**By** Zahia Clemmons, Kat Zelaya-Cordova

**Faculty Mentor:** Leanna Giancarlo

**Abstract**

In classical mechanics, atoms are treated as point masses connected by spring-like bondsthat obey Hooke’s law, which states that the restorative force of a system is proportional to the displacement as a mass oscillates about its equilibrium distance.1,2 The quantum mechanical harmonic oscillator is derived as an analog to the classic simple harmonic oscillator to more accurately describe the vibrational energy of a diatomic molecule by accounting for quantum effects of a molecule.2 The energy of the system can be approximated by using the Schrödinger

equation, resulting in a harmonic potential energy diagram with evenly spaced energy levels caused by the constant vibrational frequency in an ideal model.3 A better approximation for the vibrational energy of a real diatomic molecule is the Morse potential because it explicitly

accounts for the effects of a bond dissociating and breaking, resulting from anharmonicity.1,3 This model shows that the restorative force and the displacement are not proportional, resulting in an energy diagram that becomes asymptotic when the bond breaks.1-3 Computational chemistry was used to model the vibrational energy of dihydrogen (H2), dinitrogen (N2), and hydrofluoric acid (HF), with basis set 6-31G*, to determine the effect of mass on dissociation energy. Water (H2O) was also investigated to determine whether the parameters of the anharmonic oscillator can be applied to polyatomic systems using the same basis set. It was determined that as the mass difference increased between diatomic molecules the accuracy of the model decreased; however, polyatomic molecules could not be modeled accurately by the anharmonic oscillator using the basis set 6-31G* due to bonds breaking at shorter distances.

References:

1. Engel, T and Reid, P. Quantum Chemistry and Spectroscopy, 4th ed.; Pearson Education:

New York, 2019; Chapters 7, 8, and 15.

2. 7.5 The Quantum Harmonic Oscillator. In University Physics Volume 3, OpenStax.

https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantumharmonic-

oscillator (accessed 2023-03-03).

3. 5.3: The Harmonic Oscillator Approximates Vibrations. Chemistry LibreTexts.

https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/05%3A

_The_Harmonic_Oscillator_and_the_Rigid_Rotor/5.03%3A_The_Harmonic_Oscillator_

Approximates_Vibrations (accessed 2023-03-03).

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