17.2 Harmonic oscillator

The quantum harmonic oscillator model is crucial for understanding atomic and molecular behavior. It describes systems with restoring forces proportional to displacement, like vibrating molecules. The model's potential energy function is parabolic, assuming small displacements from equilibrium.

Solving the Schrödinger equation for a harmonic oscillator gives discrete energy levels and wavefunctions. This model introduces zero-point energy, the lowest possible energy of a quantum system. It's key for explaining molecular vibrations and interpreting spectroscopic data.

Quantum Harmonic Oscillator

Model Description and Potential Energy Function

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• A harmonic oscillator is a quantum mechanical model describing a system experiencing a restoring force proportional to its displacement from equilibrium
• The potential energy function of a harmonic oscillator is given by $V(x) = \frac{1}{2}kx^2$, where:
  • $k$ is the force constant determining the strength of the restoring force and oscillation frequency
  • $x$ is the displacement from equilibrium
• The potential energy function is parabolic, with minimum energy at the equilibrium position ($x = 0$)
• The harmonic oscillator model assumes small displacements from equilibrium, leading to a symmetric potential energy well (spring-mass system)

Approximations and Limitations

• The harmonic oscillator model is an approximation valid for small displacements from equilibrium
• Real molecular systems deviate from the parabolic potential energy function assumed by the harmonic oscillator model, especially at large displacements (anharmonicity)
• The model does not account for the dissociation of molecules at large internuclear distances (breaking of chemical bonds)

Solving for Energy Levels

Schrödinger Equation and Solutions

• The Schrödinger equation for a harmonic oscillator is given by: $\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}kx^2\right]\psi(x) = E\psi(x)$
  • $m$ is the mass of the oscillator
  • $\hbar$ is the reduced Planck's constant
  • $E$ is the energy eigenvalue
• Solutions to the Schrödinger equation yield discrete energy levels: $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$, where:
  • $n$ is the quantum number ($n = 0, 1, 2, ...$)
  • $\omega$ is the angular frequency of oscillation, related to the force constant and mass by $\omega = \sqrt{\frac{k}{m}}$

Wavefunctions and Probability Density

• The wavefunctions for a harmonic oscillator are given by: $\psi_n(x) = N_n H_n(\alpha^{1/2}x) e^{-\alpha x^2/2}$, where:
  • $N_n$ is a normalization constant
  • $H_n$ are the Hermite polynomials
  • $\alpha = \frac{m\omega}{\hbar}$
• The probability density of finding the oscillator at a given position is proportional to the square of the wavefunction, $|\psi_n(x)|^2$
• The wavefunctions have different shapes for different quantum numbers, with nodes (zero probability) at specific positions

Zero-Point Energy

Concept and Significance

• Zero-point energy is the lowest possible energy level of a quantum mechanical system, such as a harmonic oscillator
• For a harmonic oscillator, the zero-point energy is given by $E_0 = \frac{1}{2}\hbar\omega$, obtained by setting $n = 0$ in the energy level equation
• The existence of zero-point energy implies that a quantum mechanical oscillator can never have zero energy, even at absolute zero temperature

Consequences for Molecular Vibrations

• In the context of molecular vibrations, zero-point energy is the vibrational energy that molecules possess even in their ground state
• Zero-point energy contributes to the total energy of the system, affecting the stability and reactivity of molecules
• The presence of zero-point energy can affect the accuracy of thermodynamic calculations and the interpretation of spectroscopic data (IR and Raman spectroscopy)

Vibrational Spectra of Molecules

Applying the Harmonic Oscillator Model to Diatomic Molecules

• The harmonic oscillator model can describe the vibrational motion of diatomic molecules
• The reduced mass, $\mu$, of the diatomic molecule is used instead of individual atomic masses: $\mu = \frac{m_1 m_2}{m_1 + m_2}$
• The vibrational energy levels of a diatomic molecule are given by: $E_v = \left(v + \frac{1}{2}\right)\hbar\omega$, where:
  • $v$ is the vibrational quantum number ($v = 0, 1, 2, ...$)

Selection Rules and Spectral Features

• The selection rule for vibrational transitions in a harmonic oscillator is $\Delta v = \pm 1$, meaning transitions can only occur between adjacent vibrational energy levels
• The frequency of the absorbed or emitted photon during a vibrational transition is given by: $\nu = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$
• The vibrational spectra of diatomic molecules consist of a series of equally spaced lines, with spacing determined by the vibrational frequency, $\omega$ (HCl and CO)

Anharmonicity in Molecular Systems

Limitations of the Harmonic Oscillator Model

• The harmonic oscillator model assumes the restoring force is perfectly proportional to the displacement, valid only for small displacements
• In real molecular systems, the potential energy function deviates from the parabolic shape assumed by the harmonic oscillator model, especially at large displacements

Anharmonic Potential Energy Functions

• Anharmonicity refers to the deviation of the potential energy function from the ideal harmonic behavior
• Anharmonic potential energy functions, such as the Morse potential, account for the dissociation of molecules at large internuclear distances (H2 and N2)
• The energy levels of an anharmonic oscillator are not equally spaced, and spacing decreases as the vibrational quantum number increases

Spectral Consequences of Anharmonicity

• Anharmonicity leads to the appearance of overtones and combination bands in the vibrational spectra of molecules, not predicted by the harmonic oscillator model (CO2 and H2O)
• Overtones occur at frequencies that are integer multiples of the fundamental vibrational frequency
• Combination bands arise from the simultaneous excitation of multiple vibrational modes
• The inclusion of anharmonicity is necessary for accurate modeling of molecular vibrations and the interpretation of high-resolution vibrational spectra (polyatomic molecules)