L18.1 Born-Oppenheimer approximation: Hamiltonian and electronic states

TL;DR

This content discusses the scales involved in molecules and introduces the Born-Oppenheimer approximation for simplifying the problem of solving the Schrodinger equation for molecules.

Transcript

PROFESSOR: We began our introduction to molecules last time and tried to get a picture of the scales that are involved. In these objects we spoke of a lattice of nuclei and clouds of electrons in which a molecule had some scale, A. Then we had electronic energies E, electronic. We had vibrational energies. This is from the nuclei. And we had rotati... Read More

Key Insights

• 🛩️ Molecules have different energy components, with electronic energies being the largest and rotational energies being the smallest.
• 😶‍🌫️ The adiabatic approximation simplifies the problem of molecular vibrations by assuming that electronic clouds move with the vibrating nuclei in an adiabatic way.
• ❓ The Born-Oppenheimer approximation considers a fixed nuclear skeleton and solves for electronic states associated with that skeleton.
• 🧑‍🤝‍🧑 Solving the full Schrodinger equation for molecules with coupled nuclear and electronic degrees of freedom is very difficult and usually requires approximations.

Questions & Answers

Q: What are the different energies involved in molecules?

Molecules have electronic energies, vibrational energies from the nuclei, and rotational energies of the whole molecule. The electronic energies are the largest, followed by the vibrational and rotational energies.

Q: What is the adiabatic approximation in relation to molecular vibrations?

The adiabatic approximation is the assumption that as the nuclei vibrate, the electronic clouds move with them in an adiabatic way. This approximation is valid when the timescale of nuclear vibrations is much larger than any variation in the electronic configuration.

Q: What is the Born-Oppenheimer approximation?

The Born-Oppenheimer approximation is a method for simplifying the problem of solving the Schrodinger equation for molecules. It involves considering a fixed nuclear skeleton and finding electronic states associated with that fixed configuration.

Q: How are the electronic and nuclear variables labeled in the Born-Oppenheimer approximation?

The nuclei variables are labeled as P alpha (momentum) and R alpha (position), while the electron variables are labeled as p (momentum) and r (position). These variables help distinguish the different particles in the molecule.

Summary & Key Takeaways

• The content introduces the scales involved in molecules, including electronic energies, vibrational energies, and rotational energies.

• It explains the adiabatic approximation, where the electronic cloud adjusts to changes in the nuclear state, and how it simplifies solving molecules.

• The Born-Oppenheimer approximation is introduced as a way to solve molecules by considering a fixed nuclear skeleton and finding electronic states associated with it.