#### Overview of this book

Explore quantum chemical concepts and the postulates of quantum mechanics in a modern fashion, with the intent to see how chemistry and computing intertwine. Along the way you’ll relate these concepts to quantum information theory and computation. We build a framework of computational tools that lead you through traditional computational methods and straight to the forefront of exciting opportunities. These opportunities will rely on achieving next-generation accuracy by going further than the standard approximations such as beyond Born-Oppenheimer calculations. Discover how leveraging quantum chemistry and computing is a key enabler for overcoming major challenges in the broader chemical industry. The skills that you will learn can be utilized to solve new-age business needs that specifically hinge on quantum chemistry
Preface
Chapter 1: Introducing Quantum Concepts
Free Chapter
Chapter 2: Postulates of Quantum Mechanics
Chapter 3: Quantum Circuit Model of Computation
Chapter 5: Variational Quantum Eigensolver (VQE) Algorithm
Chapter 6: Beyond Born-Oppenheimer
Chapter 7: Conclusion
Chapter 8: References
Chapter 9:Glossary
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Appendix B: Leveraging Jupyter Notebooks on the Cloud

# 6.2. Vibrational frequency analysis calculations

Within the BO approximation, the total energy of a molecule is the sum of the electronic, vibrational, and rotational energy:

Molecular vibrations can be modeled like the motion of particles connected by springs, representing atoms connected by chemical bonds of variable lengths. In the harmonic oscillator approximation, the force required to extend the spring is proportional to the extension (Hooke's law). When the vibrational energy is high, the harmonic oscillator approximation is no longer valid, and neither is the concept of normal mode. We now consider the simple case of a diatomic molecule.

## Modeling the vibrational-rotational levels of a diatomic molecule

The rotational energies of a diatomic molecule are represented by a series of discrete values:

where:

• is the angular momentum quantum number.
• is the equilibrium distance.
• is the reduced Planck...