Harmonic approximation Calculator

Formula first

Overview

The harmonic approximation treats the potential energy of a chemical bond as a parabolic function relative to the displacement from its equilibrium bond length. By assuming the potential surface is locally symmetric and quadratic, the model enables the treatment of molecular vibrations as simple harmonic oscillators. This simplification is fundamental in quantum mechanical derivations of vibrational energy levels and infrared spectroscopic predictions.

Apply it well

When To Use

When to use: Use when analyzing molecular vibrations near the potential energy minimum where displacement is small.

Why it matters: It serves as the foundation for the Quantum Harmonic Oscillator model, allowing for the analytical calculation of vibrational energy transitions.

Avoid these traps

Common Mistakes

• Assuming the harmonic approximation applies equally well to bond stretching at high temperatures or energies.
• Forgetting that the approximation is strictly valid only at the bottom of the potential energy well.

One free problem

Practice Problem

Which region of a diatomic molecule's potential energy surface is most accurately described by the harmonic approximation?

Solve for: $V(r)\approx \frac{1}{2}k{r}^2$

Hint: Consider the shape of a parabola compared to a full Morse potential.

The full worked solution stays in the interactive walkthrough.

References

Sources

1. Atkins, P., & de Paula, J. (2014). Physical Chemistry (10th ed.). Oxford University Press.
2. McQuarrie, D. A. (2008). Quantum Chemistry (2nd ed.). University Science Books.
3. OpenStax University Physics Volume 3, 7.5 The Quantum Harmonic Oscillator
4. Chemistry LibreTexts, Harmonic Oscillator Approximation
5. Atkins' Physical Chemistry
6. Introduction to Quantum Mechanics by David J. Griffiths
7. NIST CODATA
8. Molecular Spectroscopy: A Textbook by Ira N. Levine