of each rotational energy level is given by $2J+1$, which arises from the $2J+1$ possible values of the magnetic quantum number $m_J$ for each value of $J$
• The rotational constant, $B = \frac{\hbar}{4\pi I}$, is a measure of the spacing between adjacent rotational energy levels and depends on the moment of inertia of the molecule
• Molecules with larger moments of inertia have smaller rotational constants and more closely spaced rotational energy levels
• Centrifugal distortion in real molecules leads to deviations from the rigid rotor model, causing a slight decrease in the spacing between rotational energy levels as $J$ increases

Rotational Transition Selection Rules

Selection Rules and Transition Frequencies

• Rotational transitions occur when a molecule absorbs or emits radiation, causing a change in its rotational energy level
• The selection rule for rotational transitions in the rigid rotor model is $\Delta J = \pm 1$, meaning that only transitions between adjacent rotational energy levels are allowed
• The selection rule arises from the fact that the transition dipole moment integral is non-zero only when the change in the rotational quantum number is $\pm 1$
• The frequency of the absorbed or emitted radiation in a rotational transition is given by $\nu = \frac{\Delta E}{h} = 2B(J+1)$, where $B = \frac{\hbar}{4\pi I}$ is the rotational constant

Transition Intensities and the Boltzmann Distribution

• The intensity of rotational transitions depends on the population of the initial rotational state, which is governed by the Boltzmann distribution at thermal equilibrium
• The population of a rotational state with energy $E_J$ is proportional to $\exp(-E_J/k_BT)$, where $k_B$ is the Boltzmann constant and $T$ is the temperature
• At higher temperatures, the population of higher rotational states increases, resulting in the appearance of transitions from these states in the rotational spectrum
• The relative intensities of rotational transitions can be used to determine the temperature of the sample and the relative populations of the rotational states